On The Uniqueness Of Solutions To One-Dimensional Constrained Hamilton-Jacobi Equations

The goal of this paper is to study the uniqueness of solutions to a constrained Hamilton-Jacobi equation{u(t) = u(x)(2) + R(x, I(t)) in R x (0,infinity),max(R) u(center dot, t) = 0 on [0,infinity),with an initial condition u(x, 0) = u(0)(x) on R. A reaction term R(x, I(t)) is given while I(t) is an unknown constraint (Lagrange multiplier) that forces maximum of u to be always zero. In the paper, we prove uniqueness of a pair of unknowns (u, I) using dynamic programming principle for a particular class of non-separable reaction R(x, I(t)) when the space is one-dimensional.