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

In this paper the problem of the stability of motion of the equilibrium solution x1 = x2… = 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…xn, t), i = 1, 2…n or x˙ = A (x,t)x. Various x1 are known as state variables and Fi (0, 0…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…αn, S1, S2…Sn. 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 V˙ is given by V˙ = x′[B′A + S˙]x = x′T(t,x)x where B x gives the gradient ∇V, and S˙ is defined as ∂S/∂t. For the equilibrium solution x1 = x2… = xn = 0 to be stable it is required that V˙ should be negative definite or negative semidefinite and V should be positive definite. These considerations determine the sufficient conditions of stability.