On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials

doi:10.1016/j.anihpc.2006.07.005

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	On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials
	Duyckaerts, Thomas 1; Zhang, Xu 2,3; Zuazua, Enrique 3
	2008
发表期刊	ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
ISSN	0294-1449
卷号	25 期号:1 页码:1-41
摘要	In this paper we prove the optimality of the observability inequality for parabolic systems with potentials in even space dimensions n >= 2. This inequality (derived by E. FernAndez-Cara and the third author in the context of the scalar heat equation with potentials in any space dimension) asserts, roughly, that for small time, the total energy of solutions can be estimated from above in terms of the energy localized in a subdomain with an observability constant of the order of exp(C parallel to a parallel to(2/3)(infinity)), a being the potential involved in the system. The problem of the optimality of the observability inequality remains open for scalar equations. The optimality is a consequence of a construction due to V.Z. Meshkov of a complex-valued bounded potential q = q(x) in R-2 and a nontrivial solution u of Delta u = q(x)u. with the decay property vertical bar u(x)vertical bar < exp(-vertical bar x vertical bar(4/3)). Meshkov's construction may be generalized to any even dimension. We give an extension to odd dimensions, which gives a sharp decay rate up to some logarithmic factor and yields a weaker optimality result in odd space-dimensions. We address the same problem for the wave equation. In this case it is well known that, in space-dimension n = 1, observability holds with a sharp constant of the order of exp(Cllallo,). For systems in even space dimensions n >= 2 we prove that the best constant one can expect is of the order of exp(C parallel to a parallel to(2/3)(infinity)) for any T > 0 and any observation domain. Based on Carleman inequalities, we show that the positive counterpart is also true when T is large enough and the observation is made in a neighborhood of the boundary. As in the context of the heat equation, the optimality of this estimate is open for scalar equations. We address similar questions, for both equations, with potentials involving the first order term. We also discuss issues related with the impact of the growth rates of the nonlinearities on the controllability of semilinear equations. Some other open problems are raised. (c) 2006 Elsevier Masson SAS. All rights reserved.
关键词	optimality Meshkov's construction observability inequality heat equation wave equation potential Carlenman inequality decay at infinity
DOI	10.1016/j.anihpc.2006.07.005
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000252802500001
出版者	GAUTHIER-VILLARS/EDITIONS ELSEVIER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/5970
专题	中国科学院数学与系统科学研究院
通讯作者	Duyckaerts, Thomas
作者单位	1.Univ Cergy Pontoise, Dept Math, F-95302 Cergy Pontoise, France 2.Acad Sinica, Acad Math & Syst Sci, Key Lab Syst & Control, Beijing 100080, Peoples R China 3.Univ Autonoma Madrid, Fac Ciencias, Dept Matemat, E-28049 Madrid, Spain
推荐引用方式 GB/T 7714	Duyckaerts, Thomas,Zhang, Xu,Zuazua, Enrique. On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials[J]. ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE,2008,25(1):1-41.
APA	Duyckaerts, Thomas,Zhang, Xu,&Zuazua, Enrique.(2008).On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials.ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE,25(1),1-41.
MLA	Duyckaerts, Thomas,et al."On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials".ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE 25.1(2008):1-41.