On the finite-time stability of two-dimensional linear systems

In this paper we extend the finite-time stability (FTS) theory to two dimensional (2D)-systems. Such class of systems plays an important role in many engineering contexts, such as digital filtering, image processing, gas absorpsion technology, as well as in other fields, like seismological data processing, thermal and industrial processes. Each independent variable of a 2D-systems attains values into a given (possibly finite) interval (for example, an infinite dimensional system depends on a time variable taking values in the interval [0, +∞], and on a space variable attaining values into a given finite interval). Due to the finite-interval definition of some (or all) of the independent variables, it is quite straightforward the idea to exploit, for 2D-systems, the FTS theory developed in the context of the classical one-dimensional system framework. To this regard, we provide a sufficient condition for the FTS, and a sufficient condition for the finite-time stabilization of both continuous and discrete-time linear 2D-system; such conditions will require the solution of feasibility problems based on linear matrix inequalities (LMIs). A numerical example illustrates the benefits of the proposed methodology.