Appendix A

Solutions of Linear Differential Equations

A . l Linear Differential Equations with Constant Coefficients

Linear diflFerential equations with constant coefficients are usually writ­ten as

2/("> + ai2/("-i) + ... + a„_i2/(i) + anV = g, (A.l)

where a^, fc = 1,..., n, are numbers, y^^^ = ^ , and g = g{t) is a known function of t. We shall denote hy D = ^ the derivative operator^ so that the differential equation now becomes

p{D)y = (D^ + a iD^- i + ... + a^_iD + an)y = g. (A.2)

If g(t) = 0, the equation is said to be homogeneous. If g{t) ^ 0, then the homogeneous or reduced equation is obtained from (A.2) by replacing g byO.

If y and y* are two different solutions of (A.2), then it is easy to show that y — y* solves the reduced equation of (A.2). Hence, if y is any solution to (A.2), it can be written as

y = y*+y\ (A.3)

where y* is any other particular solution to (A.2) and y^ is a suitable solution to the homogeneous equation. Therefore, solving (A.2) involves (a) finding all the solutions to the homogeneous equation, caUed the gen­eral solution, and (b) finding a particular solution to the given equation.

364 A. Solutions of Linear Differential Equations

The rest of these notes indicate how to solve these two problems. Given (A.l) the auxiliary equation is

p{m) = mP + aim^'^ + ... + an-im + an = 0, (A.4)

In other words, p{m) is obtained from p{D) by replacing D by m. The auxiliary equation is an ordinary polynomial of nth degree and has n real or complex roots, counting multiple roots according to their multiplicity. We will see that, given these roots, we can write the general solution forms of homogeneous Unear differential equations.

A.2 Homogeneous Equations of Order One

Here the equation is

(D - a)y = y'-ay = 0,

which has y = Ce^^ as its general solution form.

A.3 Homogeneous Equations of Order Two

Here the differential equation can be factored (using the quadratic for­mula) as

( D - m i ) ( Z ) - m 2 ) 2 / - 0 ,

where m\ and m^ can be real or complex. Examples are given in Table A.l and the solution forms are given in Table A.2.

Differential Equation

1. y"-Ay' + Ay = Q

2. y" - %' + 3y = 0

3. y" - 4y' + 5y = 0

y{t)

y{t)

y{t)

General Solution Form

= e2*(Ci + Cit)

= e2*(Di sinh t) + D2 sinh t)

= e^\Disva. t + Di cos t)

Table A.l: Examples of Homogeneous Equations of Order Two

A.4. Homogeneous Equations of Order n 365

Root

^ 1 7 ^2? real

m i = a + 6, 1712 = a — b

mi = 7722 = ^^

^ 1 7 ^ 2 J complex

m i = a + bi^ m2 = a — bi

General Solution Form

y{t) = Cie^ i* + C2e^2t

= e^*(Cie^^ + C2e-^^)

or

y(t) = e"*(Ci sinh 6t + C2 cosh bt)

y{t) = iCi+C2t)e^'

y(t) = Cie^ i* + C2e^2t

- e"^(Cie^^* + C2e-'^^)

or

y(^) 3.: e"^(Di sin 6t + D2 cos bt)

or

y(^) = e^'^lEi sm{bt + ^2)]

or

y{t) = e''^[Ficos{bt + F2)] |

Table A.2: General Solution Forms for Second-Order Linear Homogeneous Equations, Constant Coefficients

A.4 Homogeneous Equations of Order n

When (A.2) is of order n, the auxiliary equation p(m) = 0 has n roots, when multiple roots are coimted according to their multiplicity. Also, complex roots occur in conjugate pairs. The general solutions of the homogeneous equations is the sum of the solutions associated with each multiple root. They can be foimd in Table A.4 for each root and should be added together to form the general solution. First, we give some examples in Table A.3.

366 A. Solutions of Linear Differential Equations

Differential Equation

1. D 2 ( 2 ) 2 _ 4 D + 4 ) 2 / - 0

2. ( D - 3 ) 2 ( D + 5 ) 3 ( D 2 - 4 D

3. ( D 2 -2D-\- 2 ) 3 ( D 2 - 2D -

+ 5 ) 2 y = 0

- 3)^2/ = 0

General Solution Form

y(t) = Ci + C2i + e2*(C3 + C^t)

2 / ( t ) - e 3 * ( C i + C 2 t )

+e -5 t (C3 + C4t + C5t2)

+e2*[(C6+C7)sin t

^ ( C s + CgOoos t]

2/(i) = e*[(Ci -f C2* + C3t2) sin t

+(C4-f-C5t-f C6t2)cos i]

+(C7 + Cst)e^^ + (Cg + Ciot)e-^

Table A.3: Examples of Homogeneous Equations of Order n

Root

rrij, real

Complex Conjugate

aj lb bj2

Multiplicity

r , = l

rj > 1

r , = l

0 > 1

General Solution Form

yj{t) = Ce'^J^

yj{t) = (Ci 4- C2* + . . . + Crjf-J - i)e"^i*

e'^J*(Ci sin 6jt + C2 cos 6jt)

e^i*[Ci + C2t + . . . + Cr^-Ti -^ )s in 6jt]

+(Cr^- + l + Cr^+2t + . . . + C2r,-r: ' ' - ^ ) COS bjt]

Table A.4: General Solution Forms for Multiple Roots of Auxiliary Equation

A.5 Particular Solutions of Linear D.E. with Constant Coefficients

The particular solution to the inhomogeneous equation (A.2) can usually be found by guessing the form of the answer and then verifying the guess by substitution. Table A.5 shows the correct forms for guessing for various kinds of forcing fimctions g(t). Note that the form of the guess depends on whether certain nimibers are roots of the auxiliary equation. Table A.6 gives examples of differential equations along with their particular integrals.

A,5. Particular Solutions of Linear D,E, — Constant Coefficients 367

Forcing Function, g{t)

( i ) c

(2) h{t)

(3) csin qt or ccos qt

(4) ce"*

(5) ce^* sin qt or ce^* cos qt

(6) /i(i)e«*

(7) /i(t) sin ^t or /i(f) cos qt

(8) /i(t)eP*sin gt or /i(i)eP*cos gt

K

0

0

9

9

p + iq

Q

iq

p + iq

Particular Integral, y{t)

A

X

A sin qt — B cos qt

Ae<i*

AeP* sin ^t - Be^* cos ^i

Xe«*

X sin gi + y cos qfi

XeP*sin ^i + FeP^cos qt

Notat ion.

(a) In the forcing function column, p, q, and c are given constants and h{t) is a given polynomial of degree s.

(b) In the p>articular integral column, A and B are coefficients to be determined and

X = ^0 + Alt + . . . + Ast\ Y = Bo-\-Bit+ ... + Bst^

are s degree polynomials whose coefficients are to be determined.

Rules.

(a) If the number in the K column is not a root of the auxiliary equation p(m) = 0, then the proper guess for the particular integral is as shown.

(b) If the number in the K colimin is a root of the auxiliary equation of degree r, then multiply the guess in the last column by t^.

Table A.5: Particular Solution Forms for Various Forcing Functions

If the forcing function g{t) is the sum of several functions, 9^=91 + g2 + *"+9ky each having one of the forms in the table, then solve for each Qi separately and add the results together to get the complete solution.

In the next table, we wiU apply the formulas and the rules in Table A.6 to obtain particular integrals in specific examples.

368 A. Solutions of Linear Differential Equations

Differential Equation

1.2/ '"-32/" = 5

2. y'" - 3y" = l + 3t + 5t^

3. 2 / " - V + 42/ = 3 - i 2

4. y" -4y' + 4y = 2sm t

5. 2 / " -43 / '+ 42/= 5sin 2t

6. 2/" - 42/' + 4y = lOe^*

7. 2/" - 4y' + 42/ = lOe^*

8. 2/" - 42/' + 52/ = e * cos t

9. 2/"-42/ ' + 52/ = t^sin f

10. y" - 42/' + 5y = i^e^* cos t

Particular Integral

At^

t^Ao + Ait + A2t'^)

Ao + Ait + A2t^

Asm t + Bcos t

Asm 2t + Bcos 2t

Ae'^

i2(Ae2«)

t{Ae^^ sin t + Be^* cos t)

X^r_o(^r^'^ sin i + B^V cos i)

* Er=o(^r*''e2* sin « + B^fe^* cos *)J Table A.6: Particular Integrals in Specific Examples

A.6 Integrating Factor

Consider the first-order linear equation

2/' + ay = / ( i ) . (A.5)

If we multiply both sides of the equation by the integrating factor e"*, we get

d ^(ye«')=j / 'e"*+aye«* = e"V(i). (A.6)

Integrating from 0 to t we have

y{t) = y{0)e--' + T e'^(--^)/(T)rfr, (A.7) Jo

which is the complete solution (homogeneous solution plus particular solution) to the equation.

A, 7. Reduction of Higher-Order to First-Order Linear Equations 369

A.7 Reduction of Higher-Order Linear Equations to Systems of First-Order Linear Equations

Another way of solving equation (A.l) is to convert it into a system of first-order linear equations. We use the transformations

zi = y, Z2 = y^^\...,zn = y^'' ^\ (A.8)

so that (A.l) can be written as

4

Z^

0

0

0

—dn

1

0

0

-an- -1

0

1

0

-CLn- - 2 •

.. 0

.. 0

.. 1

ai

zi

^2

Zn-1

Zn

+

0

0

0

9 J

In vector form this equation reads

z' = Az + b

(A.9)

(A.10)

with the obvious definitions obtained by comparing (A.9) and (A. 10). We will present two ways of solving the first-order system (A. 10).

The first method involves the matrix exponential function e*"* defined by the power series

/2 42

E A;! (A.11)

It can be shown that this series converges (component by component) for all values of t. Also it is differentiable (component by component) for all values of t and satisfies

^(e*^) = Ae'^ = {e'^)A. (A.12)

By analogy with Section A.6, we try e^* as the integrating factor for (A. 10) to obtain

370 A. Solutions of Linear Differential Equations

(Note that the order of matrix multiphcation here is important.) Using the product rule for matrix multiphcation of fimctions, which can be shown to be vahd, the above equation becomes

dV '

Integrating from 0 to i gives

Jo

Evaluating and solving, we have

z{t) = e'^z{0) + e'^ r Jo

TA b{r)dT.

.-TA b{T)dr, (A.13)

The analogy between this equation and (A.6) is clear. Although (A.13) represents a formal expression for the solution of

(A. 10), it does not provide a computationally convenient way of getting explicit solutions. In order to demonstrate such a method we assimie that the matrix A is diagonalizable, i.e., that there exists a nonsingtilar square matrix P such that

P-^AP = A. (A.14)

Here A is the diagonal matrix

A =

Ai 0 ••• 0

0 A2 ••• 0

0 0 An

(A.15)

where the diagonal elements, Ai , . . . , A i, are eigenvalues of A, The ith colunm of P is the column eigenvector associated with the eigenvalue A (to see this multiply both sides of (A.14) by P on the left). By looking at (A.ll) it is easy to see that

p-l^tAp^^tA^

Suppose we make the following definitions:

z = Pw, z(0) = Pw{Q), z' = Pw\

(A.16)

(A.17)

A. 7. Reduction of Higher-Order to First-Order Linear Equations 371

These in turn imply

w = P-^Z, w{0) = P-h{0), w' = p-^z\ (A.18)

Substituting (A. 17) into (A. 10) gives

Pw' = APw + b,

vJ = p-^APw + p-^b,

which by using (A. 14) gives

w' = Aw + p-^b. (A.19)

Since A is a diagonal matrix, it is easy to solve the homogeneous part of (A.19), which is

w' = Aw. (A.20)

The solution is

Wi = Wi{0)e~^^^ for i = 1,..., n.

We solve (A.19) completely by multiplying through by the integrating factor e~*^:

^(e-'^w) = e'^w' - e-^^Aw = e'^^p-^b. at

Integrating this equation from 0 to ^ gives

w{t) = e^^w(0) + e^^ f e-^^p-^b{T)dT. (A.21) Jo

Using the substitutions (A.18) yields

z{t) = {Pe^^p-^)z{0) + Pe^^ f e-^^p-^b{r)dT, (A.22) Jo

which is the formal solution to (A. 10). Since well-known algorithms are available for finding eigenvalues and eigenvectors of a matrix, the solution to (A.22) can be found in a straightforward manner.

372 A. Solutions of Linear Differential Equations

A.8 Solution of Linear Two-Point Boundary Value Problems

In linear-quadratic control problems with linear salvage values (e.g., the production-inventory problem in Section 6.1) we require the solution of linear two-point boundary value problems of the form

X 11

21

An

A22

X

A +

hi

62 J (A.23)

with boundary conditions

a;(0) = XQ and A(T) = Ay. (A.24)

The solution of this system will be of the form (A.22), which can be restated as

x{t)

m Quit) Quit)

Q2l{t) Q22{t)

rr(O)

. (° . +

Rx{t)

Mt) (A.25)

where the A(0) is a vector of unknowns. They can be determined by setting

AT = Q2i{T)x{0) + Q22(T)A(0) + R^iT), (A.26)

which is a system of linear equations for the variables A(0).

A.9 Homogeneous Partial Differential Equations

A homogeneous partial differential equation is an equation containing one or more partial derivatives of an unknown function with respect to its independent variables. If the highest partial derivative appearing ex­plicitly in the equation has order n, then the partial differential equation is said to be of order n.

As we saw in the previous sections, the general solutions of ordinary differential equations involve expressions containing arbitrary constants. Similarly, the solutions of partial differential equations are expressions containing arbitrary (differentiable) functions. Conversely, when arbi­trary fimctions can be eliminated algebraically from a given expression,

A.9, Homogeneous Partial Differential Equations 373

after suitable partial derivatives have been taken, then the result is a partial differentiable equation.

Example A , l Eliminate the arbitrary function / from the expression u = f{ax — by), where a and b are non-zero constants.

