Optimal control of diffusion equation with missing data governed by Dirichlet fractional Laplacian

TL;DR

This paper addresses an optimal control problem for a diffusion equation involving a fractional Laplacian with missing data, utilizing no-regret and low-regret control methods to handle uncertainties in initial conditions.

Contribution

It introduces a novel approach to control diffusion equations with missing data governed by fractional Laplacians using Lions' no-regret and low-regret control techniques.

Findings

Characterization of optimal control using no-regret and low-regret frameworks.

Handling of uncertainties in initial conditions within infinite-dimensional spaces.

Application to diffusion equations with fractional Laplacian and Dirichlet boundary conditions.

Abstract

We consider an optimal control problem of diffusion equation with missing data governed by the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary interaction domain disjoint from the domain of the state equation. We assume that the unknown initial condition belongs to an appropriate space of infinite dimension, the so-called space of uncertainties. The key tools we used in order to characterize the optimal control is the no-regret and low-regret control developed by J.L Lions.