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We introduce a class of deterministic finite-horizon two-player nonzero-sum differential games where Player 1 uses continuous controls while Player 2 uses impulse controls. We formulate the necessary and sufficient conditions for the existence of an open-loop Nash equilibrium in this class of differential games. We specialize these results to linear-quadratic games, and show that the open-loop Nash equilibrium strategies can be computed by solving a constrained non-linear optimization problem. For a specialized canonical linear-state differential game, we obtain analytical solutions for equilibrium number, timing and level of impulses. In particular, we show that there can be at most one interior impulse in the open-loop Nash equilibrium and the level of impulse depends only on the problem parameters. When the timing of impulse is exogenously given, we recover the classical result that the open-loop Nash equilibrium and feedback Nash equilibrium coincide for deterministic linear-state differential games. When Player 2 decides the number and timing of impulses besides the size of impulses, we find the classical result does not hold, that is, open-loop and feedback Nash equilibrium are different.
This is a joint work with Puduru Viswanadha Reddy and Georges Zaccour.
Bio: Utsav Sadana is a final year PhD student in the Department of Decision Sciences at HEC Montreal, working under the supervision of Professor Georges Zaccour. He graduated from the Indian Institute of Technology (IIT) Kanpur, India, with a Bachelor of Technology in Materials Science and Engineering and a Masters in Economics. From September 2019 - February 2020, he was a visiting research scholar at the Coordinated Science Laboratory of the University of Illinois at Urbana-Champaign where he worked with Professor Tamer Basar.