Positional solutions of Hamilton‐Jacobi equations in control problems for discrete‐continuous systems

We develop a canonical global optimality theory based on operating with the set of solutions for the Hamilton‐Jacobi inequalities that parametrically depend on the initial (or final) position. These solutions, called positional L‐functions (of Lyapunov type), naturally arise in the studies of control problems for discrete‐continuous (hybrid, impulse) systems; an important prototype of such problems are classical optimal control problems with general end constraints on the trajectory. We analyze sufficient optimality conditions with this new class of L‐functions and invert the maximum principle into a sufficient condition for nonlinear problems of optimal impulse control.