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.


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