Lectur23 - Lecture XXIII Sufficiency, Interpretations, and...

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Lecture XXIII Sufficiency, Interpretations, and Several Variables I. Sufficiency A. In the calculus of variations, the first and second order necessary conditions were also sufficient. That is if any solution obeyed the Euler equation and the Legendre condition, the path was optimum. The Legendre condition basically imposed concavity on x and x’. Similar conditions hold for the optimal control formulation. B. The optimum control formulation: max ( , , ) ’( , , ) , ( ) ftxud t st x g t x u x t x t t 0 1 00 == 1. Pontryagin conditions a. Optimality condition ftx u gtx u ux (, , ) += λ 0 b. Multiplier or costate condition λλ , , ) ( , , ) =- - ft x u u xx c. State equation xg t x u , , ) = d. Transversality conditions λ λ () ’( ) t t 1 0 0 = . 2. Assume x * , u * , and λ satisfy the first order conditions. This also implies that the path is feasible. To establish sufficiency, we must establish: Df f d t t t ≡- * 0 1 0 If f is concave in (x,u), then (29 [] Dx x f u u f d t xu t t ≥- + - ** 0 1 . By the multiplier equation fg * =- - by the optimality condition x g u u g d t uu t t * = ∴≥ - -- + - λ λ 0 1 Focusing on (x * -x)(- λ ’- λ g x )
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(29 --+ - - =- ∫∫ xx x x g xd t x t xt t gd t x t t t t t t t t ** *** ’’ ()() λλ λ λ 0 1 0 1 0 1 0 1 00
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Lectur23 - Lecture XXIII Sufficiency, Interpretations, and...

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