Feedback Nash Equilibrium for Randomly Switching Differential–Algebraic Games

Abstract

As a subclass of stochastic differential games with algebraic constraints, this article studies dynamic noncooperative games where the dynamics are described by Markov jump differential-algebraic equations (DAEs). Theoretical tools, which require computing the extended generator and deriving Hamilton-Jacobi-Bellman (HJB) equation for Markov jump DAEs, are developed. These fundamental results lead to pure feedback optimal strategies to compute the Nash equilibrium in noncooperative setting. In case of quadratic cost and linear dynamics, these strategies are obtained by solving coupled Riccatilike differential equations. Under an appropriate stabilizability assumption on system dynamics, these differential equations reduce to coupled algebraic Riccati equations when the cost functionals are considered over infinite-horizon. As a first casestudy, the application of our results is studied in the context of an economic system where different suppliers aim to maximize their profits subject to the market demands and fluctuations in operating conditions. The second case-study refers to the conventional problem of robust control for randomly switching linear DAEs, which can be formulated as a two-player zero sum game and is solved using the results developed in this paper.Index Terms-Noncooperative games; differential-algebraic dynamical systems; piecewise deterministic Markov processes; infinitesimal generators; generalized HJB equation; coupled Riccati equations; minimax robust control; Leontief input-output models.

Impulse-free jump solutions of nonlinear differential–algebraic equations

“…A similar definition of nonlinear consistency projector can be found in [40], where index-1 DAEs are studied and it is assumed that they are global equivalent (actually ex-equivalent using Definition 2.6) to a semi-explicit form. Such an assumption is equivalent to the involutivity assumption of ker E (see Theorem 3.13 of [28]) when the singular points are not considered.…”

“…But it can be seen from Remarks 4.4(iii) and 4.7(iii) above, those singular points actually play important roles for the existence of impulse-free jumps. Note that under an additional Q -transformation, we can always transform the semi-explicit form in [40] into our (INWF). Moreover, we have shown a way of constructing the (Q , ψ)transformations to obtain the (INWF) and to define the nonlinear consistency projector in the proof of Theorem 4.6, those results are not discussed in [40].…”

Dynamic games for secure and resilient control system design

“…To solve game ( 3 ) with differential-algebraic equations, one can refer to Ref. [ 58 ]. For specific systems, one can adopt specific models including Markov decision processes, difference equations and partial differential equations, to describe the dynamics in the physical layer and the cyber layer.…”

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

“…Feedback control and its many branches (optimal control, controllability, observability, state observer design and separation principle, stabilization, tracking control, classification of systems into subclasses depending on their controllability/observability properties, robust control) has not been treated in detail. While these problems have received some attention in the community [287,309,526,555], there are many related questions which need to be investigated in depth, as they lead to some challenging questions at the intersection of functional analysis, optimization algorithms, numerical implementation, and control theory.…”