Trajectory based small gain theorems for parameterized systems

In this paper, trajectory based small gain theorems are developed for parameterized families of continuous-time systems. We show that the interconnection of two input-to-state stable (ISS) parameterized systems is semiglobally and practically ISS, when ISS gain functions are appropriately parameterized, and we provide explicit sufficient conditions on the parameter. A small gain theorem is also presented for the case where two parameterized systems that satisfy some input-to-state and input-to-output stability properties are interconnected with a bounded-input-bounded-state system. Obtained results are applied to several stabilization and tracking problems.