Abstract
In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear stochastic system with finite jumps. This allows us to obtain necessary and sufficient conditions assuring the existence of a sampled-data Nash equilibrium strategy, extending earlier results to a general context with more than two players. Furthermore, we provide a numerical algorithm for calculating the feedback matrices of the Nash equilibrium strategies. Finally, we illustrate the effectiveness of the proposed algorithm by two numerical examples. As both situations highlight a stabilization effect, this confirms the efficiency of our approach.
Highlights
Published: 26 October 2021Stochastic control problems governed by Itô’s differential equations have been the subject of intensive research over the last decades
This generated a rich literature and fundamental results such as the H2 and linear quadratic (LQ) robust sampled-data control problems under a unified framework studied in [1,2], classes of uncertain sampled-data systems with random jumping parameters characterized by finite state semi-Markov process analysed in [3], or stochastic differential games investigated in [4,5,6,7]
Our aim is to investigate the problem of designing a Nash equilibrium strategy in the class of piecewise constant admissible strategies of type (4), for a LQ differential game described by a dynamical system of type (1), under the performance criteria (2)
Summary
Stochastic control problems governed by Itô’s differential equations have been the subject of intensive research over the last decades. The original problem is transformed into an equivalent one which asks to find some existence conditions for a Nash equilibrium strategy in a state feedback form for a LQ stochastic differential game described by a system of Itô differential equations controlled by impulses. The problem of sampled-data Nash equilibrium strategy can be transformed in a natural way into a problem stated in discrete-time framework Such a transformation is not possible when the dynamical system contains state multiplicative and control multiplicative white noise perturbations.
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