Geodesic
Optimal control problems with upper semicontinuous Hamiltonians
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 30 (2010) no. 1, pp. 71-99.

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In this paper we give examples of value functions in Bolza problem that are not bilateral or viscosity solutions and an example of a smooth value function that is even not a classic solution (in particular, it can be neither the viscosity nor the bilateral solution) of Hamilton-Jacobi-Bellman equation with upper semicontinuous Hamiltonian. Good properties of value functions motivate us to introduce approximate solutions of equations with such type Hamiltonians. We show that the value function is the unique approximate solution.
Keywords: Hamilton-Jacobi-Bellman equation, Bolza problem, viscosity solution, bilateral solution, monotonic approximation, semicontinuous Hamiltonian 
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Misztela, Arkadiusz. Optimal control problems with upper semicontinuous Hamiltonians. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 30 (2010) no. 1, pp. 71-99. http://geodesic.mathdoc.fr/item/DMDICO_2010_30_1_a4/

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