Moreau-Yosida regularization of state-dependent sweeping processes with nonregular sets. (English) Zbl 1376.34059
The authors study the state-dependent sweeping process with nonregular moving set
\[
-\mathrm{d}x\in N(C(t,x(t)),x(t)), t\in I\equiv \left[ T_{0,}T\right],
\]
\[ x(T_{0})=x_{0}\in C\left( T_{0},x_{0}\right), \] where \(C:I\times H\rightarrow 2^{H}\) is a set-valued map with nonempty and closed values, \(H\) is a separable Hilbert space, \(N\) is the Clarke normal cone and d\(x\) is the differential vector measure associated with \(x\). Existence of continuous, bounded variation solutions is proven in the sense of differential measures (see [L. Thibault, J. Convex Anal. 23, No. 4, 1051–1098 (2016; Zbl 1360.34032)]). \(C\) is assumed to satisfy a continuity-type assumption and a noncompactness criterion, and these sets are assumed to be equi-uniformly subsmooth. The proof makes use of the Moreau-Yosida regularization (using a result from D. Bothe [Isr. J. Math. 108, 109–138 (1998; Zbl 0922.47048)]) and a reparametrization technique from V. Recupero [J. Differ. Equations 259, No. 8, 4253–4272 (2015; Zbl 1322.49014)]. The results are applied to a topic from hysteresis.
\[ x(T_{0})=x_{0}\in C\left( T_{0},x_{0}\right), \] where \(C:I\times H\rightarrow 2^{H}\) is a set-valued map with nonempty and closed values, \(H\) is a separable Hilbert space, \(N\) is the Clarke normal cone and d\(x\) is the differential vector measure associated with \(x\). Existence of continuous, bounded variation solutions is proven in the sense of differential measures (see [L. Thibault, J. Convex Anal. 23, No. 4, 1051–1098 (2016; Zbl 1360.34032)]). \(C\) is assumed to satisfy a continuity-type assumption and a noncompactness criterion, and these sets are assumed to be equi-uniformly subsmooth. The proof makes use of the Moreau-Yosida regularization (using a result from D. Bothe [Isr. J. Math. 108, 109–138 (1998; Zbl 0922.47048)]) and a reparametrization technique from V. Recupero [J. Differ. Equations 259, No. 8, 4253–4272 (2015; Zbl 1322.49014)]. The results are applied to a topic from hysteresis.
Reviewer: Daniel C. Biles (Nashville)
MSC:
| 34G25 | Evolution inclusions |
| 34C55 | Hysteresis for ordinary differential equations |
| 49J52 | Nonsmooth analysis |
| 49J53 | Set-valued and variational analysis |
Keywords:
sweeping process; normal cone; Moreau-Yosida regularization; measure differential inclusion; Play operatorReferences:
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