Solution. Taking partial derivatives, we have

Ux = af and Uy = —bf

so that fru^ + auy = abf - abf = 0. (A.27)

Here u = f{ax — by) is a, solution for the equation bux + aUy = 0. To show that any solution u = g{x, y) can be written in this form,

we set s = ax — by, and define

\s + by G{s,y) = g .y = 9{^^y)'

Then, gx = Gs% = aGs and gy = G , ^ + Gy = -bGs + Gy. Since we assume g solves the equation bux + aUy = 0, we have

Q = bgx- agy = abGg - abGs + aGy = 0, (A.28)

but this implies Gy = 0 so that G is a function oi s = ax — by only, and hence g{x, y) = G{s) = G{ax — by) is of the required form. We conclude that u = f(ax — by) is a general solution form for bux +auy = 0.

Example A.2 Eliminate the arbitrary fimctions / i and /2 from the expression u = fi{x)f2{y).

Solution. Taking partial derivatives, we have

"^x = / i / 2 , Uy = /1/2, and Uxy = / I /2

so that uuxy - UxUy = / i /2/{/2 - / i /2/1/2 = 0-

As in Example A.l we conclude that u = h{^)f2{y) is the general solu­tion form of the equation uUxy — UxUy = 0.

The subject of partial differential equations is too vast to even sur­vey here. However, Table A.7 gives general solution forms for all the homogeneous partial differential equations we will consider in this book, as well as others.

374 A. Solutions of Linear Differential Equations

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(15)

(16)

(17)

(18)

Partial Differential

Ekjuation

bux + auy = 0

XUx + yUy = 0 , X y/^0

xux — yUy == 0

Ux -\-Uy =^ au

Uz-\-Uy = au^, k ^ 1

^xx ^ yy — ^

Uxx + CL^Uyy = 0

Uxy =0

UUxy —UxUy = 0

UUxy +UxUy=0

bcux + acuy + abuz = 0

xUx + yuy 4- zuz = 0

xUx — yUy -1- xuz = 0

U^Uxyz — UxUyUz = 0

^txy^ = 0

Uxx — 0^{Uyy 4 - Uzz) = 0

Ux -\- Uy + Uz = au

General Solution Form

u = f(ax — by)

u = f{y/x)

u = fi^y)

u = h{x- y)e^ + f2{x - y)e''y

u^[{k-\){h{x-y)-ax)Y/^^-'^

^[k-l)h{x-y)-ay)Y^^^-')

u=:fi{y + ax) + hiy - ax)

u = fi{y + iax) -\- f2(y- iax), i = y/^

u = fi(x) - f2(y)

u = fi(x)f2(y)

u = Mx)/f2{y)

u =: fi{ax - by) ^ f2(by - cz) + fsicz - ax)

u = fi(x/y)-\-f2{y/z)

u = fi(xy)-\-f2{yz)

u = Mx)f2(y)f3(z)

u=-fi{x)-\-f2iy)-\-f3{z)

u = fi(y + ax) + f2{y - ax)-\- h{z + ax)

+f4{z-ax)

u = h{x- y)e^ + h{y- ^)e"^ + h{z - ^)e"' [

Note . The function fi are arbitrary differentiable functions of a single variable; a, 6, c , . . . stand for arbitrary (non-zero) constants.

Table A.7: General Solution Forms for Some Homogeneous Partial Differential Equations

A. 10 Inhomogeneous Partial Differential Equations

As in the ordinary case, an inhomogeneous partial differential equation is obtained from a homogeneous one by adding one or more forcing

A.iJ. Solutions of Finite Difference Equations 375

functions. The general case of this problem is too difficult to treat here. We consider only the case in which the forcing functions are separable, i.e., can be written as a sum of functions each involving only one of the independent variables. In solving such problems we can make use of the solutions to ordinary differential equations considered earlier.

Example A.3 Solve the partial differential equation

Ux + Uy = 3x + e^.

Solution. We know from the previous section that the general solution to the homogeneous equation is of the form f{x—y). To get the particular solutions we solve separately the ordinary equations

Ux = Sx^ and Uy = e^^

obtaining solutions x^ and e^. Therefore, the general solution to the original equation is

u = f{x-y)+x^ + e^.

Generally speaking, the above philosophy of finding particular solu­tions to separable partial differential equations (when it works) follows the same method of "dividing and conquering." Other methods involve the use of series. We will not go further here for lack of space.

A. 11 Solutions of Finite Difference Equations

In this book we will have uses for finite difference equations only in Chapters 8 and 9. For that reason we will give only a brief introduction to solution techniques for them. Readers who wish more details can consult one of several texts on difference equations; see, e.g., Goldberg (1986) or Spiegel (1971).

If f{k) is a real function of time, then the difference operator applied to / is defined as

A/(fc) = /(fc + l ) - / ( f c ) . (A.29)

The factorial power of k is defined as

A;(") = k{k - l)ik -2)...{k-{n- 1)). (A.30)

It is easy to show that Aifc("> = nfc("-i). (A.31)

376 A. Solutions of Linear Differential Equations

Because this formula is similar to the corresponding formula for the derivative d{k'^)/dk^ the factorial powers of k play an analogous role for finite differences that the ordinary powers of k play for differential calculus.

K /(fc) is a real function of time, then the anti-difference operator A~- applied to / is defined as another fimction g — A~^f{k) with the property that

A^ = f(k). (A.32)

One can easily show that

A-ifcW - ( l / (n + l))fc(^+i) + c, (A.33)

where c is an arbitrary constant. Equation (A.33) corresponds to the integration formula for powers of k in calculus.

Note that formulas (A.31) and (A.33) are similar to, respectively, differentiation and integration of the power function k"^ in calculus. By analogy with calculus, therefore, we can solve difference equations in­volving polynomials in ordinary powers of k by first rewriting them as polynomials involving factorial powers of k so that (A.31) and (A.33) can be used. We show next how to do this.

A . 11.1 Changing Polynomials in Powers of k into Facto­rial Powers of k

We first give an abbreviated list of formulas that show how to change powers of k into factorial powers of k:

fcO = fc(o) = 1 (by definition),

k'=k('\

fc4 = fc(i)+7/c(2)+6fc(3)+fcW,

k^ = fc(i) + 15fc(2) + 25fc( ) + lOfcW + fc(^).

The coefficients of the factorial powers on the right-hand sides of these equations are called Stirling numbers of the second kind, after the person who first derived them. This list can be extended by using a

A,ll. Solutions of Finite Difference Equations 377

more complete table of these nimibers, which can be found in books on difference equations cited earlier.

Example A.4 Express A; — 3fc + 4 in terms of factorial powers.

Solution. Using the equations above we have

k^ = fed) + 7fc(2) + 6fc(3) + fcW, -3k = -3fc(i), 4 = 4,

SO that

Example A.5 Solve the following difference equation in Example 8.7:

AA^ = -k + 6,X^ = 0.

Solution. We first change the right-hand side into factorial powers so that it becomes

AA^ = -fc(^) + 5.

Applying (A.33), we obtain

A = -(l/2)fc(2) + 5fcW + c,

where c is a constant. Applying the condition A = 0, we find that c= —15, so that the solution is

A = -(l/2)fc(2) + 5fc( ) - 15. (A.34)

However, we would like the answer to be in ordinary powers of fc. The way to do that is discussed in the next section.

A . 11.2 Changing Factorial Powers of k into Ordinary Powers of k

In order to change factorial powers of k into ordinary powers of fc, we make use of the following formulas:

fc(i) = k,

ki'^) = ^k + k\

fc(3) = 2k- 3A;2 + k^,

k^^) = -6k + llfc2 _ 6^3 ^ ^4^

378 A. Solutions of Linear Difkrential Equations

fc(5) = 24k - 50A;2 + 35k^ - lOJt + k^.

The coefficients of the factorial powers on the right-hand sides of these equations are called Stirling numbers of the first kind. This list can also be extended by using a more complete table of these nimibers, which can be found in books on difference equations.

Solution of Example A.5 Continued. By substituting the first two of the above formulas into (A.34), we see that the desired answer is

A' = -(1/2)A;2 + (n/2)k - 15,

which is the solution needed for Example 8.7.

EXERCISES FOR A P P E N D I X A

(A.35)

A . l If ^ = 3 2

2 3 show that A

5 0

0 2 a n d F =

1 1

1 - 1

Use (A.22) to solve (A. 10) for this data, given that ^(O) =

A.2 If A 3 3

2 4 , show that A =

6 0

0 1 a n d P =

1 3

1 - 2

Use (A.22) to solve (A. 10) for this data, given that z{0) = 0

5

A.3 After you have read Section 6.1, re-solve the production-inventory example stated in equations (6.1) and (6.2), (ignoring the control constraint ( P > 0) by the method of Section A.8. The linear two-point boundary value problem is stated in equations (6.6) and (6.7).

Appendix B

Calculus of Variations and Optimal Control Theory

Here we introduce the subject of the calculus oi variations by analogy with the classical topic of maximization and miniiir:zation in calculus; see Gelfand and Fomin (1963), Young (1969), and Leitmann (1981) for rigorous treatments of the subject. The problem of the calculus of varia­tions is that of determining a function that maximizes a given functional, the objective fimction. An analogous problem in calculus is that of de­termining a point at which a specific function, the objective function, is maximum. This, of course, is done by taking the first derivative of the function and equating it to zero. This is what is called the first-order condition for a maximum. A similar procedure will be employed to de­rive the first-order condition for the variational problem. The analogy with classical optimization extends also to the second-order maximiza.-tion condition of calculus. Finally, we will show the relationship between the maximum principle of optimal control theory and the necessary con­ditions of the calculus of variations. It is noted that this relationship is similar to the one between the Kuhn-Tucker conditions in mathematical programming and the first-order conditions in classical optimization.

We start with the "simplest" variational problem in the next section.

B . l The Simplest Variational Problem

Assume a function x : C-^[0^t] —> E^, where C^[O^T] is a class of func­tions defined over the interval [0,T] with continuous first derivatives. (For simplicity in exposition, assimie x to be a scalar fimction. The

380 B. Calculus of Variations and Optimal Control Theory

extension to a vector function is straightforward.) Assume further that a function in this class is termed admissible if it satisfies the terminal conditions

x{0) = xo and x{T) = XT- (B.l)

We are thus dealing with a fixed-end-point problem. Examples of admis­sible functions for the problem are shown in Figure B.l; see Section 6 and Chapters 2 and 3 of Gelfand and Fomin (1963) for problems other than the simplest problem, i.e., the problems with other kinds of conditions for the end points.

0 T

Figure B.l: Examples of Admissible Functions for the Problem

The problem under consideration is to obtain the admissible function X* for which the fimctional

fT J{x) = / g{x^x^t)dt

Jo (B.2)

has a relative maximum. We will assume that all first and second partial derivatives of the function g : E^ x E^ x E^ —^ E^ are continuous.

B.2 The Euler Equation

The necessary first-order conditions in classical optimization were ob­tained by considering small changes about the solution point. For the

B.2, The Euler Equation 381

variational problem, we consider small variations about the solution func­tion. Let x{t) be the solution and let

y{t)=x{t) + sri{t),

where T]{t) : C^[0,T] —» E^ is an arbitrary continuously differentiable function satisfying

r){0) = r,{T) = 0, (B.3)

and £ > 0 is a small number. A sketch of these functions is shown in Figure B.2.

Figure B.2: Variation about the Solution Function

The value of the objective functional associated with y{t) can be considered a fimction of s, i.e.,

fT V{e) = J{y) = g{x + erj, x + ef],, t)dt

Jo

However, x(t) is a solution and therefore V{e) must have a maximum at £ = 0. This means

A dV\ de e=Q

- 0 ,

where 8J is known as the variation 8J in J. Differentiating V{e) with respect to e and setting e — Q yields

8J = de

= / {9xV + 9xf])dt = 0, =0 ^0

382 B. Calculus of Variations and Optimal Control Theory

which after integrating the second term by parts provides

«5 = - ^ 1 = j ^ 9xVdt + {9iv)\o - I Jt^9i)vdt = 0. (B.4)

Because of the end conditions on 77, the expression simplifies to

= [9x- •:^9x]vdt = 0. zzo Jo dt de

We now use the fundamental lemma of the calculus of variations which states that if /i is a continuous fimction and /Q h(t)rj{t)dt = 0 for every continuous function r]{t)^ then h{t) — 0 for all t G [0, T]. The reason that this lemma holds, without going into details of a rigorous proof which is available in Gelfand and Fomin (1963), is as follows. Suppose that h(t) ^ 0 for some t G [0,T]. Since h{t) is continuous, there is, therefore, an interval (^1, 2) C [0, T] over which h is nonzero and has the same sign. Now selecting r/(t) such

r]{t) is < > 0 , te{ti,t2)

0, otherwise,

it is possible to make the integral /Q h(t)r){t)dt 7 0. Thus, by contrar-diction, h{t) must be identically zero over the entire interval [0,T].

By using the fimdamental lemma, we have the necessary condition

9x - -Tjgx = 0 (B.5)

known as the Euler equation, which must be satisfied by a maximal solution X*,

We note that the Euler equation is a second-order ordinary differen­tial equation. This can be seen by taking the total time derivative of QX and collecting terms:

^9xx + i^9xx + {9tx - 9x) = 0.

The boimdary conditions for this equation are obviously the end-point conditions x{0) = XQ and x{T) = x^-

B.3. The Shortest Distance Between Two Points on the Plane 383

Special Case (i) When g does not depend explicitly on x.

In this case, the Enler equation (B.5) reduces to

which is nothing but the first-order condition of classical optimization. In this case, the dynamic problem is a succession of static classical optimization problems.

Special Case (ii) When g does not depend explicitly on x.

The Euler equation reduces to

j^9^ = 0, (B.6)

which we can integrate as

gx = constant. (B.7)

Special Case (iii) When g does not depend explicitly on t,

Finally, we have the important special case in which g is explicitly independent of t. In this case, we write the Euler equation (B.5) as

j^{9-i9x)-9t = 0. (B.8)

But gt = 0 and therefore we can solve the above equation as

g-xgx = C, (B.9)

where C is the constant of the integration.

