Group for Research in Decision Analysis

G-2020-81

Sampled-data Nash equilibria in differential games with impulse controls

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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 conditions for the existence of sampled-data Nash equilibrium for a general class of differential games with impulse controls. We specialize our results for 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 non-linear equality constraints. By reformulating our problem as a constrained non-linear optimization problem, we compute the equilibrium timing and level of impulses. We find that the equilibrium piecewise continuous control is a linear function of the last measured state value. For linear-state differential games, we obtain analytical characterizations of equilibrium number, timing and levels of impulses in terms of the problem data, and provide an extension of our results for the case with piecewise constant time-varying problem parameters. In particular, there can be at most one impulse in the game when the problem parameters are fixed while each sampling interval can contain at most one impulse when the problem parameters differ between the sampling intervals. Using a numerical example, we illustrate our results.

, 23 pages