##
**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.

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.

Reviewer: I.Rachůnková (Olomouc)

### 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 |

### Keywords:

state constraints; existence of neighbouring trajectories relaxation; value functions; differential inclusions; Cauchy problems; optimal control
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\textit{H. Frankowska} and \textit{F. Rampazzo}, J. Differ. Equations 161, No. 2, 449--478 (2000; Zbl 0956.34012)

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