2.2 Calculus of Variations Fixed Ends
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Transcript of 2.2 Calculus of Variations Fixed Ends
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FIXED-END TIME AND FIXED-ENDPROBLEM: WITHOUT CONSTRAINTS
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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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The Second Variation (contd
) Approach 1:
From boundary conditions: =0.
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The Second Variation (contd
) Approach 1:
=
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The Second Variation (contd
) Approach 1:
=
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The Second Variation (contd
) Approach 1:
From Fundamental Theorem:
For maximum: and For minimum:
and
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The Second Variation (contd
) Approach 2Approach 2:
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The Second Variation (contd
) Approach 2Approach 2: ,
where
From Fundamental Theorem:
For maximum: For minimum:
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