Generalized solution of time dependent impulsive control system corresponding to vector-valued controls of bounded variation

Bull. Korean Math. Soc. 2000 Vol. 37, No. 2, 229-247

Chang Eon Shin and Ji Hyun Ryu Sogang University, Sogang University

Abstract : This paper is concerned with the impulsive Cauchy problem where the control function $u$ is a possibly discontinuous vector-valued function with finite total variation. We assume that the vector fields $f, g_i (i=1, \cdots, m)$ are dependent on the time variable. The impulsive Cauchy problem is of the form $$ \dot{x}(t)= f(t,x) + \sum_{i=1}^m g_i(t,x) {\dot u}_i (t),\quad t\in [0,T] ,\quad x(0)=\bar{x} \in {\R}^n ,$$ where the vector fields $f, g_i: \R \times {\R}^n \to {\R}^n $ are measurable in $t$ and Lipschitz continuous in $x$. If ${g_i}^\prime s $ satisfy a condition that $$\sum_{i=1}^m |g_i(t_2,x)-g_i(t_1,x)| \leq \phi(t_2) - \phi(t_1),\quad \forall t_1 < t_2, x \in {\R}^n,$$ for some increasing function $\phi$, then the input-output function can be continuously extended to measurable functions of bounded variation.

Keywords : impulsive control system, generalized solution