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

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 $L^p$ ($p<\mathrm{\infty }$) and $L^{\mathrm{\infty }}$ assuming that the local solution satisfies a no-gap second-order sufficient optimality condition and a strict complementarity condition.