## Filippov’s and Filippov-Ważewski’s theorems on closed domains.(English)Zbl 0956.34012

The authors extend the known Filippov and Filippov-Ważewski theorems for differential inclusions to the case when the state variable $$x$$ is constrained to the possibly nonsmooth closure of an open subset in $$\mathbb{R}^n$$. They consider Cauchy problems of the form $x'\in F(t, x),\quad x(t_0)= x_1,\quad x(t)\in\overline\Theta,\tag{1}$ where $$\Theta\subset \mathbb{R}^n$$ is an open subset and, in contrast to other authors, a time interval is unbounded, here. More precisely, the authors extend the following statements:
1. There exists a constant $$L>0$$ such that, for any pair of initial conditions $$x_1,x_2\in \mathbb{R}^n$$ and any solution $$x_1(\cdot)$$ to $x'\in F(t,x),\quad x(t_0)= x_1,\tag{2}$ defined on $$[t_0, T]$$, there is a solution $$x_2(\cdot)$$ to (2) with $$x_1$$ replaced by $$x_2$$ such that $\sup_{t\in [t_0,T]}|x_1(t)- x_2(t)|+ \int^T_{t_0}|x_1'(t)- x_2'(t)|dt\leq L|x_1- x_2|.$ 2. For any trajectory $$x(\cdot)$$ to (2) with $$F(t,x)$$ replaced by $$\overline{\text{co}} F(t,x)$$ defined on $$[t_0, T]$$ and for any $$\varepsilon> 0$$, there exists a trajectory $$x_\varepsilon(\cdot)$$ to (2) such that $\sup_{t\in [t_0,T]}|x(t)- x_\varepsilon(t)|< \varepsilon.$ Let us note that in the case when $$\Theta$$ is nonsmooth just a $$C^0$$ estimate is proved in the first type assertion. Applications to the study of regularity of value functions of optimal control problems with state constraints are discussed, too.

### MSC:

 34A60 Ordinary differential inclusions 49J24 Optimal control problems with differential inclusions (existence) (MSC2000) 34H05 Control problems involving ordinary differential equations 93C15 Control/observation systems governed by ordinary differential equations
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