# State-constrained control-affine parabolic problems I: first and second order necessary optimality conditions

## TL;DR

This paper establishes first and second order necessary optimality conditions for a control problem governed by a semilinear heat equation with integral state constraints, bilinear control terms, and a tracking cost functional.

## Contribution

It introduces novel second order necessary conditions using alternative costates and quasi-radial critical directions for complex PDE control problems.

## Key findings

- Derived second order necessary conditions for the control problem.
- Provided an example demonstrating the applicability of the theoretical results.
- Extended optimality conditions to problems with integral state 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 necessary conditions relying on the concept of alternative costates and quasi-radial critical directions. The appendix provides an example illustrating the applicability of our results.