Tham khảo tài liệu 'báo cáo hóa học: " research article solvability for a class of abstract two-point boundary value problems derived from optimal control"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 27621 17 pages doi 2007 27621 Research Article Solvability for a Class of Abstract Two-Point Boundary Value Problems Derived from Optimal Control Lianwen Wang Received 21 February 2007 Accepted 22 October 2007 Recommended by Pavel Drabek The solvability for a class of abstract two-point boundary value problems derived from optimal control is discussed. By homotopy technique existence and uniqueness results are established under some monotonic conditions. Several examples are given to illustrate the application of the obtained results. Copyright 2007 Lianwen Wang. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction This paper deals with the solvability of the following abstract two-point boundary value problem BVP x t A t x t F x t p t t x a x0 p t -A t p t G x t p t t p b Ĩ x b . Here both x t and p t take values in a Hilbert space X for t e a b F G X X X X a b X and X - X are nonlinear operators. A t a t b is a family of linear closed operators with adjoint operators A t and generates a unique linear evolution system U t s a s t b satisfying the following properties. a For any a s t b U t s e ẩ X the Banach space of all bounded linear operators in X with uniform operator norm also the mapping t s U t s x is continuous for any x e X b U t s U s t U t t for a T s t b c U t t I for a t b. 2 Boundary Value Problems Equation is motivated from optimal control theory it is well known that a Hamiltonian system in the form dH x p t x t ----- . ---- x a Xo 2 p t W y p x b ax is obtained when the Pontryagin maximum principle is used to get optimal state feedback control. Here H x p t is a Hamiltonian function. Clearly the solvability of system is crucial for the discussion of optimal control. System
