On exponential stability of switched homogeneous positive systems of degree one

In this paper, we investigate the stability problem of switched homogeneous positive systems of degree one. We first show that the positiveness of the considered system is guaranteed when each switched subsystem is cooperative. Then, an integral excitation condition is formulated to ensure the exponential stability of the switched homogeneous positive system of degree one. In particular, the proposed condition allows that some or even all switched subsystems are only Lyapunov stable rather than asymptotically stable. In the second part of this paper, we further consider the cases of sub-homogeneous systems, time-varying delay systems and discrete-time systems. In the sub-homogeneous case, we first show that the previous integral excitation condition is also sufficient such that the exponential stability is achieved. We also derive an exponential stability result by relaxing the cooperative and integral excitation conditions. In the time-varying delay case, a delay-independent exponential stability result is shown under the extended integral excitation condition. Finally, in the discrete-time case, based on the fact that each switched subsystem is order-preserving instead of cooperative, the exponential stability of the considered system is shown under a weak excitation condition.