Abstract

One dimensional dynamics with impact are described by the data of a closed interval K, a function f of time, position, and velocity, and a restitution coefficient e ∈ [0,1). When u is in the interior of K, it satisfies the ordinary differential equation u ̈ = f(t, u, u ̈ ) . When it hits the end points of K, the velocity is reversed and multiplied by e. If f is analytic with respect to its three arguments, it is proved that uniqueness holds for the forward Cauchy problem.

Highlights

  • We study the uniqueness and continuous dependence on data of solutions to a differential system with impact

  • We seek a function ‘1~E C”( [T, T’]; IR) whose derivative is of bounded variation, and a real measure p on [T, T’] which satisfy the following set of relations: u E CO([TT, ’]; K), ii = f( ., U, C) + p, in the sense of distributions, (/A~--) 10, VU E CO([T,T’]; K)

  • To apply Ascoli-Arzelb’s theorem, we show that V, is complete, and we characterize the compact subsets of V

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Summary

INTRODUCTION

We study the uniqueness and continuous dependence on data of solutions to a differential system with impact. I conjecture that if K is a closed subset of RN with analytic boundary, e E [0,11,and f is a real analytic function of (t, a, b) on a domain 0 of W x RN x RN, the solution of the forward Cauchy problem is unique for the N-dimensional generalization of (l.l)-(1.5). This theorem assumes that we have a reasonable notion of a local solution of (l.l)-(1.5).

THE PHASE SPACE
LOCAL ESTIMATES AND LOCAL EXISTENCE
UNIQUENESS
CONTINUOUS DEPENDENCE OF SOLUTIONS
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