2.2 Calculus of Variations Fixed Ends

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FIXED-END TIME AND FIXED-ENDPROBLEM: WITHOUT CONSTRAINTS

Optimal Control: Calculus of Variations 12


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Fixed-End Time and Fixed-End State Systems

Initial time

and state

are fixed Final time and state are fixed The problem is to find theoptimal function for

which the below functional is

optimum.



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Solution Steps

Step 1: Assumption of an Optimum

Step 2: Variations and Increment Step 3: First Variation

Step 4: Fundamental Theorem

Step 5: Fundamental Lemma Step 6: Euler-Lagrange Equation



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Step 1: Assumption of an Optimum

Let us assume that

is the

optimum attained for the function . Take some admissiblefunction close to

, where

is the

variation of . The function should also satisfy theboundary conditions, i.e.

.



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Step 2: Variations and Increment



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Step 2: Variations and Increment

Using Taylors series expansion ( gets cancelled):



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Step 3: First variation

Integration by parts:



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Step 3: First variation

Recollect



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Step 4: Fundamental Theorem,



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Step 6: Euler-Lagrange Equation

Euler-Lagrange Equation:

This is, in general, a nonlinear, time-varying, two-point

boundary value, second order, ordinary differential equation.



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Different cases of Euler-Lagrange Equation

E-L equation:

Case 1:is independent of but dependent on otherterms. Then

. Therefore, , which leads to

=Constant.Case 2: is independent of, but dependent on otherterms. Then

. Therefore,

.



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Example 1

Find the minimum length between any two points.



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Example 1 (contd.)

A small arc length is related by:

Or Total arc length:

E-L equation:



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Example 1 (contd.)

E-L equation:

Therefore, and may be solved from boundary conditions.



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Example 2Find the optimum of

with boundary conditions

E-L equation:

Using boundary conditions:



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The Second Variation

Approach 1:

Consider the last term:

Integration by parts:

and



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The Second Variation (contd

) Approach 1:

Consider the last term: Integration by parts: and



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) Approach 1:

From boundary conditions: =0.



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) Approach 1:

=



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) Approach 1:

=



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) Approach 1:

From Fundamental Theorem:

For maximum: and For minimum:

and



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) Approach 2Approach 2:



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) Approach 2Approach 2: ,

where

From Fundamental Theorem:

For maximum: For minimum:



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2.2 Calculus of Variations Fixed Ends