Viscosity solutions for the dynamic programming equations
This paper extends previous work on Hamilton-Jacobi equations of the form \(u+H(x,u,Du)=0\), \(u_ t+H(t,x,u,Du)=0\), by adding an unbounded nonlinear term \(\langle Ax,Du\rangle\), so that the above equations become \(u+\langle Ax,Du\rangle+H(x,u,Du)=0\) and \(u_ t+\langle Ax,Du\rangle+H(t,x,u,Du)=0\). Then it is shown that the minimum time problem for an evolution equation of the form \(\dot x+Ax=Bu\), \(x(0)=y\), \(x(\tau)=0\), is a viscosity solution (with respect to \(y\)) if it is continuous in \(y\). Moreover, with a special unbounded nonlinear term, it is shown that the Neumann problem in convex domains may be interpreted as a Hamilton-Jacobi equation with this nonlinear term.
scientific article; zbMATH DE number 39509