A simple planning problem for COVID-19 lockdown: a dynamic programming approach

TL;DR

This paper applies dynamic programming to analyze a COVID-19 lockdown planning problem modeled with a non-convex optimal control framework, providing theoretical insights into the value function and optimality conditions.

Contribution

It introduces a dynamic programming approach to a non-convex COVID-19 control problem, proving continuity of the value function and analyzing the Hamilton-Jacobi-Bellman equation.

Findings

Proved continuity properties of the value function.

Demonstrated the value function solves the HJB equation in viscosity sense.

Discussed conditions for optimality in non-convex control problems.

Abstract

A large number of recent studies consider a compartmental SIR model to study optimal control policies aimed at containing the diffusion of COVID-19 while minimizing the economic costs of preventive measures. Such problems are non-convex and standard results need not to hold. We use a Dynamic Programming approach and prove some continuity properties of the value function of the associated optimization problem. We study the corresponding Hamilton-Jacobi-Bellman equation and show that the value function solves it in the viscosity sense. Finally, we discuss some optimality conditions. Our paper represents a first contribution towards a complete analysis of non-convex dynamic optimization problems, within a Dynamic Programming approach.