A convex analysis approach to optimal controls with switching structure for partial differential equations

Abstract

Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau-Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach.

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• Nonconvex penalization of switching control of partial differential equations
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• Optimal control problems with control complementarity constraints
• Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations
• Convex regularization of discrete-valued inverse problems
• Convex relaxation of discrete vector-valued optimization problems
• L1 penalization of volumetric dose objectives in optimal control of PDEs
• Error estimates for the approximation of multibang control problems