Research Initiation: Structural Properties of Nonlinear Systems with Applications to Their Control

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System models are based on physical principles, laws, and simplifying assumptions. These models often possess specific structural characteristics which could be exploited when the system is analyzed. Two of these characteristics are considered in this research. The first concerns the study of systems which possess symmetries. The classical notion of symmetry in Hamiltonian systems is extended to systems with inputs, which are of interest to control engineers. This research will provide further information on the structure of systems with symmetries and will make direct use of this in resolving certain optimization problems. The second topic deals with a geometric basis for the study of discrete time systems and focuses on the study of invariant distributions for such a class of systems. These distributions are closely linked to the classical notion of a subsystem and induce local decompositions of the system. As such, they give a powerful tool for the study of various decoupling problems in nonlinear discrete time systems. Geometric methods are applied consistently to obtain structural information about nonlinear systems, as characterized by the presence of symmetries or invariant distributions. This information is then used to resolve certain control problems in both the continuous and discrete time domains.