A counter-example to the characterization of the discontinuous value function of control problems with reflection (Q1854678)

A Neumann boundary-value problem for a Hamilton-Jacobi equation associated with a relaxed (in the sense of Gamkrelidze) optimal control problem is under consideration. The optimality criterion is the sum of integral and terminal terms, where the terminal component \(\psi\) is a locally bounded, but not necessarily continuous function. It is obtained that the lower-semicontinuous viscosity supersolution \(v\) and upper-semicontinuous viscosity subsolution \(w\) satisfy the estimates \[ \hat{u}\left[\psi_* \right]\leq v \text{ and }u\left[\psi^*\right]\geq w, \] where \(u\) and \(\hat{u}\) are the value functions of the original and relaxed optimization problems; \(\psi_*\) and \(\psi^*\) denote, respectively, the upper- and lower-semicontinuous envelopes of the function \(\psi\). It is known for the class of problems under consideration that in case of a continuous function \(\psi\), the value function is the unique solution of the corresponding Hamilton-Jacobi equation. At the same time, a control problem with reflection is presented, where a strict inequality \(\hat{u}\left[\psi_* \right]<u_*\left[\psi^*\right]\) takes place.