The Dini Derivative and Generalization of the Direct Lyapunov Method

Abstract

The contemporary theory of stability for systems of differential equations is based on the concept of Lyapunov stability, A.M. Lyapunov’s results and their certain generalizations. Analysis of the first approximation of a system is used as a main method to study a stability of the equilibrium points. In publications this method is known as the first Lyapunov method. But this method does not allow drawing conclusions in the critical case, and then the second Luapunov method can be used, which is also known as a direct Lyapunov method.The direct Lyapunov method is based on existing function with certain properties. The function has to be positive definite. If in the certain vicinity of the equilibrium point a function derivative by virtue of the system is not positive, then it is called Lyapunov function. The existence of Lyapunov function means that the equilibrium point is stable.A role of the Lyapunov function is not only to establish the fact of the equilibrium point stability (or stronger property of asymptotic or exponential stability). It gives a lower bound of the region of attraction of an equilibrium point, which can be important in the control theory. So, to construct the Lyapunov function is a problem of importance, even if the fact of equilibrium stability has been already established.Herewith there are no universal methods to construct the Lyapunov function for the autonomous system of differential equations. To solve the problem are used various numerical methods in which it is difficult to provide the important condition —differentiability of function under construction. At the same time, in the Lyapunov method the differentiability condition is non-essential and is related only to the mathematical technique. Therefore, generalizations of the Lyapunov method, which are directed to the abandonment of strong conditions of function smoothness, are important. One of such generalizations is the use of the Dini derivative. The use of the Dini derivative allows us to construct, for example, the piecewise linear Lyapunov functions.The results connected with the Dini derivative are rarely included in standard monographs, and an objective of the present article is to make these results more accessible