Sampled-Data Nash Equilibria in Differential Games with Impulse Controls

We study a class of deterministic two-player nonzero-sum differential games where one player uses piecewise-continuous controls to affect the continuously evolving state, while the other player uses impulse controls at certain discrete instants of time to shift the state from one level to another. The state measurements are made at some given instants of time, and players determine their strategies using the last measured state value. We provide necessary and sufficient conditions for the existence of sampled-data Nash equilibrium for a general class of differential games with impulse controls. We specialize our results to a scalar linear-quadratic differential game and show that the equilibrium impulse timing can be obtained by determining a fixed point of a Riccati-like system of differential equations with jumps coupled with a system of nonlinear equality constraints. By reformulating the problem as a constrained nonlinear optimization problem, we compute the equilibrium timing, and level of impulses. We find that the equilibrium piecewise continuous control and impulse control are linear functions of the last measured state value. Using a numerical example, we illustrate our results.