$L_1$-Stochastic Stability and $L_1$-Gain Performance of Positive Markov Jump Linear Systems With Time-Delays: Necessary and Sufficient Conditions

This paper is concerned with L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -stochastic stability and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -gain performance analysis of a continuous-time positive Markov jump linear system with a time-delay. By constructing a novel linear co-positive stochastic Lyapunov functional based on the positivity of the system, and characterizing the system equation of the mathematical expectation of the markovianized states, some necessary and sufficient delay-dependent conditions for L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -stochastic stability and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -gain performance are presented in terms of linear programming. A numerical example is provided for showing the effectiveness of the obtained conditions, especially for exploring and revealing both the negative effect and the positive effect of the time-delay on stochastic stability of the positive Markov jump linear system with a time-delay.