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Sweeping process with right uniformly lower semicontinuous mappings. (English) Zbl 1437.34068

The authors consider the sweeping process \[ -\dot{x}(t)\in N^{P}\left( C(t),x(t)\right) \text{ a.e. on }\left[T_{0},T\right], \] \[ x\left( T_{0}\right) =x_{0}\in C\left( T_{0}\right) \] where \(N^{P}\) denotes the proximal normal cone in an infinite-dimensional separable Hilbert space. \(C\) is assumed to be prox regular and closed set valued, and uniformly lower semicontinuous from the right. Existence and uniqueness of solutions are proven by constructing a sequence of approximate solutions using a discretization scheme from [L. Thibault, J. Convex Anal. 23, No. 4, 1051–1098 (2016; Zbl 1360.34032)]. Next, as a corollary, the authors prove existence and uniqueness for the problem with a single-valued, bounded and measurable perturbation term. Next, the existence of solutions is proven for a set-valued perturbation which is closed and convex valued, and is scalarly upper semicontinuous with respect to both variables. Finally, using an idea from [J. Noel and L. Thibault, Vietnam J. Math. 42, No. 4, 595–612 (2014; Zbl 1315.34068)], this last result is extended to one which holds on an unbounded interval.

MSC:

34G25 Evolution inclusions
49J52 Nonsmooth analysis
49J53 Set-valued and variational analysis
34D10 Perturbations of ordinary differential equations
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