On uniform solvability of parameter-dependent lyapunov inequalities and applications to various problems

We consider the problem of finding a common quadratic Lyapunov function to demonstrate the stability of a not necessarily bounded family of matrices which incorporate design freedoms. Generically, this can be viewed as the problem of picking a family of controller (or observer) gains so that the family of closed-loop system matrices admits a common quadratic Lyapunov function. We provide several necessary and sufficient conditions for various structures of matrix families. Specifically, we consider matrix families which can be obtained through similarity transformations from a lower Hessenberg structure. Families of matrices containing a subset of diagonal matrices, which is invariant under the design freedoms, are also considered since they occur in many applications. The conditions for uniform solvability of the Lyapunov inequalities are explicitly given and involve inequalities regarding relative magnitudes of terms in the matrices. Various motivating applications of the obtained results to observer and controller designs for time-varying, switched, and nonlinear systems are highlighted.