The Piecewise-Linear Lyapunov Function Construction for Dynamical Systems of the Second Order

Abstract

The paper considers a stability problem of the equilibrium points of dynamical systems of the second order and describes an investigative technique based on the Lyapunov function construction. This method is useful for exploring the equilibrium points, which are exponentially stable. One of the methods to analyse the equilibrium point stability of the systems of ordinary differential equations is search for the Lyapunov function. It is mandatory for the found function to meet the specific conditions. So far, there is no universal technique to construct the Lyapunov function. There are developed approaches, which allow constructing the Lyapunov functions in different forms. One of well-studied technique to construct the Lyapunov function is based on the Zubov’s equation solution [1, 2]. Its disadvantage is that it is necessary to solve a partial differential equation. A number of articles offer piecewise specifying of Lyapunov function at different meshes of the state space containing equilibrium point. Publications present studies of the Lyapunov function construction for a diversity of techniques to specify the triangle meshes. The paper describes a construction method of the piecewise-linear Lyapunov function on the triangle mesh. Presents an algorithm of triangulation of the state space within which there is the equilibrium point. The suggested method is based on the solution of the linear programming problem. The variables of this problem are the values of the Lyapunov function in the nodes of the mesh and the additional constants, which ensure that function satisfies specific conditions. This method is iterative. The initial triangulation is built. If there is no solution on the current iteration, another triangulation is built, and finding a new problem solution with the new amount of the variables takes place. Implementation of this method for different values of the parameter is analyzed for a dynamical system of the second order. It is shown that the number of required iterations depends on the parameter value. A lot of iterations involve essential calculus problems. The algorithm efficiency considerably varies with respect to the eigenvalues of the system Jacobian matrix in the equilibrium point. Development of technique to construct the Lyapunov function, which is useful for wider class of dynamical systems, is planned.