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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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