## HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints: Regularity and Applications

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Federico, Salvatore

Goldys, Ben

Gozzi, Fausto

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We study a class of optimal control problems with state constraints where the
state equation is a differential equation with delays. This class includes some
problems arising in economics, in particular the so-called models with time to
build. We embed the problem in a suitable Hilbert space H and consider the
associated Hamilton-Jacobi-Bellman (HJB) equation. This kind of
infinite-dimensional HJB equation has not been previously studied and is
difficult due to the presence of state constraints and the lack of smoothing
properties of the state equation. Our main result on the regularity of
solutions to such a HJB equation seems to be completely new. More precisely we
prove that the value function is continuous in a sufficiently big open set of H
, that it solves in the viscosity sense the associated HJB equation and it has
continuous classical derivative in the direction of the present. This
regularity result is the starting point to define a feedback map in classical
sense, which gives rise to a candidate optimal feedback strategy for the
problem. The study of verification theorems and of the closed loop equation
will be the subject of a forthcoming paper.

Comment: The old paper was split in two differnet papers. This is the first part (24 pages)

Comment: The old paper was split in two differnet papers. This is the first part (24 pages)

##### Keywords

Mathematics - Optimization and Control, 49J25, 49L25