The Hamilton-Jacobi-Bellman equation with a gradient constraint
This paper deals with the existence and uniqueness of viscosity solutions for the Hamilton-Jacobi-Bellman equation \[ \max \{L^ 1u-f^ 1,...,L^ mu-f^ m,| Du| -g\}=0\quad in\quad \Omega;\quad u|_{\partial \Omega}=0. \] Here \(L^ p\), \(p=1,...,m\) are uniformly elliptic operators with smooth coefficients. For smooth \(f^ p\geq 0\), \(p=1,...,m\), and \(g\geq 0\), it is proved that there exists a viscosity solution which belongs to \(W^{2,\infty}_{loc}(\Omega)\cap W^{1,\infty}(\Omega)\) and it is unique in the class \(C^ 1(\Omega)\cap C({\bar \Omega})\) if \(g>0\). The solution is obtained as a limit of the system of approximate equations: \[ L^ p u^ p_{\epsilon}+\beta_{\epsilon}(| Du^ p_{\epsilon}|^ 2-g^ 2)+\gamma_{\epsilon}(u^ p_{\epsilon}- u_{\epsilon}^{p+1})=f^ p\quad in\quad \Omega, \] \[ u^ p_{\epsilon}|_{\partial \Omega}=0,\quad p=1,...,m,\quad u_{\epsilon}^{m+1}=u^ 1_{\epsilon}, \] where \(\beta_{\epsilon}\) and \(\gamma_{\epsilon}\) are standard penalty functions. The uniqueness proof is carried out by using this approximation again. The uniqueness of the solution in the class \(C(\Omega)\) is also considered and the author proves the affirmative results (1) for the case \(m=1\) of the above equation, (2) for the minimax equation \[ \min \{\max \{-\Delta u+u-f,u-\psi_ 1\},u-\psi_ 2\}=0\quad in\quad \Omega;\quad u|_{\partial \Omega}=0. \]
scientific article; zbMATH DE number 4086157
scientific article