Transmission conditions on interfaces for HamiltonJacobi-Bellman equations, in preparation

We establish a comparison principle for a Hamilton–Jacobi–Bellman equation, more appropriately a system, related to an infinite horizon problem in presence of an interface. Namely a low dimensional subset of the state variable space where discontinuities in controlled dynamics and costs take place. Since corresponding Hamiltonians, at least for the subsolution part, do not enjoy any semicontinuity property, the comparison argument is rather based on a separation principle of the controlled dynamics across the interface. For this, we essentially use the notion of ε–partition and minimal ε–partition for intervals of definition of an integral trajectory. Key words. Optimal control, discontinuous dynamics and cost, comparison principle, Hamilton-Jacobi equations, transmission conditions, viscosity solutions. AMS subject classifications. 49L20, 49J15, 35F21