Lectur24 - Lecture XXIV Alternative End Point Restrictions...

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Lecture XXIV Alternative End Point Restrictions and Dynamic Programming I. The Salvage Value Problem A. The problem involves a value to be assigned to a state variable at the end of the planning horizon. B. Mathematically, max ( , , ) ( ( )) ’( ) ( , , ) () ,, f txud t xt st x t g t x u x t t t t 0 1 1 0 012 1 + = = φ fixed free 1. The standard control formulation becomes [] Jf g x d t t x t x t t t t t =+ + - + λλ λ φ ’( ) ( ) ( ( ) ) 0 1 0 1 1 2. Hypothesizing alternative paths ut u t aht yta gtxud t Ja f tyta u t tgtyta u t tyta d t ty ta y t t t t (, ) (, , ) ( ) (, (, ) , () ) () (, (, ) ) () (, ) ()(,) ()(,) ((,) ) * ** = + + + - ++ 0 1 0 1 11 00 1 λλφ taking the derivative with respect to a and letting a go to zero yields: (29 ( 29 g y t a f g h t d t tyta y tayta xx a uu t t aa ( , ) () (,) ’ ((,) ) (,) 0 0 1 1 1 + + + - + λ λφ which yields the traditional optimality conditions plus -+ = ⇒= () ’ (() ) tx t y t a t a 1 1 1 0 II. Fixed Endpoint Problems A. Suppose both the initial and terminal points are fixed (as in the first formulation of the calculus of variations problem): max ( , , ) ( , , ) , ftxud t st x t g t x u x x tt t t 0 1 01 = = = free
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1. Let u * be the optimal control function and x * be the corresponding state function. Let u denote another feasible function satisfying the equation of motion and terminal conditions: [] J J J ftxu g txu x ftxu g xd t t t -= = + + - - - ** * * * * (,,) (, , ) ∆λ λ λ λ 0 1 using a Taylor series expansion (29 J f g x x f g u u d t hot xx uu t t =+ + - + + -+ λλ λ ’. . . 0 1 Labeling δ x=x-x * and δ u=u-u * as perturbations: ( 29 δλ λ δ λ δ Jf g x f g u d t t t + + + 0 1 2. Similar to before, the optimal path must satisfy fg ++ = ’0 for any δ x. The difference is that δ u must now be chosen to satisfy the terminal conditions at both sides. Kamien and Schwartz show in an appendix that if there is a feasible control u that forces the state from x(t 0 )=x 0 to x(t 1 )=x 1 , then the coefficient on δ u must be zero += λ 0 when λ obeys λ ’=-[f x + λ g x ]. 3. Our inventory problem then becomes: min ( ) ( ) ’( ) ( ) () cut cxt d t st x t u t x xT B ut T 1 2 2 0 00 0 + = = = The Hamiltonian then becomes: Htxu cxt tut (, , , ) () () + 1 2 2 Deriving the costate condition: - = ⇒= - + H x ctt c t d 22 1 Using this result in the optimality condition: λ H u t ct d c c t d c = ⇒- + = =- 20 2 0 11 2 1 2 1 1 1
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Lectur24 - Lecture XXIV Alternative End Point Restrictions...

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