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Strong Stationarity Conditions for Optimal Control of Hybrid Systems


A.B. Hempel, P.J. Goulart, J. Lygeros

vol. AUT15-06, available at

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 every continuous piecewise affine system can be written as an optimizing process 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 illustrating the potentially 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.

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% Autogenerated BibTeX entry
@TechReport { HemGou:2015:IFA_5205,
    author={A.B. Hempel and P.J. Goulart and J. Lygeros},
    title={{Strong Stationarity Conditions for Optimal Control of
	  Hybrid Systems}},
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