Abstract
We consider a class of linear-quadratic differential games with piecewise-deterministic dynamics, where the changes from one structure (for the dynamics) to another are governed by a finite-state Markov process. Player 1 controls the continuous dynamics, whereas Player 2 controls the rate of transition for the finite-state Markov process; both have access to the states of both processes. Player 1 wishes to minimize a given quadratic performance index, while Player 2 wishes to maximize the same quantity. For this class of differential games, we present results on the existence and uniqueness of a saddle point, and develop computational algorithms for solving it. Analytical (closed-form) solutions are obtained for the case when the continuous state is of dimension 1. The paper also considers the modified problem where Player 2 is also a minimizer, that is, when the underlying problem is a team, and illustrates the theoretical results in this context on a problem that arises in high-speed telecommunication networks. This application involves combined admission and rate-based flow control, where the former corresponds to control of the finite-state Markov process, and the latter to control of the continuous linear system.
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Altman, E., Başar, T., Pan, Z. (2000). Piecewise-Deterministic Differential Games and Dynamic Teams with Hybrid Controls. In: Filar, J.A., Gaitsgory, V., Mizukami, K. (eds) Advances in Dynamic Games and Applications. Annals of the International Society of Dynamic Games, vol 5. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1336-9_12
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DOI: https://doi.org/10.1007/978-1-4612-1336-9_12
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