[Paper Review] Control-constrained parabolic optimal control problems on evolving surfaces - theory and variational discretization

Morten Vierling|arXiv (Cornell University)|Jun 3, 2011

Advanced Numerical Methods in Computational Mathematics33 references3 citations

TL;DR

This paper establishes the existence and uniqueness of weak solutions for control-constrained linear-quadratic optimal control problems on evolving hypersurfaces in R^{n+1}, using vector-valued distributions and a variational discretization approach. It proves optimal convergence rates for fully discrete approximations under minimal regularity assumptions, validated through numerical examples with observed experimental orders of convergence near 2 for L² errors and 2.0 for L∞ errors.

ABSTRACT

We consider control-constrained linear-quadratic optimal control problems on evolving surfaces. In order to formulate well-posed problems, we prove existence and uniqueness of weak solutions for the state equation, in the sense of vector-valued distributions. We then carry out and prove convergence of the variational discretization of a distributed optimal control problem. In the process, we investigate the convergence of a fully discrete approximation of the state equation, and obtain optimal orders of convergence under weak regularity assumptions. We conclude with a numerical example.

Motivation & Objective

• To establish a rigorous functional analytic framework for control-constrained parabolic optimal control problems on evolving hypersurfaces.
• To prove existence and uniqueness of weak solutions for the state equation using vector-valued distributions and a W(0,T)-type solution space.
• To develop and analyze a fully discrete variational discretization scheme for the state equation with optimal convergence rates under weak regularity assumptions.
• To extend the theory to control-constrained optimal control problems and derive implementable numerical algorithms via variational discretization.
• To validate the theoretical findings with numerical experiments demonstrating optimal convergence orders.

Proposed method

• Formulate the state equation in weak form using material derivatives and surface gradients on evolving $ C^2 $-smooth hypersurfaces $ \Gamma(t) \subset \mathbb{R}^{n+1} $.
• Use pullback to a fixed domain and define weak solutions via vector-valued distributions in the sense of [LM68], ensuring existence and uniqueness under mild regularity.
• Apply variational discretization in the sense of [Hin05] to the control-constrained optimal control problem, enabling fully discrete and implementable optimization schemes.
• Derive error estimates in $ L^2(0,T;L^2(\Gamma(t))) $-like norms for the state equation under weak regularity, achieving optimal convergence rates.
• Use a discontinuous Galerkin time discretization and finite element spatial approximation, with error bounds derived via energy techniques and interpolation estimates.
• Implement a semi-smooth Newton method for the discrete optimal control problem to handle box constraints on the control.

Experimental results

Research questions

• RQ1 Under what conditions does a weak solution exist for the parabolic state equation on evolving surfaces with control constraints?
• RQ2 Can optimal convergence rates be achieved for fully discrete approximations of the state equation under minimal regularity assumptions?
• RQ3 How does variational discretization enable the construction of implementable algorithms for control-constrained optimal control problems on evolving surfaces?
• RQ4 What are the experimental orders of convergence for the discrete optimal control problem in the presence of high and low regularity data?
• RQ5 How do the error estimates and convergence behavior differ between $ L^2 $ and $ L^\infty $ norms in the discrete setting?

Key findings

• The paper proves existence and uniqueness of weak solutions for the parabolic state equation on evolving surfaces using a distributional framework, extending prior results from [DE07].
• Optimal convergence rates of order $ \mathcal{O}(h^2) $ are achieved for the $ L^2(0,T;L^2(\Gamma(t))) $-norm error in the state approximation under weak regularity assumptions.
• Numerical experiments confirm experimental orders of convergence (EOC) of approximately 2.0 for $ L^2 $-errors and 2.01–2.03 for $ L^\infty $-errors, indicating optimal convergence behavior.
• For low-regularity data, such as $ y_T = 1/(x+y)^{0.45} $, the EOC for $ L^2 $-error drops to around 1.0, consistent with $ \mathcal{O}(h) $ convergence expected under minimal integrability.
• The variational discretization approach yields fully implementable optimization algorithms, with the semi-smooth Newton method successfully solving the discrete control problem.
• The error analysis accounts for the full time-space evolution of the surface, with consistent error bounds derived via lifting and projection techniques on triangulated surfaces.