B.3 The Shortest Distance Between Two Points on the Plane

The problem is to show that the straight line passing through two points on a plane is the shortest distance between the two points. The problem

384 B. Calculus of Variations and Optimal Control Theory

can be stated as follows:

imn Jo

subject to

x(0) = xo and x{T) = XT-

Here t refers to distance rather than time. Since g = — v T + ^ does not depend explicitly on x^ we are in the second special case and the first integral (B.7) of the Euler equation is

This implies that i is a constant, which results in the solution

x{t) = Cit + C2,

where Ci and C2 are constants. These can be evaluated by imposing boimdary conditions which give Ci — {XT — xo)/T and C2 = XQ. Thus,

x{t) = XT-XQ

t + Xo, T

which is the straight line passing through XQ and XT-

B.4 The Brachistochrone Problem

The problem arises from the search for the shape of a wire along which a bead will slide in the least time from a given point to another, imder the influence of gravity; see Figure 1.1.

The Brachistochrone problem has a long history. It was first studied (incorrectly) by Galileo in 1630. The problem was correctly posed by Johann Bernoulli in 1696 and later solved by Johann Bernoulli, Jacob Bernoulli, Newton, and L'Hospital. Note that Euler deduced the Euler's equation in 1744, and we will solve the Brachistochrone problem using Euler's equation. But first we must formulate the problem.

Assimie the bead slides with no friction. Let m denote the mass of the bead, s denote the arc length, t denote the horizontal axis, x denote the vertical axis (measured vertically down), and r denote the time. Assimie 0 = 0, rr(io) = 0, T - l , x{T) = l.

B.4. The Brachistochrone Problem 385

We wish to minimize

^^ ds

Jo Jo V

where v represents velocity, and ST is the final displacement measured on the curve. We can write

ds = \ / l + x'^dt

and, from elementary physics, it is known that if v{to = 0) — 0 and a denotes the gravitational acceleration constant, then

v = V2^ ax,

Therefore, the variational problem can be stated as

1 /*! 1+x^ , 1 f' h \2aJo V

min^TT- / \l—- dt X

where x = dx/dt (note that t does not denote time), and x(0) = 0 and x{l) = 1. Since a is a constant, we can rewrite the problem as

J{x) = / g(x^x,t)dt= / \ dt> . mm

Since g does not depend explicitly on f, the problem belongs to the third special case. Using the first integral (B.9) of the Euler equation for this case, we have

i ;2[x(l+i2)]-i /2

We can reduce this to

1+x^ X

1/2

= Ci (a constant).

dx dt y xCf

To solve this equation, we separate the variables as

•^xdx dt

^ l ""x

386 B. Calculus of Variations and Optimal Control Theory

and substitute

X = (sin^ e)/Cl = (1 - cos 26)/Cl (B.IO)

The resulting expression can be integrated to yield

t=[e- (1/2) sm2e]/Cf + C2. (B.ll)

The condition ^ = 0 at ^ = 0 implies C2 = 0 providing Cf > 0. The value of Cl can be obtained in terms of the value of 6 at t = 1; let this be 9i. Then, since a; = 1 at ^ = 1, we have Ci = sin 6, where Oi satisfies

26>i - - I = sin20i-cos26>i.

This equation must be solved nimierically. An iterative numerical pro­cedure yields 0i = 1.206 and therefore Cf = 0.873. Defining 0 = 26, we can write (B.IO) and (B.ll) as

X = 0.573(1-cos 20),

t = 0 .573(0-sin0) ,

which are equations of a cycloid in the parametric form. The shape of the curve is shown in Figure 1.1 in Chapter 1.

B.5 The Weierstrass-Erdmann Corner Conditions

So far we have only considered functionals defined for smooth curves. This is, however, a restricted class of curves which qualify as solutions, since it is easy to give examples of variational problems which have no solution in this class. Consider, for example, the objective functional

min h{x) = I x'^il - xfdt\ , x ( - l ) = 0, x{l) = 1.

The greatest lower bound for J(x) for smooth x = x{t) satisfying the boimdary conditions is obviously zero. Yet there is no x € C- [—1,1] with x(—l) = 0 and x{l) = 1, which achieves this value of J(x). In fact, the minimum is achieved for the curve

x(t) 0, - 1 < < < 0,

t, 0<t<l,

B.5. The Weierstrass-Erdmann Corner Conditions 387

which has a comer (i.e., a discontinuous first derivative) at t = 0, Such a piecewise smooth extremal with corners is called a broken extremal

We now enlarge the class of admissible functions by relaxing the requirement that they be smooth everywhere. The larger class is the class of piecewise continuous functions which are continuously differentiable almost everywhere in [0,r], i.e., except at some points in [0, T].

Let x{t) be an extremal with a corner at r G [0, T]. Let us decompose J{x) as

rT PT PT

J{x) = / g{x,x^t)dt= / g{x,x,t)dt+ / g{x^x,t)dt Jo Jo JT

= Ji{x)+J2{x),

It is clear that on each of the intervals [0, r ) and {r^T], the Euler equation must hold.

To compute variations SJi and SJ2, we must recognize that the two 'pieces' oi x are not fixed-end-point problems. We must require that the two pieces of x join continuously at ^ = r; the point t = T can, however, move freely as shown in Figure B.3.

Figure B.3: A Broken Extremal with Corner at r

This will require a slightly modified version of formula (B.4) for writ­ing out the variations; see pp. 55-56 in Gelfand and Fomin (1963). Equat­ing the sum of variations

SJ = SJi +SJ2 = 0

388 B. Calculus of Variations and Optimal Control Theory

for x{t) to be an extremal and using the fact that x{t) must be continuous at t = r imphes

g^l^- = g^l^^ , (B.12)

[9 - igxlr- = [9- i9x]T+' (B.13)

These conditions are called Weierstrass-Erdmann corner conditions, which must hold at the point r where the extremal has a corner.

In each of the interval [0, r ) and (r, t], the extremal x must satisfy the Euler equation (B.5). Solving these two equations will provide us with four constants of integration since the Euler equations are second-order differential equations. These constants can be foiind from the end-point conditions (B.l) and Weierstrass-Erdmann conditions (B.12) and (B.13).

B.6 Legendre's Conditions: The Second Variation

The Euler equation is a necessary conditions analogous to the first-order condition for a maximum (or minimimi) in the classical optimization problems of calculus. The condition analogous to the second-order nec­essary condition for a maximum is the Legendre condition

9xx < 0. (B.14)

To obtain this condition, we use the second-order condition of classical optimization on function V(e) to be a maximum at e = 0, i.e.,

d^V{e)

de'^ e=0

PT

/ {9xxrj^ + '^gxxm + 9xxf]^)dt < 0. (B.15) JO

Integrating the middle term by parts and using (B.3), we can transform (B.15) into a more convenient form

/ {Qri^ + Pfi^)dt<Q, (B.16) Jo

where

Q = Q{t) = 9xx --^9xx and P = P{t) =^ g^j^.

While it is possible to rigorously obtain (B.14) from (B.16), we will only provide a qualitative argument for this. If we consider the quadratic functional (B.16) for functions r]{t) satisfying 77(0) = 0, then r]{t) will be

B.7, Necessary Condition for a Strong Maximum 389

small in [0, T] if f]{t) is small in [0, T], The converse is not true, however, since it is easy to construct r]{t) which is small but has a large derivative 7]{t) in [0,r] . Thus, Pfj^ plays the dominant role in (B.16); i.e., Pif can be much larger than Qrf but it cannot be much smaller (provided P 7 0). Therefore, it might be expected that the sign of the fimctional in (B.6) is determined by the sign of the coefficient P(t) , i.e., (B.16) implies (B.14). For a rigorous proof, see Gelfand and Fomin (1963).

We note that the strengthened Legendre condition (i.e., with a strict inequahty in (B.14)), the Euler equation, and one other condition called strengthened Jacobi condition are sufficient for a maximum. The reader can consult Chapter 5 of Gelfand and Fomin (1963) for details.

B.7 Necessary Condition for a Strong Maximum

So far we have discussed necessary conditions for a weak maximum. By weak maximum we mean that the candidate extremals are smooth or piecewise smooth functions. The concept of a strong m>axim.um, on the other hand requires that the candidate extremal need only be continuous functions. Without going into details, which are available in Gelfand and Fomin (1963), we state a necessary condition for a strong maximum. This is called the Weierstrass necessary condition. The condition is analogous to the one in the static case that the objective function be concave. It states that if the fimctional (B.2) has a strong maximum for the extremal 7 satisfying (B.l), then

E{x,x,t,v) < 0 (B.17)

along 7 for every finite v, where E is the Weierstrass Excess Function defined as

E{x^ i , t, v) = g{x^ v^ t) — g{x^ i , t) — gxi^i ^, t){'^ — x), (B.18)

Note that this condition is always met if g{x^x^t) is concave in x. The proof of (B.17) is by contradiction. Suppose there exists a r G

[0, T] and a vector q such that

E{T,x{T),x{T),q) > 0 ,

where x = x(t) is the equation of the extremal 7. It is then possible to suitably modify 7 to /? which is close to 7 in C^[0,T] such that

^J — I g{x^x^t)dt— / g{x^x^t)dt > 0, JB J-i

390 B. Calculus of Variations and Optimal Control Theory

contradicting the hypothesis that J{x) has a strong maximum for 7.

B.8 Relation to the Optimal Control Theory

It is possible to derive the necessary conditions of the calculus of varia­tions from the maximum principle. This is strongly reminiscent of the relationship between the first-order conditions of classical optimization and the Kuhn-Tucker conditions of mathematical progranmiing.

First, we note that the calculus of variations problem can be stated as an optimal control problem as follows:

max J — j g{x,u^t)dt >

subject to

X = n, x(0) = xo, x(T) = XT^

The Hamiltonian is

H(x^ u^ A, t) = g(x^ u, t) + Xu

with the adjoint variable A satisfying

A = —Hx = —Qx-

Maximizing the Hamiltonian with respect to u yields

Hu = 9x + ^=> ^ = -Qx-

Differentiating with respect to time, we have

d A

This equation with (B.20) implies

dt 9x'

9x - ^9x = 0,

(B.19)

(B.20)

(B.21)

which is the Euler equation of the calculus of variations.

B.8, Relation to the Optimal Control Theory 391

The second-order condition for the maximization of the Hamiltonian, i.e.,

Huu < 0 => pi;i < 0,

which is the Legendre condition. Again, by the maximum principle, if u is an optimal control, then

H{x,t,X,t) >H{x,v,\t),

where v is any other control. By the definition of the Hamiltonian (B.19) and equation (B.21), we have

g{x, i , t) - QxX > g(x, v, t) - g±v,

which by transposition of terms yields the Weierstrass necessary condi­tion

E(x^ ir, i, v) = g{x^ v^ t) — g(x^ i , t) — gx{v — i ) < 0.

We have just proved the equivalence of the maximum principle and the Weierstrass necessary condition in the case where Q is open. In cases when O is closed and when the optimal control is on the boundary of f2, the Weierstrass necessary condition is no longer valid, in general. The maximum principle still applies, however.

Finally, according to the maximum principle, both A and H are con­tinuous functions of time. However,

A == -gx and H = g - g±x,

which means that the right-hand sides must be continuous with respect to time, i.e., even across corners. These are precisely Weierstrass-Erdmann corner conditions.

Appendix C

An Alternative Derivation of the Maximum Principle

Recall that in the derivation of the maximum principle in Chapter 2, we assumed the twice differentiability of the return function V, Looking at (2.32), we can observe that the smoothness assumptions on the return function do not arise in the statement of the maximum principle. Also since it is not an exogenously given fimction, there is no a priori reason to assume the twice differentiabihty. In many important cases as a matter of fact, V has no derivatives at individual points, e.g., at points on switching manifolds.

In what foUows, we wiU give an alternate derivation. This proof fol­lows the course pointed out by Pontryagin et al. (1962) but with certain simplifications. It appears in Fel'dbaum (1965) and, in our opinion, it is one of the simplest proofs for the maximiim principle which is not related to dynamic programming and thus permits the elimination of assumptions about the differentiability of the return fimction V(t, x),

We select the Mayer form of the problem (2.5) for deriving the max­imum principle in this section. It will be convenient to reproduce (2.5) here as (C.l):

max {J = cx(T)} u{t)en{t)

subject to (C.l)

X = f{x,U,t), x{0) =Xo,

394 C. An Alternative Derivation of the Maximum Principle

C.l Needle-Shaped Variation

Let u*{t) be an optimal control with corresponding state trajectory x*{t). We sketch u*{t) in Figure C.l and x*{t) in Figure C.2 in a scalar case. Note that the kink in x*{t) aX t = 9 corresponds to the discontinuity in u*{t) 8itt = e.

• - > t 6 T-8 T T

Figure C.l: Needle-Shaped Variation

Let r denote any time in the open interval (0 , r ) . We select a suffi­ciently small e to insure that r — e > 0 and concentrate our attention on this small interval (r — e^r]. We vary the control on this interval while keeping the control on the remaining intervals [0, r — s] and (r, T] fixed.

Specifically, the modified control is

^-> t 0 T-s T r

Figure C.2: Trajectories x*{t) and x(t) in a One-Dimensional Case.

C.l. Needle-Shaped Variation 395

u{t) = I V eft, t e (r — s , r] ,

(C.2) u*{t), otherwise.

This is called a needle-shaped YSuYiaXion as shown in Figure C.l. It is a jump function and is different from variations in the calculus of variations (see Appendix B). Also the difference v —u* is finite and need not be small. However, since the variation is on a small time interval, its influence on the subsequent state trajectory can be proved to be 'small'. This is done in the following.

Let the subsequent motion be denoted by x{t) ^ x*{t) for t > r — s, In Figure 0.2, we have sketched x{t) corresponding to u{t).

Let 6x{t) = x{t) - a;*(i), t>T-e,

denote the change in the state variables. Obviously 6X{T — S) = 0. Clearly,

Sx{T)^e[x{s)-x\s)], (C.3)

where s denotes some intermediate time in the interval (r — e^r]. In particular, we can write (C.3) as

6x{r) = 4H^) - ^*('^)] + ^(^) = ^[ / (x(r) ,^ , r ) - / ( x* ( r ) , u* ( r ) , r ] + o(s). (C.4)

But Sx{r) is small since / is assumed to be boimded. Furthermore, since / is continuous and the difference 6X{T) = X(T) — X*{T) is small, we can rewrite (C.4) as

6x{t) « e[f{x*{T),v,T)-f{x*{r),u*iT),T)]. (C.5)

Since the initial difference SX(T) is small and since U*{T) does not change from t > T on, we may conclude that Sx{t) will be small for aU t > r . Being small, the law of variation of Sx{t) can be foimd from linear equa­tions for small changes in the state variables. These are called variational equations. From the state equation in (C.l), we have

^^El^M. = f(a:* + Sx,u*,t) (C.6)

or,

^ + ^^fix\u*,t)+fjx (C.7)

396 C. An Alternative Derivation of the Maximum Principle

or using (C.l),

-r{Sx) ^ U{x\ u*, t)6x, for t > r, (C.8) tit/

with the initial condition 6X{T) given by (C.5). The basic idea in deriving the maximum principle is that equations

(C.8) are linear variational equations and result in an extraordinary sim­plification. We next obtain the adjoint equations.

