A characterization of the value function for a class of degenerate control problems (Q2776390)

In this paper the author considers the Hamilton-Jacobi-Bellman (HJB) equation related to the optimal control problem minimizing the cost functional NEWLINE\[NEWLINEJ(x,q)=\int_0^{\tau(x,q)}f(\xi(t),q(t)) dt + g(\xi(\tau(x,q)))NEWLINE\]NEWLINE over the controls \(q\), where \(\tau(x,q)\) is the first exit time from some domain \(\Omega\). If \(f\) vanishes on parts of \(\Omega\) then a unique viscosity solution to this HJB equation will not exist. NEWLINENEWLINENEWLINEUsing the concept of singular viscosity solutions from \textit{F. Camilli} and \textit{A. Siconolfi} [``Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems'', Indiana Univ. Math. J. 48, No. 3, 1111-1131 (1999; Zbl 0939.49019)] it is shown that within this framework the HJB equation does indeed have a unique solution which coincides with the optimal value function of the related optimal control problem. NEWLINENEWLINENEWLINEAs an application, two approximation techniques, one based on a discretization method and the other based on vanishing viscosity methods are shown to converge to the right solution provided the parameters involved in these approximations are adjusted properly.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00022].