Nonconvex penalization of switching control of partial differential equations

作者：Clason Christian^*; Rund Armin; Kunisch Karl

来源：Systems & Control Letters, 2017, 106: 1-8.

DOI：10.1016/j.sysconle.2017.05.006

• Optimal control
• Switching control
• Partial differential equations
• Nonsmooth optimization
• Nonconvex analysis
• Semi-smooth Newton method

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.

• 出版日期2017-8