Nonconvex penalization of switching control of partial differential equations

Abstract

This paper is concerned with optimal control problems for parabolic partial differential equations with pointwise in time switching constraints on the control. A standard approach to treat constraints in nonlinear optimization is penalization, in particular using $L^1$-type norms. Applying this approach to the switching constraint leads to a nonsmooth and nonconvex infinite-dimensional minimization problem which is challenging both analytically and numerically. Adding $H^1$ regularization or restricting to a finite-dimensional control space allows showing existence of optimal controls. First-order necessary optimality conditions are then derived using tools of nonsmooth analysis. Their solution can be computed using a combination of Moreau-Yosida regularization and a semismooth Newton method. Numerical examples illustrate the properties of this approach.

References in corpus (3)

• A convex penalty for switching control of partial differential equations
• A convex analysis approach to optimal controls with switching structure for partial differential equations
• Optimal control of elliptic equations with positive measures

Cited by in corpus (4)

• Optimal control problems with control complementarity constraints
• Sparse and Switching Infinite Horizon Optimal Control with Mixed-Norm Penalizations
• Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations
• Relaxation schemes for mathematical programs with switching constraints