## 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
Full Text: