Stability Optimization of Positive Semi-Markov Jump Linear Systems via Convex Optimization

In this paper, we study the problem of optimizing the stability of positive semi-Markov jump linear systems. We specifically consider the problems of tuning the coefficients of the system matrices for maximizing the exponential decay rate of the system under a budget-constraint and minimizing the parameter tuning cost under the decay rate constraint. By using a result from the matrix theory on the log-log convexity of the spectral radius of nonnegative matrices, we show that the stability optimization problems are reduced to convex optimization problems under certain regularity conditions on the system matrices and the cost function. We illustrate the validity and effectiveness of the proposed results by using an example from the population biology.