Stability and stabilization of evolutionary games with time delays via matrix method

This paper studies the stability and stabilization of a class of evolutionary games (EGs) with time delays via the semi-tensor product (STP) method. First, the matrix expression of the EGs with time delays is given, and then it is converted into the form without delay equivalently. By designing Lyapunov function, a necessary and sufficient condition for the global stability of delayed EGs is obtained. To deal with the delayed EGs with multiple equilibriums, the controlled EGs with time delays is investigated. An intermittent state feedback control is proposed, in which the controller works only at certain strategy profiles. Based on the intermittent control, two necessary and sufficient conditions are provided to assure the stabilization of the controlled EGs with time delays. Some examples are worked out to illustrate the effectiveness of the theoretical results.