  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 which 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.
