Moreau-Yosida regularization of degenerate state-dependent sweeping processes (Q2139288)

The authors study the existence of Lipschitz solutions of the degenerate state-dependent sweeping process \[ -\dot{x}(t)\in N_{C\left( t,x\left( t\right) \right) }\left( A\left( x\left( t\right) \right) \right) \text{ a.e. on }\left[ 0,T\right], \] \[ x(0)=x_{0}\text{,} \] in a separable Hilbert space. The ``degeneracy'' is the addition of an operator inside the sweeping process, which has been proposed as a model for quasistatic elastoplasticity. The moving sets \(C\) are assumed to be Lipschitz with respect to the truncated Hausdorff distance, which allows for wider application than the use of the Hausdorff distance. An example is given of a moving set that is Lipschitz with respect to the truncated Hausdoff distance but not Lipschitz with respect to Hausdorff distance. The moving sets are also assumed to be subsmooth and positively \(\alpha\)-far (introduced in [\textit{T. Haddad} et al., Pac. J. Optim. 4, No. 3, 493--512 (2008; Zbl 1185.34018)]). The proof is accomplished through the adaptation of techniques from [\textit{A. Jourani} and \textit{E. Vilches}, J. Optim. Theory Appl. 173, No. 1, 91--116 (2017; Zbl 1376.34059)] and use of the Moreau-Yosida regularization technique.