Stability analysis of two-dimensional nonlinear systems using Lyapunov´s second method

In the present paper, the second method of Lyapunov is utilized to establish sufficient conditions for the global asymptotic stability of the trivial solution of nonlinear, shift-invariant 2-D (two-dimensional) systems described by the Fornasini-Marchesini second state-space model (1976, 1978) which are endowed with saturation type nonlinearities. The Lyapunov stability concepts are introduced for 2-D systems. Results for the global asymptotic stability of the null solution of 2-D Fornasini-Marchesini second model with saturation type nonlinearities are established. Several classes of Lyapunov functions are used in establishing the present results including vector norms and the quadratic form. When the quadratic form Lyapunov functions are used, the present results involve necessary and sufficient conditions under which positive definite matrices can be used to generate Lyapunov functions for the 2-D nonlinear systems considered herein