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.


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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