Distributed finite-time control for spatially interconnected systems

Compared with the classical Lyapunov asymptotic stability, finite-time stability is much more important and significant in some practical applications. The purpose of this study is to design the distributed controllers such that the controlled spatially interconnected systems are well-posed, finite-time stable/bounded and finite-time contractive. First, the authors introduce the definitions of finite-time stability/boundedness and finite-time contractiveness for spatially interconnected systems based on the given system matrices. Second, a sufficient condition is proposed to check the well-posedness, finite-time stability/bounedness and finite-time contractiveness of spatially interconnected systems. Third, the distributed controllers are designed to solve the distributed finite-time control problem. Besides, the existence condition and construction of the distributed controllers are shown. In the end, a one-dimensional heat equation example is demonstrated to verify the effectiveness of the proposed control method.