Existence of solutions to projected differential equations in Hilbert spaces

Abstract

Abstract. We prove existence and uniqueness of integral curves to the (discontinuous) vector field that results when a Lipschitz continuous vector field on a Hilbert space of any dimension is projected on a non-empty, closed and convex subset.

On smoothness of solutions to projected differential equations

“…Thus, we only need to prove the uniqueness. Assume that there exists another vector d ∈ Y satisfying equation (14).…”

“…(with closed sets C(t) ⊂ X) is well studied in the literature (see, e.g., [7,14,33] and the references therein); the sweeping process itself given by (5) without the perturbation f (•, •) has been introduced for an elasto-plastic mechanical system by J.J. Moreau in [29] who thoroughly developed its study in a series of subsequent fundamental papers. In contrast, little is known about the regularity of those solutions, beyond absolute continuity.…”

“…Applying for example [14,Theorem 3.1] and Theorem 1.1, we deduce that problem (17) has a maximal solution y : I → X, which, by Theorem 4.7, is (Ω, p + 1)continuously derivable with Ω := I \ N and where N is at most a countable subset of I. It is not hard to realize that x : I → X given by x(t) = α(t)y(t) + ξ(t) is a maximal solution of problem (5), and thanks to Proposition 1, we deduce that it is also (Ω, p + 1)-continuously derivable.…”

Multiplayer games and HIV transmission via casual encounters

“…To solve the VI problem we compute its solution set as the set of critical points of a projected dynamical system (see [2,8]) given by…”

“…It is known that the system (1) is well-defined if F is Lipschitz continuous on K, where K is a closed and convex set. Under these assumptions, solutions to this system exist and are unique through each initial point x(0) ∈ K. A projection-type algorithm can be used to compute its trajectories and its stationary points such as the ones in [2,8]. To answer the uniqueness question, we numerically explore the set of initial conditions of system (1) and study how many (and what values of) Nash strategies we uncover.…”