Optimal control of a class of reaction–diffusion systems

Eduardo Casas,Fredi Tröltzsch,Christopher Ryll

doi:10.1007/s10589-018-9986-1

Eduardo Casas, Fredi Tröltzsch + Show 1 more

Open Access

https://doi.org/10.1007/s10589-018-9986-1

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Abstract

The optimal control of a system of nonlinear reaction–diffusion equations is considered that covers several important equations of mathematical physics. In particular equations are covered that develop traveling wave fronts, spiral waves, scroll rings, or propagating spot solutions. Well-posedness of the system and differentiability of the control-to-state mapping are proved. Associated optimal control problems with pointwise constraints on the control and the state are discussed. The existence of optimal controls is proved under weaker assumptions than usually expected. Moreover, necessary first-order optimality conditions are derived. Several challenging numerical examples are presented that include in particular an application of pointwise state constraints where the latter prevent a moving localized spot from hitting the domain boundary.

Highlights

We consider a class of optimal control problems for a general system of nonlinear reaction-diffusion equations that covers a variety of particular cases with important applications in mathematical physics
Control problems of this type received increasing attention in the recent past, we refer for instance to [16, 24, 29, 33] with respect to control methods of theoretical physics or [3, 4, 13, 12, 11] from a more mathematical perspective
We prove the existence of optimal controls under weaker assumptions than usually expected: In the unconstrained case a = −∞ or b = ∞ it seems that the existence of optimal controls cannot be shown

Summary

Introduction

We consider a class of optimal control problems for a general system of nonlinear reaction-diffusion equations that covers a variety of particular cases with important applications in mathematical physics. The consideration of the state constraints is motivated by an interesting application in theoretical physics, namely the problem of preventing localized moving spots from reaching the boundary of the spatial domain. This issue is discussed in Example 3 of Section 4. The matrices E and D will be needed in our numerical examples

Control constrained problem

Optimal control problem and existence of a solution

Optimality system

A regularity result for local solutions

Examples