tailieunhanh - Báo cáo hóa học: "Approximate controllability and regularity for nonlinear differential equations"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Approximate controllability and regularity for nonlinear differential equations | Jeong et al. Advances in Difference Equations 2011 2011 27 http content 2011 1 27 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Approximate controllability and regularity for nonlinear differential equations Jin-Mun Jeong 1 Jin-Ran Kim2 and Eun-Young Ju1 Correspondence jmjeong@pknu. department of Applied Mathematics Pukyong National University Busan 608-737 Korea Full list of author information is available at the end of the article Springer Abstract In this article we deal with the existence uniqueness and a variation of solutions of the nonlinear control system with nonlinear monotone hemicontinuous and coercive operator. Moreover the approximate controllability for the given nonlinear control system is studied. Keywords nonlinear differential equation regularity reachable set degree theory approximately controllable 1 Introduction Let H and V be two real separable Hilbert spaces such that V is a dense subspace of H. We are interested in the following nonlinear differential control system on H x t Ax t g t xt Ị k t s xs ds Bu t 0 t x 0 ộũ x s ộ1 s h s 0 where the nonlinear term which is a Lipschitz continuous operator is a semilinear version of the quasi-linear form. The principal operator A is assumed to be a single valued monotone operator which is hemicontinuous and coercive from V to V . Here V stands for the dual space of V. Let U be a Banach space of control variables. The controller B is a linear-bounded operator from a Banach space L2 0 T U to L2 0 T H for any T 0. Let the nonlinear mapping k be Lipschitz continuous from R X - h 0 X V into H. If the right-hand side of the equation SE belongs to L2 0 T V then it is well known as the quasi-autonomous differential equation see Theorem of Chapter III in 1 . The problem of existence for solutions of semilinear evolution equations in Banach spaces has been established by several authors 1-3 . We refer to 2 4 5 to see the existence