Chemical waves in confined regions by Hamilton-Jacobi-Bellman theory

Abstract

Propagation of concentration disturbances in the form of (bio)chemical waves satisfying Fermat's principle of minimum time is analysed by methods of the classical variational calculus, Finsler geometry and optimal control theory. The optimal control methods are most general and effective as they can assure the minimum time and can be applied to constrained states. In particular, the dynamic programming approach leads to the Hamilton-Jacobi-Bellman equation for chemical waves, which describes (along with its characteristic set) the link between the constrained motions of wave fronts and associated ‘rays’ or extremal trajectories. The geodesic constraints due to an obstacle influence the state changes and especially the entering (leaving) conditions of a ray as the tangentiality condition for rays that begin to slide over the boundary of an obstacle. The analysis determines also the deviations of rays from straightlinearity in inhomogeneous media. It may handle complex transversality conditions. In a complex case an optimal arc of a chemical wave is composed of an internal and boundary parts.