Hamilton–Jacobi–Bellman equation for a nonlinear impulse control problem

A minimum problem for a functional of Bolza type along trajectories of nonlinear systems of differential equations governed by impulse controls with integral constraints is considered. A definition of a solution to such systems uses the closure of the set of absolutely continuous trajectories in the topology of pointwise convergence. It is shown that the value function of such a system is Lipschitz continuous and is a unique viscosity solution to a partial first order differential equation (a Hamilton–Jacobi–Bellman equation). Boundary conditions satisfied by the solution are obtained.