Hamilton-Jacobi-Bellman equation under states constraints
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.
scientific article; zbMATH DE number 1441387
scientific article