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On differential variational inequalities and projected dynamical systems – equivalence and a stability result. (English) Zbl 1163.34375
Summary: The purpose of this note is two-fold. Firstly we deal with projected dynamical systems that recently have been introduced and investigated in finite dimensions to treat various time dependent (dis)equilibrium and network problems in operations research. Here at the more general level of a Hilbert space, we show that a projected dynamical system is equivalent in finding the “slow” solution (the solution of minimal norm) of a differential variational in- equality, a class of evolution inclusions studied much earlier. This equivalence follows easily from a precise geometric description of the directional derivative of the metric projection in Hilbert space. By our approach, we can easily characterize a stationary point of a projected dynamical system as a solution of a related variational inequality. Secondly we are concerned with stability of the solution set to differential variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data. In particular, we admit perturbations of the associated set-valued maps and the constraint set, where we impose weak convergence assumptions on the perturbed set-valued maps and employ Mosco convergence as set convergence.
MSC:
34G25Evolution inclusions
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35K85Linear parabolic unilateral problems; linear parabolic variational inequalities
49J53Set-valued and variational analysis
49J45Optimal control problems involving semicontinuity and convergence; relaxation