Boundary bilinear control of semilinear parabolic PDEs: quadratic convergence of the SQP method

TL;DR

This paper studies a boundary control problem for semilinear parabolic PDEs, deriving optimality conditions and proving quadratic convergence of an SQP algorithm under certain conditions.

Contribution

It introduces a sequential quadratic programming method for boundary control of PDEs and proves its quadratic convergence under specific optimality and regularity assumptions.

Findings

Derived first-order necessary optimality conditions.

Established second-order sufficient optimality conditions.

Proved quadratic convergence of the SQP algorithm.

Abstract

We analyze a bilinear control problem governed by a semilinear parabolic equation. The control variable is the Robin coefficient on the boundary. First-order necessary and second-order sufficient optimality conditions are derived. A sequential quadratic programming algorithm is then proposed to compute local solutions. Starting the iterations from an initial point in an $L^2$-neighborhood of the local solution we prove stability and quadratic convergence of the algorithm in $L^p$ ($p < \infty$) and $L^\infty$ assuming that the local solution satisfies a no-gap second-order sufficient optimality condition and a strict complementarity condition.