Lectur22 - Lecture XXII Inequality Constraints and...

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Lecture XXII Inequality Constraints and Introduction to Optimal Control I. Inequality Constraints A. The basic setup is out standard dynamic problem max ( , , ’) () , ( ) Ftxx d t st R t x t x t x x t x t t 0 1 00 11 ≤= = Thus, the constraint R(t)<x(t) may or may not be binding at some point for the optimal path. Obviously, if the constraint is not binding, it does not affect the original formulation of the problem. If, on the other hand, the constraint is binding, it affects the optimal path. 1. Graphically, R(t) x*(t) t x(t) a. The optimal path is affected by the inequality constraint at t 1 and follows the inequality constraint through point t 2 at which point it diverges from the constraint. b. The question is then: Where does the optimal path join and leave the constraint? 2. This problem can be formulated similarly to free endpoint problem. b. Mathematically, the optimality conditions for this problem can be discussed within the context of a total variation. To define the total variation of the function, we return to the most basic derivation: m a x ( ,,’ ) Ft d t aa bb ΦΦ + + εδ εδ In this case, the scaling parameter for the variation ε is pulling double duty. It not only controls the perturbation of the optimal path in the Euler equation
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sense, but also controls the perturbation of either endpoint. By Taylor series expansion at the upper bound, we see that: ya a y a a a a y a a () ( ) ( ) ( ) ( ) +≈+ + - =+ εδ εδ εδ Letting y’(a) ε go to y * ’(a): a yy a y a y a yx y a a xy y a a a a a ( ) ( ) * ** * * +=+ ∴= + - =- εδ δ δ δφ δ φδ δ The perturbation along the feasible path then becomes: (29 dI d bF aF F b F a Fd d d dt ba a b ε δδ ∂ε =-+ - - - + - ΦΦ Φ Φ
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This note was uploaded on 11/08/2011 for the course AEB 6533 taught by Professor Moss during the Fall '08 term at University of Florida.

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Lectur22 - Lecture XXII Inequality Constraints and...

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