C.2 Derivation of the Adjoint Equation and the Maximum Principle

For this derivation, we employ two methods. The direct method, similar to that of Hartberger (1973), is the consequence of directly integrating (C.8). The indirect method avoids this integration by a trick which is instructive.

Direct method. Integrating (C.8) we get

6x{T) = 6x{r)+ f f^[x\t),u\t),t\8x{t)dt, (C.9)

where the initial condition 6X{T) is given in (C.5). Since 8x{T) is the change in the terminal state from the optimal state

x*(r) , the change in the objective function 6J must be negative. Thus,

8J = c8x{T) = C8X{T) + f cfx[x%t),u*{t),t\8x{t)dt < 0. (CIO)

Furthermore, since (C.8) is a linear homogeneous differential equation, we can write its general solution as

8x{t) = ^t,T)8x{T), (C.ll)

where the fundamental solution matrix or the transition matrix $(t, r ) G

^$(^,r) = Ux%t),u*m^t,r), ^T,T) ^ I, (C.12)

where / is an n x n identity matrix; see Appendix A.

C.2. Derivation of Adjoint Equation and the Maximum Principle 397

Substituting for 6x{t) from (C.ll) into (C.IO), we have

6J = C6X{T) + / cfa,[x*{t), u\t), t]^{t, T)8x{T)dt < 0. (C.13)

This induces the definition

X*(t) = J cf4x'{t),u*{t),t]^t,r)dt + c, (C.14)

which when substituted into (C.13), yields

SJ=^X*{T)SX{T)<0, (C.15)

But Sx{r) is supphed in (C.5). Noting that £ > 0, we can rewrite (C.15) as

X*{T)f[x*(T),v,T]-X*(r)f[x*iT),u*iT),T]<0. (C.16)

Defining the Hamiltonian for the Mayer form as

H[x, u, A, t] = A/(x, u, t), (C.17)

we can rewrite (C.16) as

if[:r*(r),u*(r),A(r),r] >ff[ar*(r),t;,A(r),r]. (C.18)

Since this can be done for almost every r , we have the required Hamil­tonian maximizing condition.

The differential equation form of the adjoint equation (C.14) can be obtained by taking its derivative with respect to r . Thus,

- C / X [ X ' ( T ) , M ' ( T ) , T ] . (0.19)

It is also known that the transition matrix has the property:

^ ^ ^ = -$( i , r ) / , [x*(r ) ,«*(r ) ,T] ,

which can be used in (C.19) to obtain

rfA(r) rT ^^ - j cU[x*{t),u*{t),t]^{t,T)U[x*{Tlu*{r),T]dt

-cU[x*{T)X{r),T]. (C.20)

398 C. An Alternative Derivation of the Maximum Principle

Using the definition (C.14) of A(r) in (C.20), we have

^ = -A(r) / , [x*(r) ,«*(r) , r]

with A(T) = c, or using (C.17) and noting that r is arbitrary, we have

A = -A/^[x*,n*,t] = -H^[x*,u\\t), X(T) = c. (C.21)

This completes the derivation of the maximum principle along with the adjoint equation using the direct method.

Indirect method. The indirect method employs a trick which simplifies considerably the derivation. Instead of integrating (C.8) explicitly, we now assume that the result of this integration yields cSx{T) as the change in the state at the terminal time. As in (C.IO), we have

6J = c6x{T) < 0. (C.22)

First, we define

A(r) = c, (C.23)

which makes it possible to write (C.22) as

SJ - c6x{T) = X{T)6x(T) < 0. (C.24)

Note parenthetically that if the objective fimction J = S{x{T)), we must define A(r) = dS[x{T)]/dx{T) giving us

* = i^^^(^) = A(r)6 (T).

Now, X(T)Sx{T) is the change in the objective fimction due to a change Sx{T) at the terminal time T. That is, A(T) is the marginal return or the marginal change in the objective function per unit change in the state at time T. But Sx{T) cannot be known without integrating (C.8). We do know, however, the value of the change SX{T) at time r which caused the terminal change Sx(T) via (C.8).

We would therefore like to pose the problem of obtaining the change 6J in the objective function in terms of the known value SX{T); see FeFdbaimi (1965). Simply stated, we would like to obtain the marginal return A(r) per unit change in state at time r . Thus,

X{T)SX{T) = 6J = X{T)SX{T) < 0. (C.25)

C,2. Derivation of Adjoint Equation and the Maximum Principle 399

Obviously, knowing A(r) will make it possible to make an inference about SJ which is directly related to the needle-shaped variation applied in the small interval (r — e^r],

However, since r is arbitrary, our problem of finding A(r) can be translated to one of finding A(t), t G [0,r], such that

X{t)6x{t) = \{T)6x{T), t e [0,T], (C.26)

or in other words,

X(t)Sx{t) = constant, A(r) = c. (C.27)

It turns out that the differential equation which X{t) must satisfy can be easily foimd. From (C.27),

^^[X(t)Sx{t)] = A ^ + XSx = 0, (C.28)

which after substituting for dSx/dt from (C.8) becomes

Xfjx + X6x = (A/^ + X)Sx = 0. (C.29)

Since (C.29) is true for arbitrary 6x, we have

A = -XU = -Ha: (C.30)

using the definition (C.17) for the Hamiltonian. The Hamiltonian maximizing condition can be obtained by substi­

tuting for 6x{r) from (C.5) into (C.25). This is the same as what we did in (C.15) through (C.18).

The purpose of the alternative proof was to demonstrate the valid­ity of the maximum principle for a simple problem without knowledge of any return function. For more complex problems, one needs compli­cated mathematical analysis to rigorously prove the maximum principle without making use of return fimctions. A part of mathematical rigor is in proving the existence of an optimal solution without which necessary conditions are meaningless; see Young (1969).

Appendix D

Special Topics in Optimal Control

In this appendix we will discuss three specialized topics. These are linear-quadratic problems, second-order variations, and singular control. These topics are referred to but not discussed in the main body of the text because of their advanced nature. While we shall not be able to go into a great detail, we will provide an adequate description of these topics and list relevant references.

D.l Linear-Quadratic Problems

An important problem in systems theory, especially engineering sciences, is to synthesize feedback controllers. These controllers provide optimal control as a function of the state of the system. A usual method of ob­taining these controllers is to solve the Hamilton-Jacobi-Bellman partial differential equation (2.19). This equation is nonlinear in general, which makes it very difficult to solve in closed form. Thus, it is not possible in most cases to obtain optimal feedback control schemes explicitly.

It is, however, feasible in many cases to obtain perturbation feedback control, which refers to control in the vicinity of an optimal path; see Bryson and Ho (1969). These perturbation schemes reqiiire the approx­imation of the problem by a linear-quadratic problem in the vicinity of an optimal path (see Section D.2), and feedback control for the approx­imating problem is easy to obtain.

A linear-quadratic control problem is a problem with Unear dynamics

402 D, Special Topics in Optimal Control

and quadratic objective function. More specifically, it is:

max u

I J = -x'^Gx + - I {x^Cx + u^Du)dt I (D.l)

subject to x = Ax + Bu, x{Q) =- XQ. ( D . 2 )

The matrices G, C, D, A, and B are in general time dependent. Fur­thermore, the matrices G, (7, and D are assumed to be negative definite and the superscript ^ denotes the transpose operation. Note that this problem is a special case of Row (c) of Table 3.3.

To solve this problem for the explicit feedback controller, we write the Hamilton-Jacobi-Bellman equation (2.19) as

0 = max[if + Vt] = max I - - ( a : ^ C x + u^Du) + V^\Ax + Bu\ -\-v\

(D.3) with the terminal boundary condition

V{x,T) = \x^Gx. (D.4)

The maximization of the maximand in (D.3) can be carried out by taking its derivative with respect to u and setting it to zero. Thus,

^M^ = ^ = (^Duf + V,B = 0=^u' = -V,B{D^)-\ (D.5) au ou

Note that (D.5) is the same as the Hamiltonian maximizing condition. Substituting (D.5) in (D.3) and simplifying, we obtain

0 = ^x^Cx + V^Ax - ^V^BD-^B^V^. (D.6)

This is a nonlinear partial differential equation of first order and it has a solution of the form

V{x,t) = lx'^S{t)x. (D.7)

Substitution of (D.7) into (D.6) yields

0 = \x^\S + 5A + ^S - SBD-^B^S + C\x. (D.8)

D.l, Linear-Quadratic Problems 403

Since (D.8) must hold for all x, it implies the following matrix differential equation

S + SA + A^S - SBD'^B^S + C = 0, (D.9)

called a matrix Riccati equation, with the boundary condition

S(T) = G. (D.IO)

A solution procedure for Riccati equations appears in Bryson and Ho (1969). Once we have the solution S(t) of (D.9) and (D.IO), the optimal feedback control can be written as

u\t) = D{t)-^B' {t)S{t)x{t), (D.ll)

A generaUzation of (D.l), which would be useful in the next section on the second variation, is to set

1 T J = -x^ Gx + jf'^^"^) C N

N^ D

X

u

dt. (D.12)

The state equation is given by (D.2). It is possible to derive the optimal control for this problem as

u*{t) = Dity^iN'^it) + B'^{t)S{t)]x{t), (D.13)

where

S+SA+A^S-{SB+N)D-^(B^S+N^)+C = 0, S{T) = G. (D.14)

For other variations and extensions of the linear-quadratic problem (D.l) and (D.2), for which explicit feedback controllers can be developed, the reader is referred to Bryson and Ho (1969).

D.1 .1 Certainty Equivalence or Separat ion Principle

Suppose equation (D.2) is changed by the presence of a Gaussian white noise w{t) and becomes

X = Ax + Bu + w,

where E[w{t)] = 0, E[w{t)w{Tf] = Q{t)6{t - r ) ,

404 D. Special Topics in Optimal Control

and x{0) is a normal random variable with

E[x{0)] = 0, E[x{0)x{Of] = Po.

Because of the presence of uncertainty in the system equation, we must modify the objective fimction in (D.12) as follows:

max < J = E ^x'^Gx + ^{x^,u^) C N '

iV^ D ^

( \ X

[u] dt\

Assume further that x cannot be directly measured and the measure­ment process is given by (13.21), i.e.,

y{t) = H{t)x{t)+v{t),

where v{t) is a white noise as defined in (12.72). The optimal control u* (t) for this linear-quadratic stochastic optimal

control problem can be shown to be given by (D.13) with x{t) replaced by its estimate x{t); see Bryson and Ho (1969). Thus,

n*(t) - D{t)-^[N^{t) + B^it)S{t)]x{t),

where S is given in (D.14) and x is given by the Kalman-Bucy filter.

'x = Ax + Bu*+w + K[y-Hx], :r(0) = 0,

K - PH^R-\

P = AP + PA^ -KHP + Q, P{0)^Po,

The above procedure has received two different names in the liter­ature. In economics it is called the certainty equivalence principle; see Simon (1956). In engineering and mathematics literature it is called the separation principle, Joseph and Tou (1961). When we call it the cer­tainty equivalence principle, we are emphasizing the fact that x(t) can be used for the purposes of optimal feedback control as if it were the certain value of the state variable x{t). Whereas the term separation principle emphasizes the fact that the process of determining the optimal control can be broken down into two steps: first, estimate x by using the optimal filter; second, use that estimate in the optimal feedback control formula for the deterministic problem.

D,2. Second-Order Variations 405

D.2 Second-Order Variations

Second-order variations in optimal control theory are analogous to the second-order conditions in the classical optimization problem of calculus. To discuss the second-order variational condition is difficult when the control variable u is constrained to be in the control set fl. So we make the simplifying assimiption that Q = R^, and thus the control u is unconstrained. As a result, we are now dealing with the problem:

max \j=f F{x, u, t)dt + ^x{T)] \ (D.15)

subject to X = f{x^ u, t), x{0) = xo> (D.16)

From Chapter 2, we know that the first-order necessary conditions for this problem are given by

A = -Ha:, A(T) - 0, (D.17)

Hu = 0, (D.18)

where the Hamiltonian H is given by

H = F + Xf, (D.19)

Since u is unconstrained, these conditions may be easily derived by the method of calculus of variations. To see this, we write the augmented objective fimctional as

J = ^x(T)] + I [Hix, u, A, t) - Xx]dt (D.20) Jo

Consider small perturbation from the extremal path given by (D.16) -(D.19) as a result of small perturbations 6x{0) in the initial state. Define the resulting perturbations in state, adjoint, and control variables by Sx{t), <5A(i), and 6u(t), respectively. These, of course, will be obtained by linearizing (D.16 - D.18) around the external path:

dux —rr = fx^x + fu^'^i 6x{0) specified, (D.21)

do

^ = -{HxJxf - SXf - (H^uSu), (D.22)

406 D. Special Topics in Optimal Control

SHu = {HuxSxf + SX{HuXf + (HuuSuf

= {HuuSxf + 6Xfu + {HuuSuf = 0. (D.23)

Alternatively, we may consider an expansion of the objective function and the state equation to second order since the first-order terms vanish about a trajectory which satisfies (D.15 - D.18), Prom Bryson and Ho (1969), this may be accomplished by expanding (D.20) to second order and all the constraints to first order. Thus, we have

^ 2 Jo

^x

8u

dt

subject to d8x IT = fxSx + fuSu^ 6x{0) specified.

