What is Optimal Control Theory? Dynamic Systems: Evolving over time. Time: Discrete or continuous.

Optimal way to control a dynamic system.

Prerequisites: Calculus, Vectors and Matrices, ODE&PDE

Applications: Production, Finance/Economics, Marketing and others.

Basic Concepts and Definitions A dynamic system is described by state equation:

where x(t) is state variable, u(t) is control variable. The control aim is to maximize the objective function:

Usually the control variable u(t) will be constrained asfollows:

Sometimes, we consider the following constraints:

(1) Inequality constraint

(2) Constraints involving only state variables

(3) Terminal state

where X(T) is reachable set of the state variables at time T.

Formulations of Simple Control Models Example 1.1 A Production-Inventory Model.

We consider the production and inventory storage of a given good in order to meet an exogenous demand at minimum cost.

Table 1.1 The Production-Inventory Model of Example 1.1

Example 1.2 An Advertising Model. We consider a special case of the Nerlove-Arrow

advertising model.

Table 1.2 The Advertising Model of Example 1.2

Example 1.3 A Consumption Model. This model is summarized in Table 1.3:

Table 1.3 The Consumption Model of Example 1.3

History of Optimal Control Theory Calculus of Variations. Brachistochrone problem: path of least time Newton, Leibniz, Bernoulli brothers, Jacobi, Bolza. Pontryagin et al.(1958): Maximum Principle.

Figure 1.1 The Brachistochrone problem

Notation and Concepts Used = “is equal to” or “is defined to be equal to” or “is identically equal to.” := “is defined to be equal to.” “is identically equal to.” “is approximately equal to.” “implies.” “is a member of.”

Let y be an n-component column vector and z be an m-component row vector, i.e.,

when n = m, we can define the inner product

If

is an m x k matrix and B={bij} is a k x n matrix, C={cij}=AB,which is an m x n matrix with components:

Differentiating Vectors and Matrices with respect to Scalars

Let f : E1Ek be a k-dimensional function of a scalar variable t. If f is a row vector, then

If f is a column vector, then

Differentiating Scalars with respect to Vectors If F: En x Em E1, n 2, m 2, then the gradients Fy

and Fz are defined, respectively as;

Differentiating Vectors with respect to Vectors If F: En x Em Ek, is a k – dimensional vector function, f

either row or column, k 2; i,e;

where fi = fi (y,z), y En is column vector and z Em is row vector, n 2, m 2, then fz will denote the k x m matrix;

fy will denote the k x n matrix

Matrices fz and fy are known as Jacobian matrices.

Applying the rule (1.11) to Fy in (1.9) and the rule (1.12) to Fz in (1.10), respectively, we obtain Fyz=(Fy)z to be the n x m matrix

and Fzy = (Fz)y to be the m x n matrix

Product Rule for Differentiation

Let x En be a column vector and g(x) En be a row vector and f(x) En be a column vector, then

Vector Norm The norm of an m-component row or column vector z is

defined to be

Neighborhood Nzo of a point is

where > 0 is a small positive real number. A function F(z): Em E1 is said to be of the order o(z), if

The norm of an m-dimensional row or column vector function z(t), t [0, T], is defined to be

Some Special Notation left and right limits

xk: state variable at time k.

uk: control variable at time k.

k : adjoint variable at time k, k=0,1,2,…,T.

: difference operator. xk:= xk+1- xk.

xk*, uk*, and k, are quantities along an optimal path.

discrete time (employed in Chapters 8-9)

sat function

bang function

impulse control

If the impulse is applied at time t, then we calculate the objective function J as

Convex Set and Convex Hull A set D En is a convex set if y, z D,

py +(1-p)z D, for each p [0,1].

Given xi En, i=1,2,…,l, we define y En to be a convex combination of xi En, if pi 0 such that

The convex hull of a set D En is

Concave and Convex Function : D E1 defined on a convex set D En is concave

if for each pair y,z D and for all p [0,1],

If is changed to > for all y,z D with y z, and 0<p<1, then is called a strictly concave function.

If (x) is a differentiable function on the interval [a,b], then it is concave, if for each pair y,z [a,b],

(z) (y)+ x (y)(z-y).

If the function is twice differentiable, then it is concave if at each point in [a,b],

x x 0 . In case x is a vector, x x needs to be a negative definite

matrix.

If : D E1 defined on a convex set D En is a concave function, then - : D E1 is a convex function.

Figure 1.2 A Concave Function

Affine Function and Homogeneous Function of Degree One : En E1 is said to be affine, if (x) - (0) is linear. : En E1 is said to be homogeneous of degree one, if

(bx) = b (x), where b is a scalar constant.

Saddle Point : En x Em E1, a point En x Em is called a saddle

point of (x,y), if

Note also that

Figure 1.3 An Illustration of a Saddle Point

Linear Independence and Rank of a Matrix A set of vectors a1,a2,…,an En is said to be linearly

dependent if pi,, not all zero, such that

If the only set of pi for which (1.25) holds is p1= p2= ….= pn= 0, then the vectors are said to be linearly independent. The rank of an m x n matrix A is the maximum number of

linearly independent columns in A, written as rank (A). An m x n matrix is of full rank if rank (A) = n.