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Optimal regularity and optimal control of a thermoelastic structural acoustic model with point control and clamped boundary conditions

Lebiedzik C., Triggiani R.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1461-1499

EN

In this paper we consider point control of a structural acoustic model with thermoelastic effects. The key feature of this paper is that the two-dimensional plate modeling the active wall of the acoustic chamber has clamped boundary conditions. For this case a new optimal regularity result has recently become available (Triggiani, 2008). Using this new result for the plate alone, we derive a sharp (optimal) regularity result for the overall coupled system of wave and thermoelastic plate equations, after overcoming a series of additional technical difficulties. This allows for the study of an optimal control problem of the coupled system.

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

Lasiecka I., Triggiani R.

Control and Cybernetics

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2008

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Vol. 37, no 4

935-969

EN

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.

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Sharp Regularity of the Second Time Derivative w_t T of Solutions to Kirchhoff Equations With Clamped Boundary Conditions

Lasiecka I., Triggiani R.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 4

753-772

EN

We consider mixed problems for Kirchhoff elastic and thermoelastic systems, subject to boundary control in the clamped Boundary Conditions B.C. ("clamped control"). If w denotes elastic displacement and theta temperature, we establish optimal regularity of {w,w_t,w_tt} in the elastic case, and of {w,w_t,w_tt,theta} in the thermoelastic case. Our results complement those presented in (Lagnese and Lions, 1988), where sharp (optimal) trace regularity results are obtained for the corresponding boundary homogeneous cases. The passage from the boundary homogeneous cases to the corresponding mixed problems involves a duality argument. However, in the present case of clamped B.C., and only in this case, the duality argument in question is both delicate and technical. In this respect, the clamped B.C. are 'exceptional' within the set of canonical B.C. (hinged, clamped, free B.C.). Indeed, it produces new phenomena which are accounted for by introducing new, untraditional factor (quotient) spaces. These are critical in describing both interior regularity and exact controllability of mixed elastic and thermoelastic Kirchhoff problems with clamped controls.

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Inverse/observability estimates for Schroedinger equations with variable coefficients

Triggiani R., Yao P.

Control and Cybernetics

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1999

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Vol. 28, no 3

627-664

EN

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.