(D.24)

(D.25)

Since we are interested in a neighboring extremal path, we must deter­mine 6u{t) so as to maximize 6^J subject to (D.25). This problem is a linear-quadratic problem discussed in the previous section. For this problem, the optimal control 6u*{t) is given by the formula (D.14), pro­vided Huu{i) is nonsingular for 0 < t < T. The case when Huuify is singular for a finite time interval is treated in Section D.3. Thus, rec­ognizing that G = ^xx, C = Hxx, N = H^u, D = Huu, A = fx, and B = fu, we have

6u%t) = H-^[Hux + f^S{t)]8x{t), (D.26)

where

S + Sfx + f^S-{Sfu+Hxu)H-^{f^S + Hux) + Hxx = 0, S{T) = $^^. (D.27)

While a ntmiber of second-order conditions can be obtained by pro­ceeding further from this manner, we shall be interested only in the concavity condition (or strengthened Legendre-Clebsch condition). It is possible to show that neighboring stationary paths exist (in a weak sense; i.e., Sx and Su are small) if

Huu{t) < 0 for 0<t<T, (D.28)

or in other words, Huu{t) is negative semidefinite. First-order conditions, conditions (D.28), and the condition that S{t) is finite for 0 < t < T

D,3. Singular Control 407

represent sufficient conditions for a trajectory to be a local maximum. We are not being specific here because in this book we would be relying mostly on the sufficiency conditions developed in Chapters 2-4, which are based on certain concavity requirements. We are stating (D.28) because of its similarity to the second-order condition for a local maximum in the classical maximization problem.

We must note that

Hu = 0 and Huu < 0 (D.29)

form necessary conditions for a trajectory to be a local maximimi.

D.3 Singular Control

In some optimization problems including some problems treated in this text, extremal arcs satisfying Hu = 0 occur on which the matrix Huu is singular. Such arcs are called singular arcs. Note that these arcs sat­isfy (D.29) but not the strengthened condition (D.28). While no general sufficiency conditions are available for singular arcs, some additional nec­essary conditions known as the generalized Legendre-Clebsch conditions have been developed. A good reference on singular control is Bell and Jacobson (1975).

We shall only discuss the case in which the Hamiltonian is linear in one or more of the control variables. For these systems, Hu = 0 implies that the coefficient of the linear control term in the Hamiltonian vanishes identically along a singular arc. Thus, the control is not determined in terms of x and A by the Hamiltonian maximizing condition Hu = 0. Instead, the control is determined by the requirement that the coefficient of these linear terms remain zero on the singular arc. That is, the time derivatives of Hu must be zero. Having obtained the control by setting dHu/dt = 0 (or by setting higher time derivatives to equal zero) along the singular arc, we must check additional necessary conditions analogous to the second-order condition (D.28). For a maximization problem with a single control variable, these conditions turn out to be

( - ' ) ' ! ; (P^Hu dfi^

< 0 , it = 0,1,2,.... (D.30)

The conditions (D.30) are called the generalized Legendre-Clebsch conditions.

408 D. Special Topics in Optimal Control

Example D . l We present an example treated by Johnson and Gibson (1963):

{—ir^? max \ j = - - I x\dt \ (D.31)

subject to

xi = X2+u^ xi{Q) = a^ (D.32)

i2 = -u, a:(0) = fe, (D.33)

xi{T) - X2{T) = 0. (D.34)

Solution. We form the Hamiltonian

H = --x\ + Xi{x2 + u)+ Mi-u), (D.35)

where the adjoint equations are

A i = x i , A2 = -Ai . (D.36)

The optimal control is bang-bang plus singular. Singular arcs must sat­isfy

if = Ai - A2 = 0 (D.37)

for a finite time interval. The optimal control can, therefore, be obtained by

^ = A i - A 2 = :ci+Ai = 0. (D.38)

Differentiating once more with respect to time t, we obtain

2 = xi + \i = X2 + u + xi = Q,

which implies u = -{xi+X2) (D.39)

along the singular arc. We now verify for the example, the generalized Legendre-Clebsch condition (D.30) for fc = 1:

Appendix E

Answers to Selected Exercises

Completely worked solutions to all exercises in this book are contained in a forthcoming Teachers' Manual, which wiU be made available to instructors by the publisher when it is ready.

Chapter 1

1.1 (a) Feasible. J = -333,333.

1.2 J = 36.

1.3 (a) C = $157,861/year.

(b) J = 103.41 utils.

(c) $15,000/year.

1.4 (b) W{20) = 985,648; J = 104.34.

1.12 imp(Gi,G2;i) = (Gi - G2)e-^*.

Chapter 2

2.2 The optimal control is

2 if 0 < t < 2 - h i 2 . 5 ,

u* (t) = { undefined if i = 2 - In 2.5,

0 i f t > 2 - l n 2 . 5 .

410 E. Answers to Selected Exercises

2.8

2.10

2.12

u* = bang(0,1; Ai - A2), where X{t) = (8e-2(*-i8), 4e-2(*-i8)).

(a) u* = bang[0,1; (giKi + 52^2)(Ai - A2)].

(c) i = T- (1/52) Hio^bi - 9ih2)/{g2 - giM-

(a) a;(100) = 30 - 20e-i° « 30.

(b) w* = 3 f o r i € [0,100].

(c) u*{t) = 3 for i s [0, 100-101n2],

2.14

2.17

2.18

3.1

3.2

3.7

3.11

3.12

0 otherwise.

(a) C*{t) = pWoe^'-P^yil - e-P^).

(b) C*it) = Kir-p).

(a) X = x + 3Aa;2, A(l) = 0, and x =-x^ + A, x{0) = 1.

X = f{x) + b{x)u, x{0) = xo, x{T) = 0.

u = [b{xfg'{x) - 2<^u{b(x)f'{x) - V{x)f{x)y\l\l(?h{x)\.

Chapter 3

X — Ml > 0, «1 — M2 > 0, Ml > 0, 1 + M2 > 0-

X = [ -1, 5].

L = F(x, v) + A/(x, M, i) + iig{x, u, t),

A = -(a/a)X - 1^, /x > 0, /x^ = 0.

A(i) = t - 1.

(a) A(i) = 10

0

I _g0.1(t-100)

-10 2 _ gO.iCic/s-ioo)

if ii: - 300,

if ii: < 300,

M*(t) = bang[0,3;A + /Lt].

The problem is infeasible for K > 300.

E. Answers to Selected Exercises 411

(b) r * = niin[0, 100-K/3],

0 for t < t**, u*{t) = I

3 iort>t**. V

3.18 11.87 minutes.

3.19 u* = - 1 , T* = 5.

3.20 tx* = - 2 , T* = 5/2.

3.29 (a) {I, P,X} = {h - p{S - Pi), S, 2(5 - Pi)}.

(b) I = h.

Chapter 4

4.1 u*{t) = -l, iJ,i = -X = 1/2-t, H2 = 7] = 0.

4.2 One solution appears in Figure 3.1. Another solution is u(t) = 1/2 for t G [0,2]. There are many others.

4.4 (a) u* = 0.

(c) u* = < 1, 0<t<l-T,

0, l-T <t<T.

(e) J = - ( 1 / 8 + 1/8K).

(f) J = - 1 / 8 .

Chapter 5

5.1 (a) u*{t) 5, t < l + 61n0.99«0.94,

0, t > 0.094.

(b) A2(t)/Ai(i) = e3(*'-«+i)/i2,^t*(f) = J

- 5 , 0 < ^ < 0.28,

0, 0.28 < t < 0.4,

5, 0.4 < i < 0.93,

0, 0.93 < t < 1.0.

412 E. Answers to Selected Exercises

5.5 u* = v* = 0 for all t.

5.7 u* = 0, V* = 4/5 for t € [0,49],

u* = 0, ;* = 0 for ^€[49,60],

J* = 34,420.

5.10 (b) /(**) = t* - 101n(l - 0.3e°-"*).

(c) t* = 1.969327, J{t*) = 19.037.

Chapter 6

6.5 Q(t) = t^ - 160*3 ^ 1740^2 _ 7350^ + 9639.

6.7 V* = sat[-F2,14; (A2 - Aip)2/?Ai].

6.9 J* = 10.56653.

6.10 y*{t) « 3e-3*, y*(t) w 1 - Se'^*.

0, 0 < i < 7/3,

2, 7/3 <t< 3,

- 1 , 3 < ^ < 13/3,

0, 13/3 <t<6.

6.12 M*(i) = {

6.13 /xi = < - f i + | , t € [ 0 , l ] ,

0, tG{0,3].

f^2= \ 0, i e [0,1.8),

- i i + f, i e [1.8,3].

7 7 = < 0, tG[0,l)U(1.8,3],

- f i + | , «€ [1,1.8).

E. Answers to Selected Exercises 413

6.14 (a) v*{t) = {

6.15 V* = <

- 1 for tG [0,1.8),

1 for i e (1.8,3].

(b) t;*(t) = 1 for iG [0,10].

- 1 for t G [0,1/2],

0 for f G (1/2, ti], where ti = 23/12,

+1 f o r i G ( i i , i i + l/2],

0 for tG (ii + 1/2,4].

6.17 u*(t) = \ 0, f o r 0 < i < i i ,

h(t-ti)/c, ioiti<t<T,

where ti=T- ^2BC/h.

Chapter 7

7.1 p* = 102.5 + 0.2G.

7.7 {u)/{pS) = {Sp)/{rj{p + S)).

7.10 G + SG = bang[0, oo; A + 1],

-X + (p + S)X = 7r'{G).

7.12 The equations corresponding to (6.28) and (6.29) can be obtained

by replacing p by p+r/r. The form of (6.30) remains imchanged.

7.17 (b)

h 1 1 Xo _, 1 . X-X^ : ln—, t2 = „ . In • rQ + 6 x^ rQ + S X — XT

7.18

T > ^ In ^<^(1 ~ ^o) ~ ^^0 + i In f i ~ rQ + 6 rQ( l — x^) — 6x^ 6 XT '

414 E. Answers to Selected Exercises

7.19

7.25

7.26

8.1

8.2

8.3

8.6

8.7

8.8

8.9

The reachable set is [xoe'^^, {xo - x)e-(*+'''5)^ + x],

where x = rQ/{W + rQ).

r 1-A 1-B

8.10

8.14

imp{A^B;t) = -

(b) J = 0.6325.

Chapter 8

(a) y = l^z = 3.

(b) y - 2, z - 10.

(a) (1,3) is a relative maximum.

(b) (2,10) is a relative maximum.

x = 50;x = 80.

(a) a: = 4 is a local maximum.

(b) a: = 8 is a local maximimn and a: = 20 is a local and a global maximum.

(a) (0, 0) is the nearest point.

(b) (1/2,1/2) is the nearest point.

(1/V^, 2/\/5) is the closest point.

(a) ( 2 ^ , 0 ) .

(b) (0,2).

(c) (0,2).

Xj = dF/dxJ for i = 1,2,. . . , n; A^+i = 1. Note that here T

denotes the terminal time, and not the transpose operation.

+1 ifA'=+i6>l,

- 1 i f A ' ^ + i J x - l ,whereA*= = (7 + A)^-'=A^

0 if iX^+^b] < 1.

u k*

E. Answers to Selected Exercises 415

l 2

Chapter 9

9.1 t' = 5.25, T = 11.

9.3 T = i* = 2.47.

9.4 t^ = 0, T = 30.

9.5 u*{t) = sat[0,1;u^{t)], where u^{t) = \2 - e005(t-34.8)|'/(i + i),

ti^S;t2-T = 34.8.

Chapter 10

10.3 X = 0.734. 10.4 (a)

X X = (-^i)W('-^i) + Sep

prX

(b) For p = 0, 5 = 220,000. Fovp = 0.1,x = 86,000. For p = oo, X = 40,000.

10.5 [g'{x) - p]\p - c{x)] - c^{x)g{x) = 0.

10.7 [g'{x) -p\\p- c{x)] - c'{x)g{x) + p = 0.

Chapter 11

11.1 A(i) = \oe^P-^^\ where

[Koe^T + 5(1 _ g/3T)/^ _ ^^j(2p - 0) Xo =

e/?T _ e2(/3-p)T

p p Zp :(-' e^').

Chapter 12

12.5 0 < T + ; ha(l - ^) = i.

12.8 ti = r /2 .

