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Uniform energy decay rates of hyperbolic equations with nonlinear boundary and interior dissipation

100%

Lasiecka I. , Triggiani R.

Control and Cybernetics

2008

tom Vol. 37, no 4

935-969

We consider the problem of uniform stabilization of nonlinear hyperbolic equations, epitomized by the following three canonical dynamics: (1) the wave equation in the natural state space L2(Ω) x H^-1(Ω), under nonlinear (and non-local) boundary dissipation in the Dirichlet B.C., as well as nonlinear internal damping; (2) a corresponding Kirchhoff equation in the natural state space [wzór), under nonlinear boundary dissipation in the 'moment' B.C. as well as nonlinear internal damping; (3) the system of dynamic elasticity corresponding to (1). All three dynamics possess a strong, hard-to-show 'boundary → boundary' regularity property, which was proved, also by invoking a micro-local argument, in Lasiecka and Triggiani (2004, 2008). This is by no means a general property of hyperbolic or hyperbolic-like dynamics (Lasiecka and Triggiani, 2003, 2008). The present paper, as a continuation of Lasiecka and Triggiani (2008), seeks to take advantage of this strong regularity property in the case of those PDE dynamics where it holds true. Thus, under the above boundary → boundary regularity, as well as exact controllability of the corresponding linear model, uniform stabilization of nonlinear models is obtained under minimal nonlinear assumptions, provided that a corresponding unique continuation property holds true. The treatment of the present paper is cast in the abstract setting (Lasiecka, 1989, 2001; Lasiecka and Triggiani, 2000, Ch. 7, 2003, 2008), which is proper for these hyperbolic dynamics and recovers the results of Lasiecka and Triggiani (2003, 2008) in the absence of the nonlinear interior damping, in particular in the linear case.

Inverse/observability estimates for Schroedinger equations with variable coefficients

100%

Triggiani R. , Yao P.

Control and Cybernetics

1999

tom Vol. 28, no 3

627-664

We consider a general Schroedinger equation defined on an open bounded domain [Omega is a 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[sup l](Omega)-energy at time t = T, or at time t = 0, is dominated by the L[sub2](Sigma)-norms of the boundary traces [...] and w[sub t] modulo 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 arbitrarily short observability time ; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditions are imposed on the controlled part of the boundary in the Neuman case. In contrast to existing literature, the first step of our method employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in [Omega is a subset of R^n] to a problem on an appropriate Riemannian 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 va.riable 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.