J. Korean Math. Soc. 1997; 34(3): 581-598
Printed September 1, 1997
Copyright © The Korean Mathematical Society.
Chang Eon Shin
Sogang University
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
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
2003; 40(6): 1031-1050
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