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Existence results of nonconvex differential inclusions. (English) Zbl 1022.34007

The author investigates the existence of solutions to nonconvex-valued differential inclusions. In the first part of the paper, he proves that, for a large class of set-valued maps, the solution sets of the following sweeping processes \[ \left\{\begin{aligned} &\dot x(t)\in-N^C(C(t);x(t)), \text{ a.e. } t\geq 0,\\ &x(0)=x_0\in C(0),\\ &x(t)\in C(t), \quad \forall t\geq 0, \end{aligned}\right. \] and \[ \left\{\begin{aligned} &\dot x(t)\in-|\dot v(t)|\partial^Cd_{C(t)}(x(t)), \text{ a.e. }t\geq 0,\\ &x(0)=x_0\in C(0), \end{aligned}\right. \] are the same. Here, \(C\) and \(v\) are absolutely continuous functions taking values in a Hilbert space \(H,\) \(N^C(C(t);x(t))\) denotes a prescribed normal cone to the set \(C(t)\) at \(x(t),\) and \(\partial^Cd_{C(t)}\) stands for a prescribed subdifferential of the distance function \(d_{C(t)}\) associated with the set \(C(t).\) In the second part of the paper, the following class of differential inclusions is considered \[ \dot x(t)\in G(x(t))+ F(t,x(t)),\text{ a.e. } t\in [0,T], \] where \(F: [0,T]\times H\to H\) is a continuous set-valued mapping and \(G: H\to H\) is an upper semicontinuous set-valued mapping such that \(G(x)\subset \partial^C g(x),\) the Clarke subdifferential of \(g\) at \(x.\) Assuming that \(g\) is locally Lipschitz and uniformly regular, the existence of viable solutions is proved.

MSC:

34A60 Ordinary differential inclusions
34G25 Evolution inclusions
49J53 Set-valued and variational analysis
49J52 Nonsmooth analysis
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