Existence Of Mild Solution And Optimal Controls For Measure Driven Evolution Equations With Nonlocal Conditions

At home and abroad,the research on the measure driven evolution equations is gradually deepening,and measure differential equations also appear in many areas of applied mathematics,such as non-smooth mechanics,game theory,etc.and covers many well-known equations.When distribution function is an absolute continuous function,measure driven evolution equation is an ordinary differential equation.When distribution function is a jump function,it is a difference equation.When distribution function is an absolute continuous and jumping function,it is an impulsive differential equation.Simulating the Zeno trajectory is another advantage of the study of such equations.Because when distribution function is a bounded variation function,it is possible to present an infinite number of discontinuous points in a finite interval.View from domestic and foreign literature,most articles discuss the existence of mild solution for measure driven evolution equations from the non-compact measure,fixed point theorem,and in the proof process,the conditions used by the stronger.In this paper,we will try to by relaxing the qualifications to study the existence and controllability of the mild solution for measure driven evolution equations.This article is divided into four sections.The first part mainly introduces the background and significance of the research and the basic knowledge of measure driven evolution equations.The main result of this paper are the second part and the third part.The second part mainly studies the existence and controllability of mild solution for measure driven evolution equations with nonlocal conditions when operator A is infinitesimal generator ofC₀-semigroup T（7）t（8）（7）t30（8）.The third part mainly studies operator A（7）t（8）is a family of linear operators,non-local under the condition of autonomy measure developing of the existence of appropriate solutions.The second part proves that the existence of mild solutions is the highlight of this paper:1.When h is Lipschitz continuous,we prove the existence of mild solution by using the fixed point theorem ofDarbo-Sadovskii.2.When h is only continuous,we discuss the existence of mild solution by using the fixed point theorem of Schauder and constructing approximate solution.The fourth part is summary of measure driven evolution equations and non-autonomous measure driven evolution equations.And it proposes the directions that need to be worked hard in the future.