Suppose a system is represented as where is the vector of variables, with If a function can be found such that then the set of accumulation points of any trajectory is contained in where is the union of complete trajectories contained entirely in the set. If we additionally have that the function is positive definite, i.e. and if contains no trajectory of the system except the trivial trajectory for, then the origin is asymptotically stable. Furthermore, if is radially unbounded, i.e. then the origin is globally asymptotically stable.
Local version
If hold only for in some neighborhood of the origin, and the set does not contain any trajectories of the system besides the trajectory, then the local version of the invariance principle states that the origin is locally asymptotically stable.
This section will apply the invariance principle to establish the local asymptotic stability of a simple system, the pendulum with friction. This system can be modeled with the differential equation where is the angle the pendulum makes with the vertical normal, is the mass of the pendulum, is the length of the pendulum, is the friction coefficient, and g is acceleration due to gravity. This, in turn, can be written as the system of equations Using the invariance principle, it can be shown that all trajectories which begin in a ball of certain size around the origin asymptotically converge to the origin. We define as This is simply the scaled energy of the system Clearly, is positive definite in an open ball of radius around the origin. Computing the derivative, Observe that. If it were true that, we could conclude that every trajectory approaches the origin by Lyapunov's second theorem. Unfortunately, and is only negative semidefinite. However, the set which is simply the set does not contain any trajectory of the system, except the trivial trajectory x = 0. Indeed, if at some time,, then because must be less than away from the origin, and. As a result, the trajectory will not stay in the set. All the conditions of the local version of the invariance principle are satisfied, and we can conclude that every trajectory that begins in some neighborhood of the origin will converge to the origin as .
History
The general result was independently discovered by J.P. LaSalle and N.N. Krasovskii, who published in 1960 and 1959 respectively. While LaSalle was the first author in the West to publish the general theorem in 1960, a special case of the theorem was communicated in 1952 by Barbashin and Krasovskii, followed by a publication of the general result in 1959 by Krasovskii.
