# Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions

## TL;DR

This paper investigates the sensitivity of optimal control problems governed by nonlinear evolution inclusions, establishing well-posedness and continuity properties, with an application to nonlinear parabolic systems.

## Contribution

It provides a general framework for analyzing the solution set and sensitivity of nonlinear evolution inclusion-based control problems, including continuity of solutions and value functions.

## Key findings

- Solution set nonemptiness and continuous selection established
- Hadamard well-posedness of the control problem proved
- Continuity of the optimal solution multifunction demonstrated

## Abstract

We consider a nonlinear optimal control problem governed by a nonlinear evolution inclusion and depending on a parameter $\lambda$. First we examine the dynamics of the problem and establish the nonemptiness of the solution set and produce continuous selections of the solution multifunction $\xi\mapsto S(\xi)$ ($\xi$ being the initial condition). These results are proved in a very general framework and are of independent interest as results about evolution inclusions. Then we use them to study the sensitivity properties of the optimal control problem. We show that we have Hadamard well-posedness (continuity of the value function) and we establish the continuity properties of the optimal multifunction. Finally we present an application on a nonlinear parabolic distributed parameter system.