Invariant Measures, Geometry, and Control of Hybrid and Nonholonomic Dynamical Systems

Abstract

Constraints are ubiquitous when studying mechanical systems and fall into two main categories: hybrid (1-sided, unilateral) and nonholonomic/holonomic (2-sided, bilateral) constraints. A hybrid constraint takes the form h(x)≥0. An example of a constraint of this nature is requiring a billiard ball to remain within the confines of a table-top. The notable feature of these constraints is that when the ball reaches the boundary of the table-top (i.e. when h(x)=0), an impact occurs; this is a discontinuous jump in the dynamics. Dynamical systems that have this phenomenon generally fall under the domain of hybrid dynamical systems. On the other hand, nonholonomic constraints take the form h(x)=0. Generally, h will depend on both the positions and velocities and cannot be integrated to only depend on the positions (when it can be integrated, the constraint is called holonomic). An example of a nonholonomic constraint is an ice skate: motion is not allowed perpendicular to the direction of the skate. It is common that these systems are studied using tools from differential geometry.

This thesis studies both hybrid and nonholonomic constraints together using the language of differential (specifically symplectic) geometry. However, due to the exotic nature of hybrid dynamics, some auxiliary results are found that pertain to the asymptotic nature of these systems. These include the idea of a hybrid limit-set, Floquet theory, and a Poincaré-Bendixson theorem for planar systems.

The bulk of this work focuses on finding (smooth) invariant measures for both nonholonomic and hybrid systems (as well as systems involving both types of constraints). Necessary and sufficient conditions are found which guarantee the existence of an invariant measure for nonholonomic systems in which the density depends only on the configuration variables. Extending this idea to hybrid nonholonomic systems requires that the impact preserves the measure as well. To build towards this, relatively simple conditions to test whether or not a differential form is hybrid-invariant are derived. In the cases where the density depends on only the configuration variables, the measure is still invariant under the hybrid dynamics independent of the choice of impacts. The billiard problem with a vertical rolling disk as the billiard ball is one such system and is therefore recurrent for any choice of compact table-top.

This thesis concludes with optimal control of hybrid systems. First, Hamilton-Jacobi is extended to the hybrid setting (nonholonomic constraints are not considered here) and the idea of completely integrable hybrid systems is introduced. It is shown that the usual billiard problem on a circular table is completely integrable. Finally, the hybrid Hamilton-Jacobi theory is extended to a hybrid Hamilton-Jacobi-Bellman theory which allows for the study of optimal control problems.