We study the Hamilton-Jacobi equation for undiscounted exit time control problems with general nonnegative Lagrangians using the dynamic programming approach. We prove theorems characterizing the value function as the unique bounded-from-below viscosity solution of the Hamilton-Jacobi equation that is null on the target. The result applies to problems with the property that all trajectories satisfying a certain integral condition must stay in a bounded set. We allow problems for which the Lagrangian is not uniformly bounded below by positive constants, in which the hypotheses of the known uniqueness results for Hamilton-Jacobi equations are not satisfied. We apply our theorems to eikonal equations from geometric optics, shape-from-shading equations from image processing, and variants of the Fuller Problem.
Publication Source (Journal or Book title)
Nonlinear Differential Equations and Applications
Malisoff, M. (2004). Bounded-from-below solutions of the hamilton-Jacobi equation for optimal control problems with exit times: Vanishing lagrangians, eikonal equations, and shape-from-shading. Nonlinear Differential Equations and Applications, 11 (1), 95-122. https://doi.org/10.1007/s00030-003-1051-8