Inverse/observability estimates for second-order hyperbolic equations with variable coefficients

	Inverse/observability estimates for second-order hyperbolic equations with variable coefficients
	Lasiecka, I ; Triggiani, R ; Yao, PF
刊名	JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
	1999-07-01
卷号	235 期号:1 页码:13-57
关键词	hyperbolic equation inverse/observability estimates exact controllability Riemannian manifold Carleman estimates
ISSN号	0022-247X
英文摘要	We consider a general second-order hyperbolic equation defined on an open bounded domain Omega subset of R-n with variable coefficients in both the elliptic principal part and in the first-order terms as well. At first, no boundary conditions (B.C.) are imposed. Our main result (Theorem 3.5) is a reconstruction, or inverse, estimate for solutions w: under checkable conditions on the coefficients of the principal part, the H-1(Omega) x L-2(Omega)-energy at time t = T, or at time t = 0, is dominated by the L-2(Sigma)-norms of the boundary traces partial derivative w/partial derivative v(A) and w(t), module an interior lower-order term. Once homogeneous B.C. are imposed, our results yield-under a uniqueness theorem, needed to absorb the lower-order term-continuous observability estimates for both the Dirichlet and Neumann case, with an explicit, sharp observability time; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditions are imposed on the controlled part of the boundary in the Neumann case. In contrast with existing literature, the first step of our method employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in Omega subset of R-n to a problem on an appropriate Riemann manifold (determined by the coefficients of the principal part), where the principal part is the Laplacian. In our second step, we employ explicit Carleman estimates at the differential level to take care of the variable first-order (energy level) terms. In our third step, we employ micro-local analysis yielding a sharp trace estimate, to remove artificial geometrical conditions on the controlled part of the boundary, in the Neumann case. (C) 1999 Academic Press.
WOS研究方向	Mathematics
语种	英语
出版者	ACADEMIC PRESS INC
WOS记录号	WOS:000086570600002
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/14870]
专题	中国科学院数学与系统科学研究院
通讯作者	Lasiecka, I
作者单位	1.Univ Virginia, Dept Math, Charlottesville, VA 22903 USA 2.Acad Sinica, Inst Syst Sci, Beijing 100080, Peoples R China
推荐引用方式 GB/T 7714	Lasiecka, I,Triggiani, R,Yao, PF. Inverse/observability estimates for second-order hyperbolic equations with variable coefficients[J]. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS,1999,235(1):13-57.
APA	Lasiecka, I,Triggiani, R,&Yao, PF.(1999).Inverse/observability estimates for second-order hyperbolic equations with variable coefficients.JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS,235(1),13-57.
MLA	Lasiecka, I,et al."Inverse/observability estimates for second-order hyperbolic equations with variable coefficients".JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 235.1(1999):13-57.