The Stability of Systems with Time Delay

As is stated in reference 2, the systematic study of differential equations with retarded arguments or time delays, has begun only in the twentieth century in connection with the needs of the applied sciences, especially the theory of automatic control. In the last 15 or 20 years the area of application of differential equations with retarded arguments has greatly expanded, and now encompasses not only many questions of physics and technology, but also areas of economics and the biological sciences. Thus the subject is presently one of the more rapidly developing areas of mathematics. In this paper we shall be concerned with investigating the stability of solutions of differential equations describing systems having delayed arguments. In Chapter 1 we will first define a differential equation with time delay, then proceed to a brief discussion of 1) basic considerations of time delay systems, 2) practical stability , and 3) the Liapunov technique. In Chapter 2 the various definitions and forms of stability will be presented from the classical or qualitative point of view. In Chapter 3 the more practical or quantitative aspects of set stability will be presented. Throughout this paper the usual Liapunov technique will be employed in obtaining the sufficient conditions for the various forms of stability