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Báo cáo hóa học: " Research Article Interior Controllability of a 2×2 Reaction-Diffusion System with Cross-Diffusion Matrix"

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: Research Article Interior Controllability of a 2×2 Reaction-Diffusion System with Cross-Diffusion Matrix | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 560407 9 pages doi 10.1155 2009 560407 Research Article Interior Controllability of a 2x2 Reaction-Diffusion System with Cross-Diffusion Matrix Hanzel Larez and Hugo Leiva Departamento de Matemdticas Universidad de Los Andes Mérida 5101 Venezuela Correspondence should be addressed to Hugo Leiva hleiva@ula.ve Received 30 December 2008 Accepted 27 May 2009 Recommended by Gary Lieberman We prove the interior approximate controllability for the following 2 x 2 reaction-diffusion system with cross-diffusion matrix ut aAu - -A 1 2u bAv 1Mf1 t x in 0 T x Q vt cAu - dAv - -A 1 2v 1Mf2 t x in 0 t x Q u v 0 on 0 T x dQ u 0 x u0 x v 0 x v0 x x e Q where Q is a bounded domain in RN N 1 u0 v0 e L2 Q the 2 x 2 diffusion matrix D a b has semisimple and positive eigenvalues 0 p1 p2 ft is an arbitrary constant M is an open nonempty subset of Q 1M denotes the characteristic function of the set M and the distributed controls f1 f2 e L2 0 T L2 Q . Specifically we prove the following statement if . 12p-i Ộ 0 where d1 is the first eigenvalue of - A then for all T 0 and all open nonempty subset M of Q the system is approximately controllable on 0 T . Copyright 2009 H. Larez and H. Leiva. 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 In this paper we prove the interior approximate controllability for the following 2 x 2 reactiondiffusion system with cross-diffusion matrix ut aAu - fi - Ỹ 2u bAv 1Mf1 t x in 0 T x Q vt cAu - dAv - -A 1 2v 1Mf2 t x in 0 t x Q u v 0 on 0 T x dQ u 0 x u0 x v 0 x v0 x x e Q 1.1 where Q is a bounded domain in RN N 1 u0 v0 e L2 Q the 2 x 2 diffusion matrix a b c d D 1.2 2 Boundary Value Problems has semisimple and positive eigenvalues p is an arbitrary constant m is an open nonempty subset of Q 1m denotes the