Chapter 13

13.5 q*{x) = j^^^,c*ix) = j^{p-r0-i^)x,

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Index

Abad, P.L., 417, 474 adjoint equation, 31, 32, 36, 230 adjoint function, 322 adjoint variables, 10, 32 adjoint vector, 30, 33 admissible control, 24 advertising model, 5, 6 affine function, 19 Agnew, C.E., 417 Alam, M., 249, 265, 417, 465 AUen, K.R., 286, 417 Amit, R., 154, 279, 418 Amoroso-Robinson relation, 187 Anderson, R.M., 418 anti-difference operator, 376 Aoki, M., 345, 418 applications to biomedicine, 295 applications to finance, 119 applications to marketing, 185 Arnold, L., 344, 347, 348, 418 Aronson, J.E., 418, 471 Arora, S.R., 243, 418 Arrow, K.J., 10, 46, 58, 81, 82,

186, 188, 289, 290, 360, 418, 456, 459

Arthur, W.B., 304, 418 Arugaslan, O., viii Arutyunov, A.V., 103, 418 Aseev, S.M., 103, 418 Aubin, J.-R, 418, 466 autocorrelation function, 343 autonomous, 52, 81

Axsater, S., 418

Baar, T., 425 backlogging of demand, 5, 348 Bagchi, A., 419 Balachandran, B., 445 Balakrishman, A.V., 455 bang function, 16 bang-bang, 42, 78, 84, 85, 87,

122, 132, 135, 190, 194, 232, 244, 247, 257, 258, 313, 322, 328, 329, 333, 334, 408

bankruptcy, 358 Bamea, A., 452 Basar, T., 308, 419, 420, 430,

446 Bass, F.M., 419 Bayes theorem, 342 Bean, J .C , 182, 419 begiiming game, 321 Bell, D.J., 42, 407, 419 Belhnan, R.E., 9, 27, 419 Bennett, R.J., 474 Bensoussan, A., 10, 58, 88, 154,

173, 182, 191, 315, 322, 324, 360, 419, 420, 444, 461, 470

bequest function, 7, 355 Berkovitz, L.D., 10, 27, 308, 420 Bernoulli, Jacob, 8, 9, 384 Bernoulli, Johann, 8, 9, 384

484 Index

Bertsekas, D.P., 236, 346, 420, 430

Bes, C , 173, 182, 420 Beyer, D., viii, 421 Bharucha-Reid, A.T., 480 Bhaskaran, S., 182, 421 bionomic eqmlibrium, 269, 314 Black, F., 355, 421 Blaquiere, A., 325, 326, 331, 421,

464, 473 BUss point, 306 BUss, G.A., 9 Boiteux problem, 241 Boiteux, M., 241, 421 Boltyanskii, V.G., 9, 27, 421,

462 Bolza, 9 Bolza form, 25, 26, 229, 235, 239 Bookbinder, J.H., vi, 10, 304,

421 boundary conditions, 61, 350 boundary interval, 105 Bourguignon, F., 421 Brachistochrone problem, 8, 9,

384 Breakwell, J.V., 190, 421 Brekke, K.A., 360, 422 Brito, D.L., 304, 422 Brock, W.A., 360, 454 Brockhoff, K., 420 broken extremal, 387 Brotherton, T., 423 Brown, R.G., 164, 422 Brownian Motion, 356 Bryant, G.F., 27, 422 Bryson, Jr., A.E., 33, 87, 105,

108, 341, 401, 403, 404, 406, 422

Buchanan, L.F., 345, 422 Bucy, R., 345, 447

Bulirsch, R., 9, 108, 422, 461, 474

Bullinger, H.J., 479 Bultez, A.V., 315, 419, 422 Burdet, C.A., 236, 239, 422 Burmeister, E., 290, 291, 422 Buskens, C., 455 Butkowskiy, A.G., 317, 422 Bylka, S., 182, 254, 423

Gaines, P., 423 calculus of variations, 8, 379 Canon, M.D., 235, 423 canonical adjoints, 33 canonical system of equations,

33 capital accumulation model, 290 Caratheodory, C., 9 Carlson, D.A., 10, 82, 423-425 Carraro, C , 315, 425 Carrillo, J., 154, 425 Case, J.H., 308, 325, 425 Cass, D., 425 Cassandras, C.G., 236, 461 cattle ranching problem, 318 Caulkins, J.P., 477 CeUina, A., 418 certainty equivalence, 403, 404 Cesari, L., 26, 425 chain of forests model, 276-278 chain of machines, 254 Chand, S., 182, 254, 425, 469 Chandra, T., 445 Chang, S., 360, 418 Charnes, A., 304, 425 Chatterjee, R., 463 Chen, S.F., 425 Cheng, F., viii Chiarella, C , 426 Chichilinsky, G., 426

Index 485

Chikan, A., 440 Chintagunta, P.K., 315, 426 Chow, G.C., 345, 426 Clark, C.W., 10, 241, 267, 268,

273, 286, 312, 314, 426 Clarke, F.H., 27, 106, 167, 427 Clemhout, S., 427 closed-loop control, 346 closed-loop Nash solution, 311 Coddington, E.A., 193, 427 Cohen, K.J., 36, 187, 427 common-property fishery re­

sources, 312 comparison lemma, 198 complimentary slackness condi­

tions, 60 computational methods, vii,

108, 236 concave function, 18, 64 Connors, M.M., 10, 427 Conrad, K., 427 Constantinides, G.M., 427 constraint of r th order, 105 constraint qualification, 60, 105,

226 constraints, 24 consumption model, 7, 8 consumption-investment prob­

lem, 355 contact time, 105 continuous wheat trading model,

164 control of pest infestations, 295 control trajectory, 2, 24 control variable, 2, 24 convex combination, 17 convex function, 18, 64 convex huU, 17 convex set, 17 CorelDRAW, vii

Cottle, R.W., 430 Cowling, K., 444 critical points, 218 Crouhy, M., 173, 182, 419 Cruz, J.B., Jr., 472 CuUum, CD. , 235, 423 current-value adjoint variables,

70 current-value formulation, 58,

65, 239 current-value functions, 67 current-value Hamiltonian, 66 current-value Lagrange multipli­

ers, 67 current-value Lagrangian, 66 current-value maximum princi­

ple, 68, 111 cycloid, 9 Cyert, R.M., 36, 187, 427

D'Autimie, A., 428 Dantzig, G.B., 418, 427 Darrat, A.F., 456 Darrough, M.N., 427 Dasgupta, P., 279, 428, 474, 476 Davis, B.E., 119, 129, 152, 360,

428 Davis, M.H.A., 346, 428 Davis, R.E., 456 Dawid, H., 428 Day, G., 463 DDT, 299, 300, 303 Deal, K.R., 311, 315, 337, 428 decision horizon, 173, 175, 177,

179, 180 Deger, S., 428 Deissenberg, C , 428, 429 Deistler, M., 476 derivative operator, 363 derived Hamiltonian, 45

486 Index

Derzko, N.A., 45, 130, 279, 315, 317, 360, 420, 429, 466, 469

DeSarbo, W., 304, 445 Dhrymes, P.J., 213, 429 difference equation, 229, 341 diflference operator, 375 differential games, 308 differentiation with scalars, 12 differentiation with vectors, 12,

13 diffusion process, 344 Dirac delta function, 323, 344 direct adjoining method, 100 direct contribution, 35 discount factor, 6 discount rate, 6 discrete maximum principle,

217, 228, 229 discrete-time optimal control

problem, 228, 229 distributed parameter maximimi

principle, 317 distributed parameter systems,

315 Dixit, A.K., 429 Dobell, A.R., 290, 291, 422 Dockner, E.J., 10, 289, 304, 308,

315, 429, 430, 432, 446 Dohrmann, C.R., 236, 430 Dolan, R.J., 430, 445 Dorfman, R., 430 Dornoff, R.J., 474 Drews, W., 430 Dreyfus, S.E., 420 dual variables, 36 Dubovitskii, A.Y., 430 Dunn, J .C , 236, 430 Durrett, R., 347, 431

dynamic efficiency condition, 291

dynamic programming, 27, 345, 393

economic applications, 289, 360 economic interpretation, 34, 69,

138, 291, 322 educational policy, 21 eigenvalues, 370, 371 eigenvectors, 370, 371 El-Hodiri, M., 431 EUashberg, J., 304, 431, 445, 475 Elton, E., 119, 431 Elzinga, D.J., 119, 152, 428 ending correction, 158 ending game, 321 entry time, 105 environmental management, 315 EOQ, 153 epidemic control, 295 equilibrimn relation, 36 Erickson, G.M., 431 Erickson, L.E., 10, 154, 315, 443 Euler, 8, 384 Euler equation, 380, 382, 387,

388 EXCEL, vii, 48-50 exhaustible resource model, 279 exit time, 105

factorial power, 375 Fan, L.T., 154, 304, 431, 443 Farley, J.U., 475 Feichtinger, G., vi, viii, 10, 33,

58, 69, 81, 99, 106, 108, 113, 154, 167, 185, 194, 211, 289, 304, 322, 427-429, 431-434, 438-440, 442, 446, 451, 453, 455,

Index 487

457^59, 463-465, 470, 473, 475^78, 480

Feinberg, F.M., 434 Fel'dbaum, A.A., 31, 393, 398,

434 Ferreira, M.M.A., 103, 434 Ferreyra, G., 434 Filar, J., 315, 425 Filipiak, J., 434 filtering, 339, 341 finite diflference equations, 375 first-order pure state con­

straints, 106, 108 Fischer, T., 434 Fisher, A.C., 461 fishery management, 315 fishery model, 268, 312 fishing mortality function, 314 fixed-end-point problem, 70 Fleming, W.H., 346, 360, 434 Fletcher, R., 434 Fomin, S.V., 379, 380, 382, 387,

389, 437 forecast horizons, 173 forest fertilization model, 287 forest thinning model, 273, 276 forgetting coefficient, 5 Forster, B.A., 435 Fourgeaud, C , 435 Francis, P.J., 295, 435 Frankena, J.F., 435 free-end-point problem, 70 Friedman, A., 308, 435 Fromovitz, S., 228, 455 Fruchter, G.E., 315, 435 fuU-rank condition, 20, 60, 105,

106 Fuller, D., 280, 435 fundamental lemma, 382 Funke, U.H., 435

Furst, E., 476

Gaandolfo, G., 436 Gaimon, C , 154, 249, 304, 322,

331, 425, 435, 436 Galileo, 384 Gamkrelidze, R.V., 9, 436, 462 Gandolfo, G., 432 Gaskins, Jr., D.W., 436 Gaugusch, J., 437 Gaussian, 341-343 Geismar, N., viii Gelfand, I.M., 379, 380, 382,

387, 389, 437 general discrete maximum prin­

ciple, 234 general inequality constraints,

97 generalized bang-bang, 87, 132 generalized derivative, 344 generalized Legendre-Clebsch

condition, 152, 407, 408 Geoffrion, A.M., 453 Gerchak, Y., 437 Gfrerer, H., 437 Gibson, J.E., 408, 445 Gihman, LI., 344, 353, 437 Gillessen, W., 455 Girsanov, I.V., 437 Glad, S.T., 437 goal level, 319 GOAL SEEK, 48, 49 Goh, B.-S., 108, 267, 437 Goh, C.J., 476 Goldberg, S., 375, 437 Golden Path, 82 Golden Rule, 82, 93 Goldstine, H.H., 437 goodwill, 5, 186

488 Index

goodwill elasticity of demand, 188

Gopalsamy, K., 437 Gordon's formula, 145 Gordon, H.S., 268, 269, 437 Gordon, M.J., 145, 437 Gould, J.P., 191, 210, 214, 437 Green's theorem, 185, 196, 198,

205, 211, 214, 270, 296, 297, 305, 314

Grimm, W., 438, 459 Gross, M., 438 Gruber, M., 119, 431 Gruver, W.A., 448

Hadley, G., 10, 58, 289, 438 Hahn, M., 438 Halkin, H., 27, 235, 438 Hamalainen, R.P., 315, 438 Hamilton, 9 Hamilton-Jacobi equation, 349 Hamilton-Jacobi-BeUman equa­

tion, 27, 30, 345, 349 Hamiltonian, 30, 35, 60, 230,

291, 397 Hamiltonian maximizing condi­

tion, 30, 34, 397 Han, M., 438 Hanson, M., 457 Hanssens, D.M., 438 Harris, F.W., 153, 438 Harris, H., 303, 439 Harrison, J.M., 357, 439 Hartberger, R.J., 27, 396, 430,

439 Hartl, R.F., viii, 10, 26, 27, 33,

58, 62, 69, 81, 89, 99, 100, 105, 106, 108, 113, 115, 154, 167, 174, 185, 194, 211, 239, 243, 303,

304, 317, 322, 428, 432, 433, 438-442, 453, 473

Harvey, A.C., 442 Haunschmied, J., 304, 433 Haurie, A., 10, 82, 173, 308, 315,

317, 322, 420, 424, 425, 438, 442

Haussmann, U.G., 442 Heal, G.M., 279, 428, 443, 476 Heaps, T., 241, 443 Heckman, J., 443 Heineke, J.M., 427 Hestenes, M.R., 10, 58, 62, 443 higher-order constraints, 105 HJB equation, 30, 31, 354 HMMS model, 153 Ho, Y.-C., 33, 87, 105, 308, 311,

341, 401, 403, 404, 406, 422, 443, 459, 473

Hochman, E., 360, 464 Hoffmann, K.H., 443 Hohn, F., 174, 457 Holly, S., 443 Holt, C.C., 153, 443 Holtzman, J.M., 235, 443 homogeneous equation, 363 homogeneous equations of order

n, 365 homogeneous equations of order

one, 364 homogeneous equations of order

two, 364 homogeneous function of degree

one, 19 homogeneous partial differential

equations, 374 Horsky, D., 443 Hotelling, H., 279, 443 Hung, N.M., 442 Hurst, Jr., E.G., 10, 58, 154, 419

Index 489

Hwang, C.L., 154, 443 Hyun, J.S., 438

Ijiri, Y., 153, 165, 444 Ilan, Y., 154, 418 imp, 17 impulse control, 16, 125, 202,

204, 322, 324, 325 impulse control model, 88 impulse Hamiltonian, 326 impulse maximum principle,

326, 332 impulse stochastic control, 360 imputed value, 219 indirect adjoining method, 98,

100, 104,111 indirect contribution, 35 infinite horizon, 6, 80 inhomogeneous partial differen­

tial equation, 374 instantaneous profit rate, 25 interior interval, 105 Intriligator, M.D., 290, 444, 456 inventory control problem, 117 loffe, A.D., 444 Isaacs, R., 9, 133, 308, 444 isoperimetric profit constraint,

214 Ito stochastic differential equa­

tion, 344, 345 Ivanilov, Y.P., 304, 480

Jabrane, A., 424 Jacobi, 9 Jacobson, D.H., 42, 108, 407,

419, 444, 451 Jacquemin, A.P., 188, 191, 444 Jagpal, S., 444 Jain, D., 315, 426 Jamshidi, M., 444

Jazwinski, A.H., 444 Jedidi, K., 304, 445 Jennings, L.S., 445 Jeuland, A.P., 430, 445 Jiang, J., 445 Johnson, CD. , 408, 445 Jones, P., 445 J0rgensen, S., viii, 10, 154, 267,

289, 304, 308, 315, 429, 430, 433, 440, 445, 446

Joseph, P.D., 341, 404, 446 jirnip conditions, 103, 108 jump Markov processes, 360 junction times, 105

Kaitala, V.T., 315, 433, 438, 446 Kalaba, R.E., 419 Kalish, S., 304, 435, 446, 447 KaU, P., 439, 458, 470 Kahnan filter, 339, 340, 342 Kahnan, R.E., 341, 345, 447 Kaknan-Bucy filter, 339, 345 Kamien, M.I., 10, 248, 253, 289,

303, 447 Kamien-Schwartz model, 249 Kaplan, W., 447 Karatzas, I., 340, 344, 347, 359,

360, 447 Karreman, H.F., 421 Keeler, E., 448 Keller, H.B., 299, 448 Kemp, M.C., 10, 58, 289, 426,

