Properties of mild solution for the first order inhomogeneous time varying singular distributed parameter systems in hilbert space

First of all, expression to mild solution of the time varying singular distributed parameter system is given via the theory of GE mild evolution operator in Hilbert space. Then the sufficient condition concerning the continuity of the mild solution, and the sufficient condition concerning the existence and uniqueness of the solution are obtained.

1 Introduction

Singular distributed parameter systems are described by generalized partial differential equation, generalized integral equation and generalized integral-differential equation or abstract generalized differential equation in infinite dimensional space. Singular distributed parameter systems are much more often encountered than the ordinary distributed parameter systems. They appear in the study of the temperature distribution in a composite heat conductor, voltage distribution in electromagnetically coupled superconductive circuits, signal propagation in a system of electrical cables [1]–[3] etc. There is an essential distinction between singular and ordinary distributed parameter systems [1]–[19], under disturbance, not only singular distributed parameter systems lose stability, but also great changes take place in their structure, such as leading to impulsive behavior etc. Along with the rapid development of high and new technology, basic researches about singular distributed parameter systems are more and more important.