Strong Stationarity Conditions for Optimal Control of Hybrid Systems

TL;DR

This paper derives necessary and sufficient optimality conditions for hybrid systems with complementarity constraints, enabling more effective control solutions by leveraging classical nonlinear programming conditions.

Contribution

It introduces structural assumptions that ensure classical KKT conditions are both necessary and sufficient for local optimality in hybrid systems with complementarity constraints.

Findings

KKT conditions are necessary and sufficient under certain assumptions.

Optimal control of piecewise affine systems can be formulated as a linear complementarity problem.

Nonlinear programming approaches outperform mixed-integer formulations in simulations.

Abstract

We present necessary and sufficient optimality conditions for finite time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush-Kuhn-Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine system can be written as the optimizer of a mathematical program which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with piecewise affine dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation.