Stabilize-then-optimize: Feedback transformations as preconditioners in optimal control

Jul 16, 2026

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Many numerical methods for optimal control rely on eliminating the state through the control-to-state map, which directly determines convergence behavior via the conditioning of the resulting linear systems. This work demonstrates that feedback transformations can reformulate the optimal control problem to reduce the norm of the (feedbacked) control-to-state map, yielding a substantial improvement in condition numbers. The approach is developed constructively for ordinary and partial differential equation models across parabolic, hyperbolic, and elliptic classes, with numerical experiments validating efficiency and robustness.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages22

Reading Time55 min

Introduction

Control-to-state maps and reduced formulations

Feedback transformations and reformulation

Conditioning impact on numerical solvers

arXiv :2607 . 11835v1 [math .OC] 13 Jul 2026 STABILIZE-THEN-OPTIMIZE: FEEDBACK TRANSFORMATIONS AS PRECONDITIONERS IN OPTIMAL CONTROL TILL PREUSTER1 , ANTON SCHIELA2 , MANUEL SCHALLER1 , AND MARTIN STOLL1 ABSTRACT. Many numerical algorithms for optimal control leverage an elimination of the state via the control-tostate map such as condensed approaches or preconditioned conjugate gradient methods for the optimality system. As such, the norm of the control-to-state map directly enters the convergence estimates for these methods, e.g., via the condition number of the associated linear system. In this work we show that using feedback transformations one may reformulate the optimal control problem to decrease the norm of the (feedbacked) control-to-state map, leading to a drastic improvement of the involved condition numbers. We illustrate the abstract approach for ordinary and partial differential equations such as parabolic, hyperbolic or elliptic equations. For each of these problem classes we provide a constructive method to improve solution operator norms via feedbacks. Further, we showcase the efficacy of the method by means of various numerical examples with elliptic, parabolic and hyperbolic partial differential equations. Keywords: PDE-constrained optimal control, preconditioning, feedback stabilization, conjugate gradient methods 1. INTRODUCTION Optimal control of differential equations is a well-established field of mathematical analysis, numerical methods and scientific computing. Problems subject to ordinary or partial differential equations (ODEs or PDEs) have been thoroughly analyzed and numerically approached [8, 12, 21] . The fundamental structure in these systems is that pairs of optimization variables (x, u) satisfying the PDE may be characterized via a control-to-state map. This structure is useful analytically but may also be used to derive efficient numerical algorithms as the control-to-state map immediately provides access to a projection onto the feasible set. Many PDE-constrained optimal control problems may be viewed as a realization of the abstract linearquadratic optimal control problem (OCP) (x,um)×U 12 ∥Cx − yref∥2Y + α2 ∥u∥2U s.t. Ax − Bu = f, (1) where (i) X is a Banach space and Y , U , P are Hilbert spaces, (ii) A ∈ L (X , P ∗ ) is continuously invertible and encodes some ODE or PDE, (iii) B ∈ L (U , P ∗ ) is an input operator and C ∈ L (X , Y) is an observation operator, (iv) yref ∈ Y is a reference output, f ∈ P ∗ is a forcing term, and α > 0 is a regularization parameter. Here, the constraint could correspond either to a static problem that is, an elliptic PDE for which A is usually a second-order (elliptic) operator or a time-dependent problem such as a parabolic or hyperbolic partial differential equation, where the constraint operator A involves space and time derivatives. Such linear-quadratic problems occur, e.g., in in each step of an SQP-method applied to nonlinear problems. One of the simplest approaches to solve (1) is to eliminate the PDE-constraint using invertibility of A, that is, leveraging the control-to-state map u 7→ A−1Bu. Hence, setting x = A−1(Bu+f) one may eliminate the state from the optimal control problem and consider the reduced or condensed problem. Then, standard approaches such as gradient methods may be applied [8, 21] . Alternatively, one may keep the constraint and introduce a Lagrange multiplier and derive suitable first-order optimality conditions in function space. Then, these could 1 JUNIOR PROFESSORSHIP NUMERICAL MATHEMATICS, FACULTY OF MATHEMATICS, CHEMNITZ UNIVERSITY OF TECHNOLOGY, GERMANY, MAIL: {TILL . PREUSTER ,MANUEL . SCHALLER ,MARTIN . STOLL }@MATH . TU-CHEMNITZ . DE 2 INSTITUTE OF MATHEMATICS, UNIVERSITY OF BAYREUTH, GERMANY, MAIL: ANTON . SCHIELA @UNI-BAYREUTH . DE Till Preuster and Manuel Schaller acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– Project-ID 531152215

FAQs

Why does the control-to-state map affect convergence in optimal control algorithms?

The norm of the control-to-state map enters the convergence estimates, often through the condition number of the linear system solved in preconditioned conjugate gradient or condensed approaches.

What is the main idea of the paper’s stabilize-then-optimize approach?

Apply a feedback transformation to reformulate the optimal control problem, decreasing the norm of the (feedbacked) control-to-state map and thereby improving the conditioning of the systems involved.

How is the method adapted for different PDE types?

The paper provides constructive feedback-based ways to improve operator norms for parabolic, hyperbolic, and elliptic problem classes, and then demonstrates effectiveness through numerical examples.

Jul 16, 2026

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