Finite-Region Boundedness and Stabilization of 2D Continuous-Time Roesser Models

This paper investigates the finite-region stability (FRS) and stabilization problems for 2D continuous-time systems described by a Roesser model. We first establish a novel set of FRS and finite-region boundedness (FRB) conditions, extending the L2-based concept on finite-time stability from 1D systems to the 2D continuous domain with a new condition based on the generalized state vector of the 2D continuous-time system in contrast with the norm-based condition found in the literature. Sufficient conditions are then derived to guarantee that the system state remains within a predefined quadratic region over a finite-time horizon. Furthermore, the framework is extended to analyze FRB under two distinct classes of external disturbances. Finally, a complete procedure for state feedback stabilization is provided, with all sufficient conditions for FRS and stabilization expressed entirely in terms of numerically tractable Linear Matrix Inequalities (LMIs) enabling controller design that ensures closed-loop finite-region performance under both disturbance classes. The effectiveness and feasibility of the proposed approach are demonstrated through numerical examples.