The lp Induced Norm and the Small-Gain Theorem for Discrete-time Stochastic Systems

This paper discusses $l_{p}$ induced norms and a small-gain theorem for discrete-time stochastic systems. The induced norm of systems generally depends on the associated input space of the systems. At the beginning, we consider a general stochastic $l_{p}$ space and introduce the primitive $l_{p}$ induced norm based on the space. However, the primitive $l_{p}$ induced norm can actually be confirmed not to reflect the nature of the randomness behind the stochastic systems, which leads us to overestimating its value. Hence, we take a proper subspace of the stochastic $l_{p}$ space and introduce another $l_{p}$ induced norm based on the subspace. We demonstrate with examples that the norm based on the subspace successfully reflects the nature of the system randomness. Then, we give the small-gain theorem based on the norm.