2.3 Calculus of Variations Fixed Ends With Constraints

8/11/2019 2.3 Calculus of Variations Fixed Ends With Constraints

1/26

FIXED-END TIME AND FIXED-ENDPROBLEM: WITH CONSTRAINTS

Optimal Control: Calculus of Variations 37


2/26


3/26

Approach 1: Direct methodAs a necessary condition for extrema, we have

Without constraint case: optimum is where

.For with constraint case,

and

are not arbitrary, but

related as:

Arbitrarily choose one of two variables, say , as anindependent variable. Then becomes dependent variable.Then provided .



4/26

Approach 1: Direct methodTherefore,

As is arbitrary and can not be zero, .Or,

Alternatively,



5/26

Approach 1: Direct methodThe following two equations solved simultaneously to find a

solution of the problem:

Some Facts:

First equation is also know as Jacobian of and . Tedious to solve for higher order problems.



6/26

Approach 2: Lagrange Multiplier methodAn augmented Lagrangian function is formed: Where, is Lagrange multiplier, a parameter to be determined.With , .Therefore, necessary condition for extrema is that

Since and are functions of both and :



7/26

Approach 2: Lagrange Multiplier method

Since both and are not independent, let isindependent. Then is dependent differential.Further, let is so chosen that one of the coefficient becomeszero. Let the coefficient of

is made zero by choosing a value

of as , that is Therefore, .Since is independent variable:



8/26

Approach 2: Lagrange Multiplier method

Further,

Combining the three equations:

To be solved simultaneously

By eliminating from first two equations: The same can be extended for multiple constraints, i.e.

,

,



9/26

Extrema of Functionals with Constraints

subject to the constraint (plant or system equations)

with fixed end-point conditions:



10/26

Six Steps for Solution

Step 1: Lagrangian

Step 2: Variations and Increment

Step 3: First Variation

Step 4: Fundamental Theorem

Step 5: Fundamental Lemma

Step 6: Euler-Lagrange Equation



11/26

Step 1 : Lagrangian

where

is the Lagrange multiplier and



12/26

Step 2 : Variations and Increment



13/26


14/26

Step 3 : First Variations (contd)



15/26

Step 4 : Fundamental Theorem

vanishes, i.e.



16/26

Step 5 : Fundamental Lemma

Then the first variation

becomes zero.



17/26


18/26

ExampleMinimize the performance index

with boundary conditions and and subject tothe condition (plant equation)



19/26

Solution by Direct method

; ; ; Replacing

from the condition in

, one gets

Now, one can minimize in straight forward way.



20/26


; ; ; Invoke Euler-Lagrange equation:



21/26


; ; ; Solution of Euler-Lagrange equation:

Simplifying,

Solution of the above is:

The constants and can be determined using boundaryconditions as:

and



22/26


23/26

Solution by Lagrange Multiplier Method

; ; ; The condition can be written as:

Now, we form an augmented functional:



24/26


; ; ;

Invoke the optimality conditions:



25/26


; ; ;

Invoke the optimality conditions:



26/26


; ; ;

;

;

From last two equations:

Replacing above in the first equation:

Then the same solution prevails as the direct method.


2.3 Calculus of Variations Fixed Ends With Constraints

Documents

Transcript of 2.3 Calculus of Variations Fixed Ends With Constraints