Geometric approach to feedback control of differential affine variational inequalities

The research described in this proposal deals with feedback control of differential variational inequalities. These are nonlinear input/output dynamical systems that are subject to inequality constraints. The distinguishing feature of the involved inequality constraints is that they are induced by the so-called variational inequalities. Since variational inequalities have the capability of capturing many types of inequality constraints that appear in various contexts, differential variational inequalities are encountered in many real world applications from engineering as well as operations research and biology. Given a differential variational inequality that is subject to external disturbances, the problem of designing feedback controllers for disturbance decoupling, output regulation, and tracking will be investigated. Existing methods for smooth nonlinear systems are not applicable in the context of differential variational inequalities since these systems do not satisfy the required smoothness properties. Therefore, for feedback control of differential variational inequalities an entirely new approach needs to be developed. Within the proposal, we concentrate on differential affine variational inequalities for which both the systems differential equations and the variational inequalities are given by affine functions. As shown in recent work, differential affine variational inequalities enjoy certain strong geometric properties. This proposal aims at exploiting these geometric properties in order to develop a feedback control theory. This will not only provide a natural extension to the existing literature on feedback control theory for smooth nonlinear systems but will also be relevant in the context of piecewise affine dynamical systems.