The Converse of the Stability Theorems

Abstract

The principal theorems of the direct method (secs. 25 and 42) furnish sufficient conditions for stability and instability. They do not say whether the given conditions are also necessary, i.e. whether from known stability properties we can infer the existence of suitable Liapunov functions, Only the case of a linear autonomous system has been completely settled by the results of sec. 27: It is known that in case of asymptotic stability a suitable Liapunov function always exists, in fact as a quadratic form. In the general case the answer was found relatively late. The first result is due to K. P. PERSIDSKII [1] and refers to simple, non-uniform stability. Persidskii was able to show that a Liapunov function can always be found which satisfies the conditions of Theorem 42.1. However, he only considered differential equations in R n , The corresponding theorems, usually called converse theorems, on asymptotic stability could not be discovered until the concept of uniform stability had been recognized and clearly defined. After all, the so-called Second Stability Theorem of Liapunov (25.2, resp. 42.4) yields not merely asymptotic stability but considerably more, namely uniform asymptotic stability. It is thus impossible to prove the converse of the theorem in its original weaker form. Once the basic concepts had been clarified the conditions for the existence of Liapanov functions were carefully studied and the converse problem can now be considered as essentially settled.