Hölder estimates for trajectories of differential inclusions and HJB equations with state constraints (Q354381)

It is considered the differential inclusion \[ \dot{x}\in F(t,x), \tag{1} \] together with the state constraint \[ x(t)\in\Omega,\quad t\in [0,T], \tag{2} \] where \(F(t,x)\subset \mathbb{R}^n \) is a closed and nonempty compact set for every \((t,x)\in [0,T]\times\mathbb{R}^n\), \(F(\cdot,x)\) is measurable for every \(x\in\mathbb{R}^n\), \(F(t,\cdot)\) is Lipschitz continuous. The following problem is investigated: Suppose \(x^*(\cdot)\) is a given Carathéodory solution of (1),(2). Is there an \(\varepsilon_{0} >0\) such that for all \(x_{0}\in \mathrm{Int}(\Omega)\), \(|x_{0}-x^{*}(0)|<\varepsilon_{0}\), there exists a solution \(x(\cdot)\) of (1) satisfying \(x(0) =x_{0}\) such that \(\|x-x^{*}\|_{L^{\infty}([0,T])}\leq \omega(\varepsilon),\) where \(\varepsilon=|x_{0}-x^{*}(0)|\) and \(\omega:[0,\infty)\to [0,\infty)\) is such that \[ \lim_{\varepsilon\to 0}\omega(\varepsilon)=\omega(0)=0? \] Under some assumptions the existence of such solutions of (1) is proved.