This paper aims to study a family of deterministic optimal control problems in infinite-dimensional spaces. The peculiar feature of such problems is the presence of a positivity state constraint, which often arises in economic applications. To deal with such constraints, we set up the problem in a Banach lattice, not necessarily reflexive: a typical example is the space of continuous functions on a compact set. In this setting, which seems to be new in this context, we are able to find explicit solutions to the Hamilton--Jacobi--Bellman (HJB) equation associated to a suitable auxiliary problem and to write the corresponding optimal feedback control. Thanks to a type of infinite-dimensional Perron--Frobenius theorem, we use these results to gain information about the optimal paths of the original problem. This was not possible in the infinite-dimensional setting used in earlier works on this subject, where the state space was an L2 space.

State Constrained Control Problems in Banach Lattices and Applications / Calvia, Alessandro; Federico, Salvatore; Gozzi, Fausto. - In: SIAM JOURNAL ON CONTROL AND OPTIMIZATION. - ISSN 0363-0129. - 59:6(2021), pp. 4481-4510. [10.1137/20M1376959]

State Constrained Control Problems in Banach Lattices and Applications

Calvia, Alessandro
;
Gozzi, Fausto
2021

Abstract

This paper aims to study a family of deterministic optimal control problems in infinite-dimensional spaces. The peculiar feature of such problems is the presence of a positivity state constraint, which often arises in economic applications. To deal with such constraints, we set up the problem in a Banach lattice, not necessarily reflexive: a typical example is the space of continuous functions on a compact set. In this setting, which seems to be new in this context, we are able to find explicit solutions to the Hamilton--Jacobi--Bellman (HJB) equation associated to a suitable auxiliary problem and to write the corresponding optimal feedback control. Thanks to a type of infinite-dimensional Perron--Frobenius theorem, we use these results to gain information about the optimal paths of the original problem. This was not possible in the infinite-dimensional setting used in earlier works on this subject, where the state space was an L2 space.
2021
Optimal control in infinite dimension, dynamic programming, state constraints, Hamilton--Jacobi--Bellman equation, Banach lattice, AK model of economic growth
State Constrained Control Problems in Banach Lattices and Applications / Calvia, Alessandro; Federico, Salvatore; Gozzi, Fausto. - In: SIAM JOURNAL ON CONTROL AND OPTIMIZATION. - ISSN 0363-0129. - 59:6(2021), pp. 4481-4510. [10.1137/20M1376959]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11385/212196
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