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29 March 2024
 
  » arxiv » math.OC/0311269

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Bounded-From-Below Solutions of the Hamilton-Jacobi Equation for Optimal Control Problems with Exit Times: Vanishing Lagrangians, Eikonal Equations, and Shape-From-Shading
Michael Malisoff ;
Date 16 Nov 2003
Journal NoDEA Nonlinear Differential Equations and Applications, Volume 11, Number 1, pp. 95-122, February 2004 DOI: 10.1007/s00030-003-1051-8
Subject Optimization and Control MSC-class: 35F20, 49L25 | math.OC
AbstractWe 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.
Source arXiv, math.OC/0311269
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