English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study a class of optimal control problems with state constraints where thestate equation is a differential equation with delays. This class includes someproblems arising in economics, in particular the so-called models with time tobuild. We embed the problem in a suitable Hilbert space H and consider theassociated Hamilton-Jacobi-Bellman (HJB) equation. This kind ofinfinite-dimensional HJB equation has not been previously studied and isdifficult due to the presence of state constraints and the lack of smoothingproperties of the state equation. Our main result on the regularity ofsolutions to such a HJB equation seems to be completely new. More precisely weprove 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 hascontinuous classical derivative in the direction of the present. Thisregularity result is the starting point to define a feedback map in classicalsense, which gives rise to a candidate optimal feedback strategy for theproblem. The study of verification theorems and of the closed loop equationwill be the subject of a forthcoming paper.

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