Minimal time function and viscosity solutions
Two theorems of \textit{O. Hàjek} [Funkc. Ekvacioj, Ser. Int. 20, 97-114 (1978; Zbl 0371.49003)] are generalized. It is proved that, if \(V(A,\Gamma)\) is the set of points which can be steered to the origin along a solution of the control system \(x'=Ax-c\), if c(t)\(\in \Gamma\), \(\Gamma\) is a compact subset of R n, \(0\in intrelco \Gamma\), and if a rank condition holds, then the minimal time function T(.) is a viscosity solution of the Bellman equation \[ \max \{<DT(x),\gamma -Ax>: \gamma \in co \Gamma \}-1=0,\quad x\in V(A,\Gamma)\setminus \{0\}, \] and of the Hàjek equation \[ 1-\max \{<DT(x),\exp [-AT(x)]>: \gamma \in co \Gamma \}=0,\quad x\in V(A,\Gamma). \]
scientific article; zbMATH DE number 4032476