About least-squares type approach to address direct and controllability problems

TL;DR

This paper presents a least-squares approach to approximate null controls for PDEs, notably the Stokes system, avoiding duality methods and ensuring strong convergence of control sequences.

Contribution

It introduces a novel least-squares formulation for controllability problems that guarantees convergence without relying on duality, applicable to systems like Stokes and Navier-Stokes.

Findings

Constructs strong convergent sequences of controls for Stokes system.

Avoids ill-posedness of dual methods by not using duality arguments.

Provides a framework for approximating controls in PDE systems.

Abstract

- We discuss the approximation of distributed null controls for partial differential equations. The main purpose is to determine an approximation of controls that drives the solution from a prescribed initial state at the initial time to the zero target at a prescribed final time. As a non trivial example, we mainly focus on the Stokes system for which the existence of square-integrable controls have been obtained in [Fursikov \& Imanuvilov, Controllability of Evolution Equations, 1996]) via Carleman type estimates. We introduce a least-squares formulation of the controllability problem, and we show that it allows the construction of strong convergent sequences of functions toward null controls for the Stokes system. The approach consists first in introducing a class of functions satisfying a priori the boundary conditions in space and time-in particular the null controllability condition at time T-, and then finding among this class one element satisfying the system. This second step is done by minimizing a quadratic functional, among the admissible corrector functions of the Stokes system. We also discuss briefly the direct problem for the steady Navier-Stokes system. The method does not make use of any duality arguments and therefore avoid the ill-posedness of dual methods, when parabolic type equation are considered.