[Paper Review] Galerkin approximations of nonlinear optimal control problems in Hilbert spaces

Mickaël D. Chekroun, Axel Kröner+1 more|arXiv (Cornell University)|Apr 3, 2017

Stability and Controllability of Differential Equations63 references22 citations

TL;DR

This paper establishes convergence of Galerkin approximations for nonlinear optimal control problems in Hilbert spaces by identifying minimal, checkable conditions under which the value functions of the approximations converge uniformly to the true value function. The approach leverages Trotter-Kato semigroup theory and applies to semilinear heat equations on compact manifolds and energy balance climate models on the sphere, ensuring convergence for broad classes of nonlinear control strategies and cost functionals.

ABSTRACT

Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated for the first time in terms of optimal control of energy balance climate models posed on the sphere $\\mathbb{S}^2$.

Motivation & Objective

• To establish sufficient conditions for convergence of Galerkin approximations in nonlinear optimal control problems within Hilbert space frameworks.
• To extend existing convergence results from linear to nonlinear evolution equations, including semilinear parabolic PDEs and nonlinear delay differential equations.
• To ensure uniform convergence of value functions associated with Galerkin-approximated optimal control problems under minimal, verifiable assumptions.
• To demonstrate applicability to real-world systems such as energy balance climate models on the sphere and semilinear heat equations on compact manifolds.
• To provide a general framework applicable beyond eigenfunction-based Galerkin bases, enabling use with non-ideal spectral bases.

Proposed method

• Utilizes the Trotter-Kato approximation framework from $C_0$-semigroup theory to analyze convergence of Galerkin approximations.
• Establishes double uniform convergence—over time and admissible controls—of Galerkin states to the true controlled state, ensuring value function convergence.
• Introduces Assumption (A7) as a key condition that generalizes standard local Lipschitz conditions to handle state- and control-dependent nonlinearities.
• Applies the framework to semilinear heat equations on compact Riemannian manifolds, including $\mathbb{S}^2$, via orthogonal projections onto finite-dimensional subspaces.
• Adapts the method to energy balance models (EBMs) on the sphere, modeling geoengineering and greenhouse gas mitigation as optimal control problems.
• Employs a weak topology approach and compactness arguments to prove existence and convergence of optimal controls under standard functional analytic conditions.

Experimental results

Research questions

• RQ1 Under what minimal, checkable conditions does the value function of a Galerkin approximation converge to the true value function in nonlinear optimal control problems in Hilbert space?
• RQ2 How can the Trotter-Kato framework be extended to ensure uniform convergence of controlled trajectories and value functions in the presence of nonlinear state and control dependencies?
• RQ3 To what extent does the Galerkin approximation framework remain valid when the basis functions are not eigenfunctions of the linear operator?
• RQ4 Can the convergence theory be applied to climate models such as energy balance models on the sphere, particularly for geoengineering and emission mitigation strategies?
• RQ5 What error estimates can be derived for the convergence of value functions and optimal controls in this nonlinear, infinite-dimensional setting?

Key findings

• The value function of the Galerkin-approximated optimal control problem converges pointwise to the true value function under Assumption (A7), which generalizes standard Lipschitz conditions to nonlinear terms.
• Uniform convergence of controlled Galerkin states over time and admissible controls is established as the key mechanism ensuring value function convergence, formalized in Theorem 2.1 and Corollary 2.1.
• The framework applies to semilinear heat equations on compact manifolds, including $\mathbb{S}^2$, with convergence guaranteed under standard assumptions on the nonlinearities and control sets.
• The method extends to systems of nonlinear delay differential equations, broadening applicability beyond PDEs to models of complex dynamical systems like ENSO.
• Error estimates for the convergence of value functions are derived, providing quantitative bounds under the proposed assumptions.
• The approach is valid even when Galerkin bases are not eigenfunctions of the linear operator, enabling use with more flexible, problem-specific basis functions.