[Paper Review] Infinite Dimensional Control Problems with Positivity State Constraints: a Banach Lattice Approach

Alessandro Calvia, Salvatore Federico+1 more|arXiv (Cornell University)|Sep 23, 2020

Stochastic processes and financial applications23 references1 citations

TL;DR

This paper introduces a Banach lattice framework for infinite-dimensional optimal control problems with positivity state constraints, leveraging an infinite-dimensional Perron-Frobenius theorem to explicitly solve the HJB equation for an auxiliary problem and thereby derive optimal trajectories for the original problem—offering a novel approach beyond traditional L² settings.

ABSTRACT

This paper is devoted to studying a family of deterministic optimal control problems in an infinite dimension. The difficult feature of such problems is the presence of positivity state constraints, which arise very often in economic applications (our main motivation). To deal with such constraints we set up the problem in a Banach space with a Riesz space structure (i.e., a Banach lattice) and not necessarily reflexive, like $\mathcal{C}^0$. In this setting, which seems to be new in this context, we are able, using a type of infinite-dimensional Perron-Frobenius Theorem, to find explicit solutions of the HJB equation associated to a suitable auxiliary problem and to use such results to get information about the optimal paths of the starting problem. This was not possible to perform in the previously used infinite-dimensional setting where the state space was an $\mathrm{L}^2$ space.

Motivation & Objective

• To address the challenge of positivity state constraints in infinite-dimensional optimal control problems, common in economic applications.
• To develop a new mathematical framework using non-reflexive Banach lattices, such as C⁰, instead of L² spaces.
• To establish a connection between the structure of the state space and solvability of the Hamilton-Jacobi-Bellman (HJB) equation.
• To derive explicit solutions for the HJB equation of an auxiliary problem using spectral-theoretic tools.
• To use these solutions to infer optimal control paths for the original constrained control problem.

Proposed method

• Formulate the control problem in a Banach lattice space with a Riesz space structure, enabling order-theoretic analysis.
• Apply an infinite-dimensional version of the Perron-Frobenius theorem to analyze the spectral properties of the linear operator in the HJB equation.
• Construct an auxiliary optimal control problem whose HJB equation admits explicit solution due to the lattice structure.
• Use the solution of the auxiliary problem to characterize the value function and optimal trajectories of the original problem.
• Leverage the order structure of the Banach lattice to preserve positivity constraints throughout the solution process.
• Establish the link between the auxiliary problem’s solution and the optimal paths of the original problem via duality and comparison principles.

Experimental results

Research questions

• RQ1 Can positivity constraints in infinite-dimensional optimal control problems be effectively managed using a Banach lattice framework rather than L² spaces?
• RQ2 How can an infinite-dimensional Perron-Frobenius theorem be applied to solve the HJB equation in such settings?
• RQ3 What structural properties of the state space enable explicit solution of the HJB equation in the presence of constraints?
• RQ4 In what way does the Banach lattice structure improve the solvability of control problems compared to standard L² formulations?
• RQ5 How can solutions of an auxiliary problem inform the optimal control paths of the original constrained problem?

Key findings

• The use of a Banach lattice structure enables the explicit solution of the HJB equation for an auxiliary problem, which was not feasible in previous L²-based frameworks.
• The infinite-dimensional Perron-Frobenius theorem provides a key analytical tool to characterize the principal eigenfunction and associated solution structure.
• Optimal control paths for the original problem are derived by leveraging the solution of the auxiliary problem through order-theoretic and comparison arguments.
• The framework successfully handles positivity state constraints without requiring reflexivity of the state space, extending applicability to C⁰-type spaces.
• The approach provides a constructive method for identifying optimal trajectories in settings where standard variational techniques fail due to lack of smoothness or reflexivity.
• The results demonstrate that the lattice structure of the state space is instrumental in unlocking explicit solutions where they were previously out of reach.