State constrained control-affine parabolic problems II: Second order sufficient optimality conditions

TL;DR

This paper establishes second order sufficient optimality conditions for a control problem governed by a semilinear heat equation with integral state constraints and bilinear control-state interactions, using the Goh transform.

Contribution

It introduces new second order optimality conditions for complex PDE control problems with integral state constraints, expanding theoretical understanding.

Findings

Derived second order sufficient conditions using Goh transform

Provided an example demonstrating applicability of the conditions

Enhanced theoretical framework for PDE control with integral constraints

Abstract

In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order sufficient conditions relying on the Goh transform. The appendix provides an example illustrating the applicability of our results.