Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 1997; 34(3): 581-598

Printed September 1, 1997

Copyright © The Korean Mathematical Society.

Generalized solutions of implusive control systems corresponding to controls of bounded variation

Chang Eon Shin

Sogang University

Abstract

This paper is concerned with the impulsive control problem
$$
\dot x(t)=f(t,x)+g(t,x)\dot u(t),\quad t\in [0,T],\quad x(0)=\bar x,
$$
where $ u $ is a possibly discontinuous control function of bounded
variation, $ f:\R\times\R^n\mapsto \R^n $ is a bounded and Lipschitz
continuous function, and $ g:\R\times\R^n\mapsto \R^n $ is
continuously differentiable w.r.t. the variable $ x $ and satisfies
$ |g(t,\cdot)-g(s,\cdot)|\le \phi(t)-\phi(s), $ for some increasing
function $ \phi $ and every $ s x_u $ is Lipschitz continuous when $ u $ ranges in the set of step
functions whose total variations are uniformly bounded, where $ x_u $
is the solution of the impulsive control system corresponding to $ u. $
We also define the generalized solution of the impulsive control system
corresponding to a measurable control function of bounded variation.

Keywords: impulsive control, generalized solution

MSC numbers: 49N25

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