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
Summary
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).
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