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1997•Journal of the Korean Mathematical SocietyRequires access

GENERALIZED SOLUTIONS OF IMPULSIVE CONTROL SYSTEMS CORRESPONDING TO CONTROLS OF BOUNDED VARIATION

Chang Eon Shin

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Abstract

This paper is concerned with the impulsive control problem ˙ x.t/D f.t; x/C g.t; x/ ˙ u.t/; t2 (0; T ); x.0/DN x ; where u is a possibly discontinuous control function of bounded variation, f : RR n 7! R n is a bounded and Lipschitz continuous function, and g : RR n 7! R n is continuously differentiable w.r.t. the variable x and satisfies jg.t;/ g.s;/j.t/ .s/; for some increasing function and every s < t: We show that the map u7! xu is Lipschitz continuous when u ranges in the set of step functions whose total variations are uniformly bounded, where xu 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.

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What this paper is about

This paper is concerned with the impulsive control problem ˙ x.t/D f.t; x/C g.t; x/ ˙ u.t/; t2 (0; T ); x.0/DN x ; where u is a possibly discontinuous control function of bounded variation, f : RR n 7! R n is a bounded and Lipschitz continuous function, and g : RR n 7! R n is continuously differentiable w.r.t. the variable x and satisfies jg.t;/ g.s;/j.t/ .s/; for some increasing function and every s < t: We show that the map u7! xu is Lipschitz continuous when u ranges in the set of step functions whose total variations are uniformly bounded, where xu 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.

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Available abstract

This paper is concerned with the impulsive control problem ˙ x.t/D f.t; x/C g.t; x/ ˙ u.t/; t2 (0; T ); x.0/DN x ; where u is a possibly discontinuous control function of bounded variation, f : RR n 7! R n is a bounded and Lipschitz continuous function, and g : RR n 7! R n is continuously differentiable w.r.t. the variable x and satisfies jg.t;/ g.s;/j.t/ .s/; for some increasing function and every s < t: We show that the map u7! xu is Lipschitz continuous when u ranges in the set of step functions whose total variations are uniformly bounded, where xu 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.

Key concepts: Bounded variation, Bounded function, Mathematics, Lipschitz continuity, Differentiable function, Function (biology), Continuous function (set theory), Combinatorics

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