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Deterministic state-constrained optimal control problems without controllability assumptions

In the setting of optimal control problems of ordinary differential equations with state constraints, the paper aims at characterizing the value function using the Hamilton-Jacobi-Bellamn (HJB) equation. The answer is known when one assumes either the inward or the outward pointing qualification condition. In the present paper no controllability assumption is done. The value function can be discontinuous and the problem may be unfeasible, resulting in an infinite value function.

A first reduction consists in mapping the real line to a bounded interval, after which to an unfeasible problem is associated the value say 1. It appears then that the value function is the unique solution of the HJB equation with some elaborated boundary conditions. In order to get a more practical characterization (especially when having in view numerical methods), the authors consider the HJB equation on a slightly enlarged domain, with 'classical' boundary conditions. They show that any l.s.c. bounded viscosity solution of the latter converges punctually to the original value function.

Deterministic state-constrained optimal control problems without controllability assumptions
Olivier Bokanowski, Nicolas Forcadel and Hasnaa Zidani
ESAIM: COCV 17 (2011) 995-1015
http://dx.doi.org/10.1051/cocv/2010030