Analysis and Control of Layered Media

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0205148Hansen The focus of the proposed research is to analyze controllability and stabilization properties of systems of partial differential equations (PDE's) that model layered elastic media. Specific layered elastic systems we are concerned with include constrained-layer damping models, laminated plates, plates with surface-mounted and/or embedded actuators and general multilayer plate theory. For a given layered structure, generally there are many different PDE systems of varying complexity that could be used to model the vibrations. Often the various PDE's are connected through a perturbation, which may be singular. The central concern will be to determine the dependence of stability and controllability upon the perturbation parameters that connect the possible models. This way model dependence, accuracy, and robustness properties of control and feedback strategies can be tested. The plan is to apply this type of analysis to practical control applications such as optimal design for constrained-layer damping and the stabilization of elastic structures using surface-mounted and embedded piezoelectric actuators. Several theoretical challenges present themselves in attempting to develop control methods that are useful for multilayer plates. In particular, it is hoped to develop a theory of Carleman multipliers that is applicable to boundary control of layered systems. The objective is to obtain the sharpest possible results regarding boundary control. Understanding the modeling, control and stabilization of layered elastic systems is becoming increasingly important due to the rapid advances in material sciences, fabrication of composite materials, and speed of processing. The models considered are motivated by practical real-world applications involving composite layered structures. These include the design of ship hulls and airplane fuselage, sound-proofing in auto bodies and the design of sporting goods such as skis and tennis rackets. Often, it is possible to utilize a composite construction to achieve a far superior design compared to what is possible with a single material. Developing a theory for control and feedback of these structures is a challenging problem due the complicated nature of the coupling between the layers. On the other hand, for a model to be useful in practical engineering applications, various crude approximations need to be made to reduce the complexity. Thus, a large part of the goal is to quantify the connection between an accurate, but complex mathematical model and a simplified, but more useful engineering model. Specific problems of practical interest which will be addressed include: feedback control of vibrating structures using strategically placed actuators, optimal design and stability analysis for damping in layered and laminated composites.