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minimum time problem

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Degenerate Eikonal equations with discontinuous refraction index
Pierpaolo Soravia
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 12 / Issue 2 / April 2006

Published online by Cambridge University Press:

22 March 2006, pp. 216-230

Print publication:

April 2006
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We study the Dirichlet boundary value problem for eikonal type equations of ray light propagation in an inhomogeneous medium with discontinuous refraction index. We prove a comparison principle that allows us to obtain existence and uniqueness of a continuous viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value function of the generalized minimum time problem with discontinuous running cost.