tailieunhanh - Handbook of mathematics for engineers and scienteists part 150

Handbook of mathematics for engineers and scienteists part 150. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | . Calculus of Variations and Optimal Control 1011 . Necessary conditions for extremum. Pontryagin maximum principle. Suppose that d x u t0 11 is an optimal in the strong sense process in the optimal control problem and moreover the functions p 1 . pn and fi i 0 1 . m and their partial derivatives are continuous in x in a neighborhood of the Cartesian product of t x t g r by U and the functions fi i 0 1 . m are continuously differentiable in a neighborhood of the point t0 x t0 t1 iX t1 the smoothness condition . Then there exist Lagrange multipliers Xi i 0 . m and pk Pk t g PC 1 T k 1 . n which are not zero simultaneously such that the Lagrange function z. 1 m n m A Xifi t x u Pi xi t -pi t x u dt Xi i t0 x t0 t1 x t1 satisfies the following conditions 1. The conditions of stationarity with respect to x . the Euler equations nm dpi d k X dfk I O II p JPk .Xi. i 1 2 . n for all t T dt dxi dxi k 1 i k 0 i where all derivatives with respect to xk are evaluated at t x Û . 2. The conditions of transversality with respect to x Pi tk -1 k Xk Z t. k 0 1 i 1 2 . n where all derivatives with respect to xi tk k 0 1 are evaluated at t0 x t0 t1 ix 1 . 3. The condition of optimality with respect to u max H t x u p H t x Û p Û U nm where H t x u p 22 pipi t x u - 22 Xifi t x u is the Pontryagin function. 4. The conditions of stationarity with respect to tk only for movable endpoints of the integration interval Atk tk 0 k 0 1 . 5. The complementary slackness conditions AiBi cb 0 i 1 2 . m . 6. The nonnegativity conditions Xi 0 i 1 2 . m . The Pontryagin maximum principle permits one to obtain all necessary conditions in calculus of variations the Euler equations the Legendre condition the Weierstrass condition and the Lagrange multiplier rule. 1012 Calculus of Variations and Optimization . Mathematical Programming . Linear Programming . Statement of linear programming problem. The general mathematical programming problem is the problem of .