9803181 Bloch In this project the proposer will continue research into the dynamics and control of nonlinear mechanical systems in finite- and infinite-dimensions. In particular, research is proposed in the following areas: integrable Hamiltonian systems in finite- and infinite-dimensions, the stabilization and control of nonlinear mechanical systems, the analysis and control of nonholonomic systems with constraints, the optimal control of systems with constraints, and the dynamics and control of infinite-dimensional systems and systems with complex dynamics. The proposer will analyze two basic classes of integrable systems -- the Toda lattice in finite- and infinite-dimensions, and the generalized rigid body equations. The proposer's view of these systems is inspired by work in applied analysis -- optimization and control problems. The proposer will also investigate the dynamics of nonholonomic systems. Such systems, which are mechanical systems subject to nonintegrable constraints such as rolling, are not Hamiltonian in general but nonetheless conserve energy. This leads to very rich behavior including the possibility of asymptotic stability. With his collaborators the proposer has been developing an energetic approach to the control of nonlinear mechanical systems which leaves the system in classical Lagrangian form despite the presence of feedback. This enables the classical theory of stability, which is normally applied to autonomous systems, to be applied to classes of controlled systems. Several other problem in the control and dynamics of finite- and infinite-dimensional systems are discussed, such as the analysis of the structure and stability of interconnected systems including linked finite- and infinite-dimensional systems. In broad terms the proposer will study the behavior and control of various mechanical systems that are important in physics and engineering. Engineering applications include aerospace systems and robotics. He will study certain model systems that can be solved explicitly. Such systems are important for the light they shed on the behavior of more complicated systems which can only be simulated on a computer. He will also consider the optimization of the behavior of various mechanical control systems. This optimization is important not only for obtaining desirable system behavior but for reducing cost. He will study various methods for controlling the motion of such systems as robots and satellites. In the modeling of satellites he will consider the importance of flexible appendages such as antennae in the stability of their motion. He will also analyze the motion of systems in fluids -- for example the motion of underwater vehicles. In general the hope is to obtain a better understanding of the dynamics and control of a large class of engineering and physical systems.
