A numerical method for constructing Lyapunov functions and computer-aided analysis of stability of nonlinear dynamic systems (Q1914333)

The authors study the stability property of the equilibrium state of dynamic systems described by nonlinear differential equations or differential inclusions of a special type (the ``selective-linear'' ones). The proposed analysis is based on classical Lyapunov techniques. For a given set \(G\) \((0 \in \text{int} G)\) and a region \(\Gamma = \{x : 0 < \varepsilon \leq |x |\leq R\}\) the authors construct the invariant domain \(\Phi_0\) such that any solution of the system that emerges in \(\Gamma\), falls into \(\Phi_0\) in a finite time and then remains there. The conditions for the existence of such invariant sets \(\Phi_0\) are given. The limit properties of the sequence of Lyapunov functions \(v_m (x)\) defined on the sets \(\Gamma_m = \{G \backslash B_m\}\) with \(\text{diam} B_m \to 0\) are also under consideration. The paper contains an algorithm for numerical construction of Lyapunov functions in a region \(\Gamma\) proposed on the basis of the above limit analysis of the functions \(v_m (x)\) designed for discrete \(h\)-net approximations \(\Gamma_h\) (with a step \(h > 0)\) of the region \(\Gamma\). The construction of Lyapunov functions for two second-order nonlinear systems are presented via examples.