Impulsive differential equations

Abstract

In this chapter we discuss first order impulsive differential equations. Many physical situations are modelled by problems of this kind, for example problems in optimal control theory and problems in threshold theory in Biology. The last ten years or so have seen major developments in the theory of impulsive differential equations. In this chapter we present some of the more advanced results to date in the existence theory of nonlinear first order impulsive differential equations with variable times. Let k be a positive integer and T ∈ (0, ∞]. In section 15.3 we establish existence results for the impulsive differential equation (IDE), $$\left\{ {\begin{array}{*{20}c} {y' = f(t,y){\text{for}}{\kern 1pt} {\text{ a}}{\text{.e}}{\text{. }}{\kern 1pt} {\text{t}} \in {\text{[0,t), t}} \ne \tau {\text{i(y(t)),}}} \\ {y(t^ + ) = I_i (y(t^ - )){\text{ }}if{\text{ }}{\kern 1pt} t = \tau _i (y(t)),\,i = 1,.....,k,} \\ {y(0) = y_0 .} \\ \end{array} } \right.$$ (1.1)