438, 448 Kendrick, D.A., 448 Keon, J.W., 476 Khmelnitsky, E., 10, 448, 449,

454 Kilkki, P., 273, 274, 448 Kirakossian, G.T., 304, 440 Kirby, B.J., 448

490 Index

Kirk, D.E., 31, 448 Klein, C.F., 448 Kleindorfer, G.B., 234, 449 Kleindorfer, P.R., 174, 182, 234,

448, 449, 470 Kleinschmidt, P., 440 Knobloch, H.W., 449 Knowles, G., 449 Kogan, K., 10, 448, 449, 454 Kopel, M., 428 Kort, P.M., 10, 154, 267, 289,

303, 304, 433, 440, 441, 446, 478

Kortanek, K., 304, 425 Kotowitz, Y., 449 Kozlowski, J., 480 Krabs, W., 443 Kraft, D., 108, 422 Krarup, J., 430 Krauth, J., 304, 441 Kreindler, E., 449 KreUe, W., 420, 449 Krichagina, E., 449 Kriebel, C.H., 234, 449 Kriendler, E., 108 Krouse, CO. , 129, 449, 450 Krutilla, J.V., 447 Kugelmann, B., 450 Kuhn, H.W., 472 Kuhn-Tucker conditions, 218,

220, 228 Kuhn-Tucker constraint qualifi­

cation, 226, 227 Kumar, S., viii Kurawarwala, A.A., 450 Kurcyusz, S., 103, 450 Kurz, M., 10, 46, 58, 81, 82, 188,

289, 290, 418 Kushner, H.J., 450 Kydland, F.E., 450

L'Hospital, 384 Lagrange, 8 Lagrange form, 25 Lagrange multipliers, 57, 218 Lagrangian, 57, 60 Lagtmov, V.N., 450 Lakhani, C , 465 Lansdowne, Z.F., 206, 450 Lasdon, L.S., 450 Leban, R., 10, 289, 450, 451 Leclair, S.R., 445 Lee, E.B., 129, 450 Lee, S.C, 213, 469 Lee, W.Y., 450 left and right limits, 15 Legendre, 8 Legendre's conditions, 388 Legey, L., 304, 450 Lehoczky, J.P., 130, 359, 447,

450, 469 Leibniz, 8 Leitmann, G., 27, 267, 308, 379,

425, 437, 438, 442, 451, 457, 473

Leizarowitz, A., 424, 425 Leland, H.E., 451 Lele, M.M., 108, 444, 451 Lele, P.T., 243, 418 Lenclud, B., 435 Leonard, D., 10, 289, 451 Leondes, C.T., 422, 473, 479 Lesourne, J., 10, 289, 360, 419,

420, 450, 451 Lev, B., 436 Levine, J., 451 Levinson, N.L., 193, 427 Lewis, T.R., 451, 452 Li, G., 154, 452 Lieber, Z., 174, 182, 438, 449,

452

Index 491

LigneU, J., 452 LiUen, G.L., 304, 446 line integral, 197 linear differential equations, 363 linear independence, 20 linear Mayer form, 25, 26, 239 linear programming, 87, 132 linear-quadratic case, 85 linear-quadratic problems, 401 linearly independent, 20 Lintner, J., 452 Lions, J.L., 88, 317, 324, 360,

420, 444, 452, 470 Little, J.D.C., 452 little-o notation, 15 Liu, P.-T., 418, 421, 428, 451,

452 Loewen, P.D., 106, 427 logarithmic Brownian Motion,

356 Long, N.V., 10, 289, 308, 315,

426, 430, 448, 451, 452 long-rim stationary equilibrium,

82 Loon, P.J.J.M., van, 453 Lou, S., 449, 453 Lucas, Jr., R.E., 304, 453 Luenberger, D.G., 228, 453 Luhmer, A., 433, 453 limiped parameter systems, 315 Lundin, R.A., 174, 182, 453 Luptacik, M., 303, 441, 453, 463 Luus, R., 454 Lynn, J.W., 249, 417

Macki, J., 454 Magat, W.A., 454 Magill, M.J.P., 454 Mahajan, V., 430, 447, 452, 454 Maimon, O., 10, 448, 449, 454

maintenance and replacement model, 241, 242, 248, 331

Majumdar, M., 454 Malinowski, K., 115, 454, 472 Malliaris, A.G., 360, 454 Mangasarian, O.L., 46, 64, 226-

228, 455 Manh-Hung, N., 279, 455 Mantrala, M.K., 345, 458 MAPI (Machinery and Applied

Products Institute), 241 MAPLE, 150 marginal cost, 36, 187 marginal cost equals marginal

revenue, 36 marginal return, 398 marginal revenue, 36 Markus, L., 450 martingale problems, 360 Martirena-Mantel, A.M., 455 Marzano, P., 432 Masse, P., 241, 455 Mate, K., 443 Mathematica, 150 mathematical requirements, 1 Mathewson, P., 449 matrix Riccati equation, 345,

403 Matsuo, H., 450 Maurer, H., viii, 108, 115, 118,

455 maximized Hamiltonian, 114 maximum, 388 maximum likelihood estimate,

342 maximum principle, 23, 33, 34,

57, 58, 67, 104,113, 217, 393

May, R.M., 418

492 Index

Mayer form, 25, 397 Mayne, D.Q., 27, 108, 236, 422,

456, 460, 462 McCabe, J.L., 451 McCann, J.M., 454 McEneaney, W.M., 470 McGuire, T.W., 304, 469 Mclntyre, J., 108, 456 McNicoU, G., 304, 418 McShane, E.J., 10 measurement noise, 340 Meech, J.A., 445 Mehlmann, A., 304, 305, 308,

429, 433, 441, 456 Mehra, R.K., 304, 456 Mehrez, A., 456 Merton, R.C., 355, 456 Mesak, H.I., 456 Michel, P., 428, 435, 457 middle game, 321 Miele, A., 196, 457 MiUer, M.H., 130, 457 MiUer, R.E., 457 Milyutin, A.A., 430 minimax solution, 308, 309 minimum fuel problem, 238 Mirman, L.J., 452, 457 Mirrlees, J., 457 miscellaneous applications, 303 Mischenko, E.F., 9, 462 MitcheU, A., 457 Mitra, T., 454 Mitter, S.K., 459 mixed constraints, 59, 104, 106 mixed inequality constraints, 3,

57, 58 mixed optimization technique,

258 model type (a), 84, 85, 87, 243,

244

model type (b), 85-87 model type (c), 85 model type (d), 85 model type (e), 85, 86 model type (f), 85, 86, 251 model types, 83, 85 modeling "tricks", 86 Modigliani, F., 130, 153, 174,

443, 457 Moiseev, N.N., 457 Monahan, G.E., 457 Mond, B., 457 Moore, E.J., 476 Morey, R.C., 454 Morton, T.E., vi, 174, 182, 254,

259, 264, 453, 457, 458, 469

Motta, M., 324, 458 Muller, E., 430, 452, 454, 458 Mulvey, J.M., 469 Murata, Y., 458 Murray, D.M., 236, 458 Muth, J.F., 153, 443 Muzicant, J., 322, 458

Naert, P.A., 315, 419, 422 Nahorski, Z., 458 Naik, P.A., 345, 458 Nash solutions, 308 Naslund, B., 10, 58, 154, 241,

278, 287, 419, 458 natural resources, 267, 360 necessary condition, 31, 33, 228 Neck, R., 458 needle-shaped variation, 394,

395 neighborhood, 15 Nelson, R.T., 304, 458 Nepomiastchy, P., 304, 459 Nerlove, M., 186, 188, 459

Index 493

Nerlove-Love advertising model, 186

Neuman, C.P., 186, 477 Neustadt, L.W., 10, 455, 459 Newton, 8, 384 Nguyen, D., 459 Nissen, G., 322, 420 nonlinear programming, 217,

218, 227 nonzero-simi games, 310 norm, 15 Norstrom, C.J., vi, 125, 153,

170, 459 Norton, F.E., 345, 422 notation, 10 Novak, A., 304, 433, 434, 441,

459

Oakland, W.H., 304, 422 Oberle, H.J., 438, 459 objective function, 2, 25, 398 Oettli, W., 422 Oguztoreli, M.N., 459 oil driller's problem, 324 0ksendal, B.K., 344, 360, 422,

459 Okuguchi, K., 426 Olsder, G.J., 419, 459 one-sector model, 291 one-sided constraints, 71 Oniki, H., 460 open access fishery, 269 open-loop Nash solution, 310 optimal consumption of an ini­

tial investment, 89 optimal control problem, 25 optimal control theory, 1, 379 optimal economic growth mod­

els, 289, 293 optimal financing model, 129

optimal long-run stationary equilibriimi, 82, 189

optimal path, 25 optimal thinning, 274 optimal trajectory, 25 order of the constraint, 105 Oren, S.S., 460 Osayimwese, I., 460 overdraft, 124 Ozga model, 214 Ozga, S., 214, 460

Paiewonsky, B., 108, 456 Palda, K.S., 186, 460 Pantoja, J.F., 236, 460 parametric linear programming,

87 Parlar, M., 437, 460 Parrish, B., 417 Parsons, L.J., 438 partial differential equations,

372 partial fractions, 350 particular integral, 367 particular solutions, 366, 367 path of least time, 8 Pauwels, W., 460 Pekelman, D., 154, 174, 182,

241, 259, 460, 461 Pepyne, D.L., 236, 461 Pesch, H.J., 9, 450, 461 pessimal solution, 146 Peterson, D.W., 461 Peterson, F.M., 461 Peterson, R.A., 454 Petrov, lu.P., 461 phase diagram, 192, 293, 301 Phelps, E.S., 437 Pierskalla, W.P., 241, 461

494 Index

Pindyck, R.S., 279, 360, 429, 461, 462, 478

Pitchford, J.D., 435, 452, 462 Pliska, S.R., 357, 439 Pohjola, M., 462 Polak, E., 108, 235, 423, 456, 462 pollution control model, 299, 302 Pontryagin, L.S., 9, 10, 23, 27,

76, 106, 393, 462 Powell, S.G., 460 predator-prey relationships, 267 Prescott, E.G., 450 Presman, E., 462, 463 price elasticity of demand, 187 price shield, 174, 175, 177 principle of optimality, 27, 346,

358 product rule for differentiation,

14 production fimction, 292 production planning model, 153,

339 production smoothing, 154 production-inventory model, 4,

153, 154, 234 Proth, J.-M., 173, 182, 419, 425 Prskawetz, A., 463 pure constraints, 116 pure state variable inequality

constraints, 3, 97, 98, 104

Pytlak, R., 108, 463

quasiconcave fimction, 64 quasiconvex function, 64

Rajagopalan, S., 154, 452 Raman, K., 360, 463 Rampazzo, F., 324, 458 Ramsey, P.P., 69, 289, 306, 463

rank of a matrix, 20 Rao, R.C., 315, 463 Rapoport, A., 464 Rapp, B., 241, 464 Rausser, G.C., 360, 461, 464 Raviv, A., 304, 464 Ravn, H.F., 458 Ray, A., 464 reachable set, 3, 59 Reeves, CM., 434 regional allocation of invest­

ment, 52 Reinganum, J.F., 464 Rempala, R., 174, 464 Richard, S.F., 427, 464 Ringbeck, J., 304, 464 Ripper, M., 304, 450 Rishel, R.W., 346, 434, 465 Roberts, S.M., 33, 465 Robinett, R.D., 236, 430 Robinson, B., 465 Robson, A.J., 322, 465 Rockafeller, R.T., 465 Roxin, E.G., 421, 423, 452 Russak, B., 465 Russell, D.L., 427, 465 Riistem, B., 443 Ruusunen, J., 315, 438 Ryu, Y., viii

saddle point, 19, 309 Sage, A.P., 317, 465 Salukvadze, M.E., 465 salvage value, 3, 25, 87 Samaratunga, G., 466 Samuelson, P.A., 465 Sargent, T.J., 453 Sarma, V.V.S., 249, 265, 417,

465 Sasieni, M., 466

Index 495

sat fiinction, 16 Sawyer, A., 345, 458 Scalzo, R.C., 466 Schaefer, M.B., 268, 466 Schijndel, G.-J.C.Th.,van, 304,

466 SchiUing, K., 466 Schmalensee, R., 451, 452 Scholes, M., 355, 421 Schubert, U., 303, 453 Schultz, R. L., 438 Schwartz, N.L., 10, 248, 253,

289, 303, 447 Schwodiauer, G., 476 second-order differential equa­

tions, 359 second-order variations, 388, 405 Segers, R., 430 Seidman, T.I., 315, 466 Seierstad, A., 10, 27, 45, 58, 62,

64, 81, 99, 113, 289, 466 Selten, R., 308, 466 Sen, S.K., 428, 447 Sengupta, J.K., 477 separation principle, 403, 404 Sethi, S.P., 10, 26, 27, 45, 58,

62, 89, 99,100,105, 106, 108, 113, 115, 119, 129, 130, 150, 153, 154, 173, 174, 182, 185, 190, 194-196, 202, 206, 213-215, 236, 239, 241, 247, 254, 259, 264, 270, 271, 276, 278, 279, 295, 298, 303, 304, 311, 315, 317, 322, 337, 340, 347, 351, 352, 354, 357-360, 420-423, 425, 427-429, 433, 441, 442, 445, 447, 449, 450, 453, 461-464, 466-471,

476, 480 Sethi-Morton model, 254, 264,

331 shadow price, 10, 35, 219 Shapiro, A., 472 Shapiro, C , 103, 472 Shell, K., 289, 425, 472 Shipman, J.S., 33, 465 shooting method, 182 short-selling, 124 Shreve, S.E., 340, 344, 346, 347,

359, 360, 420, 447 Shtub, A., 449 Siebert, H., 472 Silva, G.N., 324, 472 Simaan, M., 472 Simon, H.A., 153, 404, 443, 472 Simon, L.S., 443 simple cash balance problem,

120 simplest variational problem,

379 Singh, M.G., 468, 472 singular arcs, 407 singular control, 42, 132, 140,

297, 407 Skiba, A.K., 472 Skorohod, A.V., 344, 353, 437 Smith, B.L.R., 444 Smith, M., 445 Smith, R.L., 182, 419 Smith, V.K., 447 Smith, V.L., 472 Snower, D.J., 472 sole owner fishery resource

model, 268 Soliman, M.A., 304, 475 Solow, R.M., 279, 472, 473 Soner, H.M., 360, 434, 450 Sorenson, H., 341, 473

