Bounded-From-Below Solutions of the Hamilton-Jacobi Equation for Optimal Control Problems with Exit Times: Vanishing Lagrangians, Eikonal Equations, and Shape-From-Shading

摘要

We study the Hamilton-Jacobi equation for undiscounted exit time controlproblems with general nonnegative Lagrangians using the dynamic programmingapproach. We prove theorems characterizing the value function as the uniquebounded-from-below viscosity solution of the Hamilton-Jacobi equation which isnull on the target. The result applies to problems with the property that alltrajectories satisfying a certain integral condition must stay in a boundedset. We allow problems for which the Lagrangian is not uniformly bounded belowby positive constants, in which the hypotheses of the known uniqueness resultsfor Hamilton-Jacobi equations are not satisfied. We apply our theorems toeikonal equations from geometric optics, shape-from-shading equations fromimage processing, and variants of the Fuller Problem.