Abstract: | The relationship between optimal control problems and
Hamilton-Jacobi-Bellman equations is well known 9]. In fact
the value function, defined as the infimum of the cost functional,
satisfies in the viscosity sense an appropriate Hamilton-Jacobi-Bellman
equation. In this paper we consider several control problems such that
the cost functional associated to each problem depends explicitly on the
value functions of the other problems. This leads to a system of
Hamilton-Jacobi-Bellman equations. This is known, in economic context
14] cap XI, as an externality problem. In these problems may
occur a lack of uniqueness of the value functions. We give conditions to
ensure existence, uniqueness of the value functions and an implicit
integral representation formula. Moreover, under uniqueness assumption,
we prove that the variational solutions of the associated
Hamilton-Jacobi system converge asymptotically to the value functions.
We prove also an uniqueness theorem in the case of viscosity solutions
of Hamilton-Jacobi-Bellman system. |