Strong Stationarity Conditions for Optimal Control of Hybrid Systems

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.Comment: 30 pages, 4 figure

On Interferometric Duality in Multibeam Experiments

We critically analyze the problem of formulating duality between fringe visibility and which-way information, in multibeam interference experiments. We show that the traditional notion of visibility is incompatible with any intuitive idea of complementarity, but for the two-beam case. We derive a number of new inequalities, not present in the two-beam case, one of them coinciding with a recently proposed multibeam generalization of the inequality found by Greenberger and YaSin. We show, by an explicit procedure of optimization in a three-beam case, that suggested generalizations of Englert's inequality, do not convey, differently from the two-beam case, the idea of complementarity, according to which an increase of visibility is at the cost of a loss in path information, and viceversa.Comment: 26 pages, 1 figure, substantial changes in the text, new material has been added in Section 3. Version to appear in J.Phys.