Control Problems of Systems of Strongly Coupled Partial Differential Equations

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9804056LasieckaThe present proposal aims at investigating mathematical aspects, as well as control and optimization properties, of models of interactive systems, which are described by strongly coupled partial differential equations, and which are subject to boundary and point control action. These are both the mathematically most challenging and the physically most relevant cases. The systems chosen are complex and realistic enough to serve as benchmark problems for the technological applications described below. Specific issues to be examined for the entire structure, and/or a component thereof, include: (1) well-posedness and regularity of solutions; (2) exact controllability in the oscillatory case for systems such as shells; (3) uniform stability of the structure solutions as well as, in the non-linear case, a corresponding study of the asymptotic behavior; (4) optimal control theory and min-max game theory with non-definite quadratic cost, as well as robust stabilization. While in recent years these problems have reached a considerable level of maturity for single classes of dynamics - diffusive and oscillatory - their study to strongly coupled systems with components of both diffusive and/or oscillatory type is just at its infancy.Control, optimization and stability problems for dynamical systems have long played a well-recognized and critical role on a broad range of diverse applications. In recent years, the emerging technology of smart materials and structures has brought forth to the front of scientific investigation the pressing need to control, optimize and stabilize "interactive structures", whose components' dynamical behavior is governed by partial differential equations. Recent civilian and military research has convincingly demonstrated smart materials/structures to be a laboratory reality. These new structural concepts actively damp noise and vibration, suppress flutter at the trailing edges of airfoils, suppress the airborne or water borne acoustic signatures of ground vehicles, aircraft, and submarines, and reduce the stressful noise levels within them. A realistic problem of keen, present interest to both the civilian and military industry is the problem of reducing, or eliminating, the unwanted noise field in a three dimensional cabin with curved walls, modeled by shells, by means of smart materials such as piezo-electric patches or memory alloys bonded to, or embedded in, an elastic wall. When wired with an appropriate voltage (control), the resulting bending moment generated by the elastic wall is expected to produce acoustic waves that counter the unwanted noise, thus leading to their elimination.