Second-Order Analysis and Numerical Approximation for Bang-Bang Bilinear Control Problems

TL;DR

This paper analyzes second-order conditions for bang-bang controls in bilinear optimal control problems, demonstrating local quadratic growth and providing error estimates for finite-element discretizations of such controls.

Contribution

It establishes second-order sufficient conditions for bang-bang controls and analyzes their implications for numerical approximation.

Findings

Second-order conditions guarantee local quadratic growth for bang-bang controls.

No quadratic growth occurs for controls that are not bang-bang.

Finite-element discretization error estimates are derived for bang-bang controls.

Abstract

We consider bilinear optimal control problems, whose objective functionals do not depend on the controls. Hence, bang-bang solutions will appear. We investigate sufficient second-order conditions for bang-bang controls, which guarantee local quadratic growth of the objective functional in $L^1$. In addition, we prove that for controls that are not bang-bang, no such growth can be expected. Finally, we study the finite-element discretization, and prove error estimates of bang-bang controls in $L^1$-norms.