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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.Comment: 29 pages, 0 figures, accepted for publication in NoDEA Nonlinear  Differential Equations and Applications on July 29, 200

nonlinear differential equations and applications

Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing lagrangians, eikonal equations, and shape-from-shading