Hamilton-Jacobi-Bellman equation under states constraints (Q1973945)

This paper is concerned with the uniqueness of discontinuous solutions of the Hamilton-Jacobi-Bellman equation \[ \begin{cases} -{\partial V\over\partial t} (t,x)+ H(t,x,{\partial V\over\partial x}(t, x))= 0,\\ V(t,x)= \psi(x)\text{ when }g(T,x)\leq 0,\end{cases} \] where \(H(t,x,p)= \sup_{v\in F(t,x)}\langle p,v\rangle\), arising in Mayer's problem under state constraints \[ \min\left\{\psi(x(T))\left|\begin{aligned} & x'(t)\in F(t,x(t))\text{ a.e. in }[0,T],\\ & g(t,x(t))\leq 0,\quad t\in[0,T],\\ & x(0)= \xi_0,\\ & x(\cdot)\in AC(0,T),\end{aligned}\right. \right\}, \] where \(T> 0\), \(F:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n\), \(g: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}\), \(\xi_0\in \mathbb{R}^n\), \(\psi: \mathbb{R}^n\to \mathbb{R}\cup\{+ \infty\}\). The author used two types of discontinuous solutions, bilateral solutions and contingent solutions, and showed that they coincide with the value function \(V\) for state constraints, that are backward invariant and forward viable.