Nonconvex penalization of switching control of partial differential equations

Christian Clason, Armin Rund, Karl Kunisch

Published 2016-05-31Version 1

A standard approach to treat constraints in nonlinear optimization is penalization, in particular using $L^1$-type norms. Applying this approach to the pointwise switching constraint $u_1(t)u_2(t)=0$ 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.