Smoothing Derivatives of Functions and Applications

Introduction. The primary purpose of this note is to construct a C* Lyapunov function for a continuous vector field which has an asymptotically stable invariant set. Since there already exist techniques for constructing continuous Lyapunov functions with continuous derivatives by the vector field, our problem is simply to smooth such a function in such a way that the desired derivative properties are preserved. An extremely simple smoothing procedure is to average the function by a convolution integral. Unfortunately, while it is well known that such averaging provides a uniform approximation for continuous directional derivatives, it seems very difficult to show that the continuous derivatives by an arbitrary continuous vector field are preserved (if they are). In the first section, we show that the derivative is preserved in the case where the function satisfies a Lipschitz condition. Thus the problem comes to approximating our given function by one which satisfies a Lipschitz condition, still preserving the derivative of course. This problem is solved in the second section by using an averaging technique which was developed by Kurzweil [3]. Unfortunately, while the convolution provides a close approximation for the derivative, the Kurzweil averaging only allows for a one-sided approximation of the derivative. However, this is enough for our purposes, and in the third section we apply these results to the converse Lyapunov problem. In the final section of this paper we apply these results to obtain a topological version of the Hirsch-Cairns smoothing theorem. We owe special gratitude to Charles C. Pugh for his careful reading of the manuscript, especially §2. Because of his diligence, the reader will be spared many ambiguities and typographical errors, and in addition one rather annoying technical error which appeared in the original manuscript.