496 Index

Sorger, G., viii, 10, 182, 254, 289, 308, 315, 423, 425, 430, 433, 434, 453, 469, 473

Sothmann, B., 459 Southwick, L., 303, 473 special topics, 401 Spence, M., 299, 448, 473 Speyer, J.L., 108, 444 Spiegel, M.R., 375, 473 Spremarm, K., 473 Sprzeuzkouski, A.Y., 154, 473 Spulber, P.F., 452, 457 Srinivasan, V., 186, 473 Sriskandarajah, C , viii Staats, P.W., 295, 469 Stalford, H., 473 standard adjoint variables, 65 standard Hamiltonian, 65 standard Lagrangian, 65 standard multipliers, 65 Starr, A.W., 308, 311, 473 starting correction, 158 state equation, 24 state trajectory, 2, 24 state variable, 2, 24 static efficiency condition, 291 stationarity assumption, 81 Stein, R.B., 459 Steinberg, R., 431 Steindl, A., 453, 473 Steiner, P.O., 430 Stepan, A., 473 Stern, L.E., 473 Stiglitz, J.E., 474 Stirling numbers of the first

kind, 378 StirUng numbers of the second

kind, 376

stochastic advertising problem, 352

stochastic calculus, 347 stochastic differential equations,

339, 344 stochastic manufacturing prob­

lems, 360 stochastic optimal control, 339,

345, 346 stochastic production planning

model, 347 stockout cost, 5 Stoer, J., 422, 474 stopping time, 357 Stoppler, S., 10, 154, 429, 474 Strauss, A., 454 Streitferdt, L., 466 strengthened Jacobi condition,

389 strengthened Legendre condi­

tion, 389 strengthened Legendre-Clebsch

condition, 406 strictly concave function, 18, 64 strictly convex function, 64 strong forecast horizon, 174,

177, 179 strong maximum, 389, 390 sufficiency conditions, 44, 46, 64,

113, 228 sufficiency transversality condi­

tion, 159 summary of transversality con­

ditions, 75 Suo, W., viii, 462, 469, 470 surveys of appfications, 10 Sutinen, J.G., 428, 452 Swan, G.W., 295, 474 Sweeney, D.J., 417, 474 switching curves, 78

Index 497

switching point, 138 switching time, 80 Sydsaeter, K., 10, 27, 45, 58, 62,

64, 81, 99,113, 289, 466, 474

synthesis of optimal controls, 76, 133

system, 2 system noise, 340 Szego, G.P., 472

Takayama, A., 36, 289, 431, 474 Taksar, M.I., 449, 450, 469, 470 Tan, K.C., 474 Tapiero, C.S., 10, 254, 264, 304,

322, 360, 420, 470, 474, 475

Taylor, J.G., 108, 304, 475 Teichroew, D., 10, 427 Teng, J.-T., 182, 475, 476 Teo, K.L., 108, 445, 476 Terborgh, G., 241, 476 terminal conditions, 32, 69 terminal inequality constraints,

59 terminal time, 4, 25, 62 Thepot, J., 304, 451, 476 Thisse, J., 444 Thompson's maintenance

model, 331 Thompson, G.L., 45, 119, 153,

165, 174, 182, 234, 241, 249, 259, 264, 304, 311, 315, 317, 322, 331, 337, 347, 351, 418, 428, 429, 436, 444, 449, 470, 475, 476

Tihomirov, V.M., 444 time-optimal control problem,

76

Tintner, G., 477 TitU, A., 472 Tolwinski, B., 308, 442, 477 total contribution, 35 Tou, J.T., 341, 404, 446 Toussaint, S., 477 TPBVP, 33, 48, 50, 182, 291 Tracz, G.S., 477 Tragler, G., 477 transition matrix, 396 transversality condition, 32, 62,

67, 69, 75, 81 Treadway, A.B., 303, 477 Troch, I., 477 Tsurumi, H., 214, 477 Tsurumi, Y., 214, 477 Tu, P.N.V., 10, 303, 477 Tuominen, M.P.T., 452 Turner, R.E., 185, 477 Turnovsky, S.J., 435, 452, 462,

477 turnpike, 82, 158, 189, 207 two person zero-sum games, 308 two-point bovmdary value prob­

lem, 33, 48, 372 two-reservoir system, 116 Tzafestas, S.G., 322, 432, 477

Udayabhanu, V., 470 Uhler, R.S., 478 utility of consiunption, 7, 289,

290, 355

Vaisanen, U., 273, 274, 448 Valentine, F.A., 10, 478 value function, 27, 346 Van Hilten, O., 10, 289, 478 Van Loon, P.J.J.M., 10, 289, 478 Vanthienen, L., 174, 478 Varaiya, P.P., 304, 308, 450, 478

498 Index

variational equations, 395 Veinott, A.F., 418 Venezia, I., 475 Verheyen, P.A., 304, 478 Verma, B., 445 Vickson, R.G., 26, 27, 58, 62, 99,

100, 105, 106, 108, 113, 115, 154, 280, 420, 435, 442, 460, 478, 480

Vidal, R.V.V., 458 Vidale, M.L., 186, 194, 195, 478 Vidale-Wolfe advertising model,

194, 353 Vilcassim, N.J., 315, 426 Villas-Boas, J.M., 478 Vincent, T.L., 267, 437 Vinokurov, V.R., 478 Vinter, R.B., 103, 108, 324, 434,

463, 472, 478 Voelker, J.A., 241, 461 Vousden, N., 303, 452, 479

Wagner, H.M., 153, 254, 479 Wagner-Whitin framework, 258 Wagner-Whitin solution, 262 Wan, F.Y., 473 Wan, Jr., H.Y., 427 Wang, C.-S., 304, 431 Wang, P.K.C., 479 warehousing constraint, 175 Warga, J., 479 Warnecke, H.J., 479 Warschat, J., 479 weak forecast horizon, 174, 175 weak maximum, 389 Weierstrass, 9 Weierstrass necessary condition,

389, 391 Weierstrass-Erdmann corner

conditions, 388

Weinstein, M.C., 279, 479 Weitz, B., 463 Weizsacker, C.C. von, 479 Welam, U.P., 479 WeU, K.H., 438 Wensley, R., 463 Westphal, L.C., 479 wheat trading model with no

short-selling, 170 Whitin, T.M., 153, 254, 479 Whittle, P., 479 Wickwire, K., 10, 295, 479 Wiegand, M., 118, 455 Wiener process, 344, 346, 356 Wiener, N., 479 Wind, Y., 430, 447, 454, 475 Wirl, F., 434, 480 Wolfe, H.B., 186, 194, 195, 478 Wong, K.H., 108, 476 Wonham, W.M., 480 Wright, C , 303, 480 Wright, S.J., 236, 480 Wunderlich, H.J., 479

Yakowitz, S.J., 236, 458 Yan, H., viii, 469, 480 Yang, J., 480 Yang, T.H., 108, 462 Yeh, D., viii Yin, G., 360, 462, 468, 470, 471,

480 Young, L.C., 379, 399, 480

Zaccour, G., 154, 446 Zalkin, J.H., 461 Zarrop, M.B., 443 Zeckhauser, R.J., 279, 299, 448,

479 Zeidan, V., 480 Zemel, E., 457

Index 499

Zhang, H., viii, 462, 463, 470, 471

Zhang, Q., viii, 340, 360, 453, 462, 463, 468, 470, 471, 480

Zhou, X., 466, 471, 480 Ziemba, W.T., 469, 480 Zimin, I.N., 304, 480 Ziolko, M., 480 Zionts, S., 303, 471, 473 Zoltners, A.A., 445, 481 Zowe, J., 103, 450 Zuckermann, D., 475

List of Figures

1.1 The Brachistochrone Problem 9 1.2 A Concave Function 18 1.3 An Illustration of a Saddle Point 19

2.1 An Optimal Path in the State-Time Space 28 2.2 Optimal State and Adjoint Trajectories for Example 2.1 . 38 2.3 Optimal State and Adjoint Trajectories for Example 2.2 . 39 2.4 Optimal Trajectories for Examples 2.3 and 2.4 41 2.5 Optimal Control for Example 2.5 44 2.6 Solution of TPBVP by EXCEL 50 2.7 Water Reservoir of Example 2.12 53

3.1 State and Adjoint Trajectories in Example 3.3 73 3.2 Minimum Time Optimal Response for Problem (3.63) . . 79

4.1 State and Adjoint Trajectories in Example 4.1 102 4.2 Infeasible State Space and Optimal State Trajectory

for Example 4.3 109 4.3 Adjoint Trajectory for Example 4.3 I l l 4.4 Two-Reservoir System of Exercise 4.6 116

5.1 Optimal Policy Shown in (Ai, A2) Space 123 5.2 Optimal Policy Shown in ( , A2/A1) Space 124 5.3 Adjoint Variables and Lagrange Multipliers for

Example 5.1 128 5.4 Case A: g<r 133 5.5 Case A: g > r 134 5.6 Optimal Path for Case A: g <r 139 5.7 Optimal Path for Case B: g > r 143 5.8 Solution for Exercise 5.10 150 5.9 Adjoint Trajectories for Exercise 5.11 151

502 List of Figures

6.1 Optimal Production and Inventory Levels 161 6.2 Solution of Example 6.1 with IQ = 10 163 6.3 Solution of Example 6.1 with IQ = 50 164 6.4 Solution of Example 6.1 with Io=^30 165 6.5 The Price Trajectory (6.49) 168 6.6 Adjoint Variable, Optimal Policy and Inventory in the

Wheat Trading Model 169 6.7 Adjoint Trajectory and Optimal Policy for the Wheat

Trading Model 173 6.8 Decision Horizon and Optimal Policy for the Wheat

Trading Model 175 6.9 Optimal Policy and Horizons for the Wheat Trading

Model with Warehouse Constraint 177 6.10 Optimal Policy and Horizons for Example 6.3 179 6.11 Optimal Policy and Horizons for Example 6.4 180 6.12 The Flow Chart for Exercise 6.9 183

7.1 Optimal Policies in the Nerlove-Arrow Model 189 7.2 A Case of a Time-Dependent Turnpike and the Nature

of Optimal Control 190 7.3 Phase Diagram of System (7.18) for Problem (7.13) . . . . 192 7.4 Feasible Arcs in ( , a;)-Space 197 7.5 Optimal Trajectory for Case 1: XQ < x^ and x^ > XT - - • 200 7.6 Optimal Trajectory for Case 2: XQ < x^ and x^ < XT - - - 200 7.7 Optimal Trajectory for Case 3: XQ > x^ and x^ > XT - - - 201 7.8 Optimal Trajectory for Case 4: XQ > x^ and x^ < XT • • - 201 7.9 Optimal Trajectory (Solid Lines) 202 7.10 Optimal Trajectory When T Is Small in Case 1: XQ < x^

8indxT<x^ 203 7.11 Optimal Trajectory When T Is Small in Case 2: XQ > x^

andxT<x' 203 7.12 Optimal Trajectory for Case 2 of Theorem 7.1 for Q = oo 204 7.13 Optimal Trajectories for x{0) <x 208 7.14 Optimal Trajectory for x(0) >x 209

8.1 Shortest Distance from a Point to a Semi-Circle 224 8.2 Graph of Example 8.5 224 8.3 Kuhn-Tucker Constraint Qualification 226 8.4 Discrete-Time Conventions 229 8.5 Sketch of x^ and A 233

List of Figures 503

9.1 Optimal Maintenance and Machine Resale Value 247 9.2 Sat Function Optimal Control 249

10.1 Optimal Policy for the Sole Owner Fishery Model 271 10.2 Singular Usable Timber Volume x{t) 275 10.3 Optimal Policy for the Forest Thinning Model when

XQ < x{to) 276 10.4 Optimal Policy for the Chain of Forests Model when

T>i 277 10.5 Optimal Policy for the Chain of Forests Model when

T<i 279 10.6 The Demand Function 281 10.7 The Profit Function 282 10.8 Optimal Price Trajectory for T > f 285 10.9 Optimal Price Trajectory for T < f 285

11.1 Phase Diagram for the Optimal Growth Model 293 11.2 Optimal Trajectory when XT > x^ 298 11.3 Optimal Trajectory when XT <x^ 298 11.4 Food Output Function 300 11.5 Phase Diagram for the Pollution Control Model 302

12.1 Region D with Boundaries Fi and r2 316 12.2 A Partition of Region D 320 12.3 Solution of Equation 12.52 323 12.4 Boundary of No-Drilling Region 328 12.5 Drilling Time 329 12.6 Value o f - Q + A(t+)[ l -x( t ) ] 330 12.7 Optimal Maintenance Policy 333 12.8 Replacement Time ti and Maintenance Policy 335 12.9 The Case ^1-^2 336

13.1 Autocorrelation Function for a Scalar Process 343 13.2 A Sample Path of Xt with Xo = a;o > 0 and 5 > 0 . . . . 352

B.l Examples of Admissible Functions for the Problem . . . . 380 B.2 Variation about the Solution Function 381 B.3 A Broken Extremal with Corner at r 387

504 List of Figures

C.l Needle-Shaped Variation 394 C.2 Trajectories x*{t) and x{t) in a One-Dimensional Case. . . 394

List of Tables

1.1 The Product ion-Inventory Model of Example 1.1 4 1.2 The Advertising Model of Example 1.2 6 1.3 The Consumption Model of Example 1.3 8

3.1 Summary of the Transversality Conditions 75 3.2 State Trajectories and Switching Curves 78 3.3 Objective, State, and Adjoint Equations for Various

Model Types 85

5.1 Characterization of Optimal Controls 135

A.l Examples of Homogeneous Equations of Order Two . . . 364 A.2 General Solution Forms for Second-Order Linear

Homogeneous Equations, Constant Coefficients 365 A.3 Examples of Homogeneous E]quations of Order n 366 A.4 General Solution Forms for Multiple Roots of Auxiliary

Equation 366 A.5 Particular Solution Forms for Various Forcing Functions . 367 A.6 Particular Integrals in Specific Examples 368 A.7 General Solution Forms for Some Homogeneous Partial

Differential Equations 374