[Paper Review] Some applications of weighted norm inequalities to the analysis of optimal control problems

Harbir Antil, Enrique Otárola+1 more|arXiv (Cornell University)|May 14, 2015

Advanced Numerical Methods in Computational Mathematics45 references3 citations

TL;DR

This paper applies Muckenhoupt weights, weighted Sobolev spaces, and weighted norm inequalities to simplify the Hilbert space-based analysis and numerical discretization of optimal control problems involving nonuniformly elliptic equations and pointwise tracking objectives. It demonstrates that this framework enables more straightforward convergence analysis and generalization compared to prior approaches.

ABSTRACT

The purpose of this work is to illustrate how the theory of Muckenhoupt weights, weighted Sobolev spaces and weighted norm inequalities can be used in the analysis and discretization of optimal control problems. We consider: a linear quadratic optimal control problem where the state solves a nonuniformly elliptic equation; a problem with a pointwise tracking objective and one where the control is the amplitude of forces, modeled as point masses. For all three examples we propose and analyze numerical schemes. While some of these problems might have been considered before in the literature, our approach allows for a simpler, Hilbert space-based, analysis and discretization and further generalizations.

Motivation & Objective

• To develop a Hilbert space-based framework for analyzing optimal control problems with nonuniformly elliptic PDEs.
• To simplify the analysis and discretization of control problems involving pointwise tracking and control as point masses.
• To extend the applicability of weighted norm inequalities to optimal control settings for improved convergence and generalization.
• To provide a unified approach that generalizes prior results with cleaner theoretical foundations.

Proposed method

• Utilizes Muckenhoupt weights to handle nonuniform ellipticity in the state equation.
• Applies weighted Sobolev spaces to define function spaces with appropriate integrability and regularity for the state and adjoint variables.
• Employs weighted norm inequalities to control error estimates in the analysis of numerical schemes.
• Derives stability and convergence results for Galerkin-type discretizations using the weighted framework.
• Designs and analyzes numerical schemes for three distinct control problems: tracking with nonuniform ellipticity, pointwise tracking, and control via point masses.
• Establishes equivalence between variational formulations and weak solutions in weighted spaces, enabling robust discretization.

Experimental results

Research questions

• RQ1 How can weighted norm inequalities be systematically applied to analyze the convergence of numerical schemes in optimal control problems with nonuniformly elliptic operators?
• RQ2 Can the use of weighted Sobolev spaces lead to a simpler Hilbert space-based analysis compared to classical methods?
• RQ3 What is the role of Muckenhoupt weights in stabilizing the solution and discretization of control problems with irregular coefficients?
• RQ4 How does the proposed framework generalize existing results for pointwise tracking and control via point masses?
• RQ5 To what extent can the weighted framework improve the robustness and convergence of finite element discretizations?

Key findings

• The use of Muckenhoupt weights enables a more natural and simplified Hilbert space analysis for optimal control problems with nonuniformly elliptic PDEs.
• Weighted Sobolev spaces provide the appropriate functional setting to handle irregular coefficients and ensure well-posedness of the state and adjoint equations.
• Weighted norm inequalities lead to sharper and more robust error estimates in the numerical discretization of the control problems.
• The proposed numerical schemes converge under weaker assumptions than classical approaches, enhancing their applicability.
• The framework allows for straightforward generalization to problems with pointwise tracking and control as point masses, with minimal modification.
• The analysis reveals that the weighted approach unifies the treatment of different control problem types under a common theoretical foundation.