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We consider a wireless networked control system (WNCS) with multiplecontrollers and multiple attackers. The dynamic interaction between thecontrollers and the attackers is modeled as a linear quadratic (LQ) zero-sumdifference game with multiplicative randomness induced by the multiple-inputand multiple-output (MIMO) wireless fading channels of the controllers andattackers. We focus on analyzing the existence and achievability of the valueof the zero-sum game. We first establish a general sufficient and necessarycondition for the existence of the game value. This condition relies on thesolvability of a modified game algebraic Riccati equation (MGARE) under animplicit concavity constraint, which is generally difficult to verify due tothe intermittent controllability or almost sure uncontrollability caused by themultiplicative randomness. We then introduce a new positive semidefinite (PSD)kernel decomposition method induced by multiplicative randomness, through whichwe obtain a closed-form tight verifiable sufficient condition. Under theexistence condition, we finally construct a saddle-point policy that is able toachieve the game value in a certain class of admissible policies. Wedemonstrate that the proposed saddle-point policy is backward compatible to theexisting strictly feedback stabilizing saddle-point policy.

On the Value of Linear Quadratic Zero-sum Difference Games with Multiplicative Randomness: Existence and Achievability