Dynamical systems and variational inequalities

Projected dynamical systems in a complementarity formalism

“…Specializing to the stationary points, we obtain as a corollary the classical result which states that, under mild conditions, variational inequalities may be rewritten as mixed nonlinear complementarity problems [8,Proposition 2.2]. Moreover, we obtain a proof of existence and uniqueness of solutions of projected dynamical systems that is independent of the original proof by Dupuis and Nagurney [6] and in particular does not use the Skorokhod problem (see [28]). Complementarity systems have already been used extensively in the engineering literature (see, for instance, [15,19,25]) and the establishment of a relation between the domains of projected dynamical systems and of complementarity systems makes it possible to compare and transfer analytic and computational techniques between the two.…”

“…In [6] the same result is stated under the assumption that K is a convex polyhedron (i.e. an intersection of finitely many closed half-spaces).…”

“…In this section we recall the definition of projected dynamical systems (PDS) [6,18]. The defining ingredients are a closed convex set K, which usually corresponds to the constraint set of a particular application, and a vector field F whose domain contains K. The projected dynamics is described by the equation x(t)= -F(x(t)) on the interior of K, but on the boundary a modification is applied to prevent the solution from leaving the constraint set.…”

“…In [6] it is even assumed that K is a convex polyhedron. Assumptions 4.4 and 4.5 are used in [18] to prove existence and uniqueness of solutions to the projected dynamical system specified by F andK.…”

“…One is the class of projected dynamical systems introduced by Dupuis and Nagurney [ 6) and further developed by Nagurney and Zhang [ 18). These systems are described by differential equations of the form…”