Advanced applications of micromachined devices and systems call for precise control of their dynamic behavior with respect to both space and time. The design and control of such systems require consideration of distributed-parameter models. Since the physical size of such systems is of the order of 10-3 meter to 10-6 meter, it is necessary to take into account the minute nonlinear forces and moments in modeling. The resulting mathematical models are in the form of nonlinear partial differential and/or integral equations. The objective of this proposed study is to develop applicable theories for the modeling and control of such systems referred hereafter as nonlinear micro-distributed systems. The initial effort will be focused on the modeling of specific micro-distributed systems derived from applications, in particular, systems with free boundaries; bifurcation phenomena and parameter estimation in micromachined elastic structures including deformable mirrors with nonlinear electrostatic or magnetostatic actuators, and elastic structures coupled with liquid films. The subsequent efforts will be focused on the design, stabilization and control of micro-distributed systems. They will include the optimal design of actuator patterns; the determination of attainable shapes in micromachined deformable mirrors using electrostatic actuation; the derivation of stabilizing feedback controls for vibration damping and precise alignment in micromachined elastic structures using electromagnetic, piezoelectric, or thermal actuation; and for laminar flow via boundary surface perturbations. Emphasis will be placed on the development of applicable mathematical theory, efficient computational algorithms, and practical implementation for micro-distributed systems. The proposed study represents a new research direction for distributed-parameter control system theory. Preliminary studies in mathematical modeling already led to new nonlinear partial differential equations which are not only useful for control development, but they are also interesting from the mathematical standpoint. The results have potential applications in micro-sensors, micro-imaging systems, and adaptive optical systems. ***
