tailieunhanh - Handbook of mathematics for engineers and scienteists part 149

Handbook of mathematics for engineers and scienteists part 149. 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. | 1004 Calculus of Variations and Optimization Let x t e C2 to ti be an extremal of problem with Ao 1 . the Euler equation is satisfied on this extremal for the Lagrangian m L t x x t fo t x x 2 Aifi t x x t with some Lagrange multipliers Ai. Legendre condition If an extremal provides a minimum resp. maximum of the functional then the following inequality holds Lx txt 0 resp. Lxtxt 0 to t ti . Strengthened Legendre condition If an extremal provides a minimum resp. maximum of the functional then the following inequality holds Lx x o resp. Lx tx t 0 to t ti . The equation . .It d I .r . . t . . . xLxx xtLx tx dt .xLx tx xtLx tx t Ligi o i 1 9i -d fi x fi x dt 1 is called the inhomogeneous Jacobi equation for isoperimetric problem on the extremal x t xi are Lagrange multipliers i 1 2 . n . Suppose that the strengthened Legendre condition is satisfied on the extremal x t . A point t is said to be conjugate to the point to if there exists a nontrivial solution of the Jacobi equation such that i gi t h t dt o i 1 2 . m o where h t is an arbitrary smooth function satisfying the conditions h o h T o. We say that the Jacobi condition resp. strengthened Jacobi condition is satisfied on the extremal x t if the interval to t1 resp. the half-interval to t1 does not contain points conjugate to to. A point t is conjugate to to if and only if the matrix H t ho T JJ g1 t ho t dt hm T ftT0 g1 t hm t dt V ftT0 gm t ho t dt ftTg gm t hm t dt is degenerate. . Calculus of Variations and Optimal Control 1005 Necessary conditions for weak minimum maximum Suppose that the Lagrangians fi t x x fi i 0 1 . m in problem are sufficiently smooth. If x t g C2 to t1 provides a weak minimum resp. maximum in problem and the regularity condition is satisfied . the functions gi t are linearly independent on any of the intervals to t and t t1 for any t then x t is an extremal of problem and the Legendre .