An optimal control approach to nonlinear wave speed selection in reaction-diffusion equations

Abstract

Travelling wave solutions of reaction-diffusion equations are widely used to model the spatial spread of populations and other phenomena in biology and physics. In this article, we reinterpret the classical variational principle approach through an optimal control formulation, in order to obtain a lower bound on the invasion speed of travelling wave solutions in systems of nonlinear partial differential equations. We begin by analysing single-species models, where the evolution of the density is governed by a scalar equation with a density-dependent diffusion term and a nonlinear reaction term. We show that for any admissible test function, maximising with respect to the parameter of interest yields a bound on the travelling wave speed. We apply this framework to several examples, including the porous-Fisher equation, and examine when nonlinear selection mechanisms dominate over the classical linear marginal stability criterion. Extending this approach, we then consider multi-species systems of reaction-diffusion equations and, reframed as Pontryagin-type optimality systems, we derive analogous bounds on the travelling wave speed using a variational framework under weak coupling. Finally, we employ numerical simulations to confirm the accuracy of the predicted wave speeds across a range of illustrative examples.