Geodesic
On a~solution to the Cauchy problem for the Hamilton--Jacobi equation with state constraints
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 2, pp. 191-208

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The Cauchy problem for the Hamilton–Jacobi equation, which appears in molecular biology for the Crow–Kimura model of molecular evolution, is considered. The notion of a continuous generalized solution of this problem with state constraints is introduced on the basis of the viscosity (and/or minimax) approach. A construction of the generalized solution of the problem is proposed, which uses the value function in an auxiliary optimal control problem with a given target set. It is shown that the generalized solution in the considered problem is not unique. The research is based on the generalized method of characteristics for the Hamilton–Jacobi equations in the Dirichlet problem.
Keywords: Hamilton–Jacobi equations, method of characteristics, viscosity solutions, minimax solutions, optimal control, value function. 
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     author = {N. N. Subbotina and L. G. Shagalova},
     title = {On a~solution to the {Cauchy} problem for the {Hamilton--Jacobi} equation with state constraints},
     journal = {Trudy Instituta matematiki i mehaniki},
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N. N. Subbotina; L. G. Shagalova. On a~solution to the Cauchy problem for the Hamilton--Jacobi equation with state constraints. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 2, pp. 191-208. http://geodesic.mathdoc.fr/item/TIMM_2011_17_2_a16/