On the Global Stability of a Class of Nonlinear Time-Varying Systems

Abstract

In this paper the problem of the stability of motion of the equilibrium solution x1 = x2 [[ellipsis]] = xn = 0 is studied, in the sense of Lyapunov, for a class of systems represented by a system of differential equations dxi /dt = Fi (x1 , x2 [[ellipsis]]xn , t), i = 1, 2[[ellipsis]]n or ẋ = A (x,t)x . Various x1 are known as state variables and Fi (0, 0[[ellipsis]]0, ∞) = 0. The various elements of square matrix A (x , t) are functions of time as well as functions of state variables x . Two different methods for generating Lyapunov functions are developed. In the first method the differential equation is multiplied by various state variables and integrated by parts to generate a proper Lyapunov function and a number of matrices α, α1 [[ellipsis]]αn , S 1 , S 2 [[ellipsis]]S n . The second method assumes a quadratic Lyapunov function V = [x ′ S (x ,t)x ], x ′ being the transpose of x . The elements of S (x ,t) may be functions of time and the state variables or constants. The time derivative V̇ is given by V̇ = x ′ [B ′ A + Ṡ ]x = x ′ T (t,x )x where B x gives the gradient ∇V, and Ṡ is defined as ∂S /∂t. For the equilibrium solution x1 = x2 [[ellipsis]] = xn = 0 to be stable it is required that V̇ should be negative definite or negative semidefinite and V should be positive definite. These considerations determine the sufficient conditions of stability.