Boundary and localized null controllability and corresponding minimal norm control blow up rates of thermoelastic and structurally damped systems.

摘要

In this dissertation we consider the problem of null controllability for elastic operators under square root damping. These partial differential equations are described by analytic semigroups on the basic space of finite energy. Because of the underlying parabolicity for these systems, the null controllability problem is appropriate for consideration. Initially we will consider linear, homogeneous structurally damped and thermoelastic systems influenced by source controls of localized support. We will show that, in this setting, the state variables can be steered to the zero state by iterations of controllers acting on appropriate finite dimensional systems. In this work, key usage is made of the diagonalization of the spatial operators which is available in the case of hinged boundary conditions. Moreover, the control strategy in [A. Benabdallah, M. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal. 7 (2002) 585-599] is critically adapted to our present needs. In particular, this strategy hinges upon the availability of a Carleman's estimate for linear combinations of eigenfunctions of the Dirichlet Laplacian.; In the process of constructing a localized control for homogeneous systems, the dependence on the terminal time T > 0 is analyzed to provide a bound for the minimal norm controller. Due to the infinite speed of propagation inherent to these systems, we provide an exponential bound for the asymptotics of the minimal energy function Emin (T) as T &drarr; 0. Though the bound given here is not optimal, but "unsharp by epsilon, the estimates given here are of interest and will be used throughout the paper.; Once localized null controllability has been established for the homogeneous version of these systems, we show localized null controllability for the nonhomogeneous system. Further, we provide estimates for the minimal energy function corresponding to the new system. By an embedding technique, we then extend these results to the corresponding boundary controllability problem for these systems.; Finally, having established null controllability for the nonhomogeneous problem, we then consider the problem of localized null controllability for a structurally damped elastic system with non-Lipschitz, but monotone, nonlinearity in place. In order to obtain this result, we will first show uniform stability for the solution in the absence of controls. After showing local controllability for the nonlinear system by localized controls, we combine the result of uniform stability to establish global localized null controllability for the nonlinear system. The goal for this problem is to provide results for the existence of exact null controls for a nonlinear system.