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<h1 class="title is-1 publication-title">Operator Methods in Systems and Control: Theory and Applications
</h1>
<div class="is-size-5 publication-authors">
<a href="https://www.ce.cit.tum.de/itr/team/hoischen/" target="_blank">Nicolas Hoischen*</a>,</span>
<span class="author-block">
<a href="https://www.ce.cit.tum.de/itr/team/beier/" target="_blank">Max Beier*</a>,</span>
<span class="author-block">
<a href="https://www.ce.cit.tum.de/itr/bevanda/" target="_blank">Petar Bevanda</a>,</span>
<span class="author-block">
<span class="author-block">
<a href="https://www.ce.cit.tum.de/itr/hirche/" target="_blank">Sandra Hirche</a>
</span>
<span class="author-block">
<a href="https://faculty.sist.shanghaitech.edu.cn/faculty/boris/" target="_blank">Boris Houska</a>
</span>
</div>
<div class="is-size-5 publication-authors">
<span class="author-block">Open invited track @ IFAC World Congress
2026<br>Submission Code: wh36m</span>
</div>
<span class="corresponding-authors"><small><sup>*</sup>corresponding authors: <a
href="mailto:nicolas.hoischen@tum.de">nicolas.hoischen@tum.de</a>,
<a href="mailto:max.beier@tum.de">max.beier@tum.de</a></small></span>
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<span>Description PDF</span>
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</section>
<!-- Paper abstract -->
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<h2 class="title is-3">Abstract</h2>
<div class="content has-text-justified">
<p>
The classical notion of state is a set of quantities in memory that completely describe the evolution of
a system in time. This allows one to work in the so-called state-space, which helped turn much of the
control theory into practical algorithms we use for estimation and control. Although the minimal
structure of the state is often well known for rigid body dynamics, for example, in terms of position
and velocities, there is a plethora of algorithms that lack such clearly defined quantities of interest.
</p>
<p>
With the increase in complexity of problems we are trying to tackle as a community, be it in unknown
systems where only data is available or systems with distributed parameters (modeled by PDEs), such as
fluids, properly choosing the state variables or state-space in advance comes with great challenges. On
one hand, it may be infeasible and impractical to find a suitable state representation of known and
complex high-dimensional nonlinear systems. On the other hand, even with a classical minimal state-space
representation available, exploiting data-driven models with highly nonlinear dynamics is nontrivial, as
optimization-based control and estimation lead to non-convex optimization problems that admit no generic
global solution schemes.
</p>
<p>
By abstracting the minimal state variables to observable functions of a different set of variables
(perhaps not a minimal state in the classical sense), we can model a dynamical system by the operator
acting on observables. Remarkably, these operators are linear operators, a property highly desired by
control engineers as it allows one to reformulate non-convex into convex optimization problems or even
argue about robustness or input-output properties using linear system analysis techniques.
</p>
</div>
</div>
</div>
</div>
</section>
<!-- End paper abstract -->
<!-- Motivation and Introduction -->
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<h2 class="title is-3">Motivation and Introduction</h2>
<div class="content has-text-justified">
<p>
The landscape of systems and control theory is undergoing a paradigm shift, driven by the increasing
complexity of real-world problems—from large-scale networks and distributed parameter systems to
data-driven applications in robotics, fluid dynamics, and energy systems. Traditional state-space
models, while foundational, face inherent limitations when confronted with high-dimensional, nonlinear,
or partially observable dynamics.
</p>
<p>
There is, however, an alternative description of nonlinear dynamical systems, which contrasts with the
classical representation of trajectories in the (immediate) state-space. This description relies on the
so-called Koopman (or composition) operator. Even though the dynamical system is nonlinear, its
(infinite-dimensional) representation in terms of the Koopman operator is linear. Therefore, this
operator-theoretic representation offers a tantalizing possibility to study nonlinear dynamical systems
via linear and, in particular, spectral techniques, while being globally valid in the whole state-space.
</p>
<!-- Operator Logic Figure -->
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<figure class="image">
<img src="static/images/ologic.png" alt="Operator Logic Diagram"
style="max-width: 800px; margin: auto;">
<figcaption class="caption" style="margin-top: 1rem;">
Figure 1: Modeling of a system by its operator acting on the space of observable functions and its
estimation based on measurements (data).
</figcaption>
</figure>
</div>
<p>
The approach also lends itself to machine learning, thereby meeting the current trend and need for
data-driven methods that are required for analysis of and controller design for complex dynamics
emerging in real-world applications. For these reasons, we have witnessed a surge of interest in
operator-based approaches, for applications in robotics <a href="#shi2024koopman">[Shi et al., 2024]</a>
and control <a href="#bevanda_koopman_2021">[Bevanda et al., 2021]</a>, <a
href="#brunton_modern_2022">[Brunton et al., 2022]</a>, with top-tier
conferences holding dedicated workshops <a href="#CDC2024WorkshopsOperator">[Bevanda et al., 2024]</a>,
<a href="#CDC2025WorkshopsOperator">[Bevanda et al., 2025]</a>, <a
href="#RSS2024KoopmanOperatorsWS">[Abraham et al., 2024]</a>, which the
authors of this proposal contributed to. Successful examples of the operator framework include:
data-driven system identification <a href="#iacob_koopman_2024">[Iacob et al., 2024]</a>, analysis <a
href="#mezic_koopman_2020">[Mezić, 2020]</a>, longterm forecasting <a
href="#kostic2024consistent">[Kostić et al., 2024a]</a>,
classical feedback control <a href="#strasser2025overview">[Sträßer et al., 2025]</a>, and optimal
control <a href="#houska2025convex">[Houska, 2025]</a>. Breakthroughs in machine learning - such as
extended dynamic mode
decomposition (EDMD) <a href="#williams_data-driven_2015">[Williams et al., 2015]</a>, kernel-based
methods <a href="#kostic_learning_2022">[Kostić et al., 2022]</a>, and neural representation learning
for linear operators <a href="#ryu_operator_2024">[Ryu et al., 2024]</a>, <a
href="#kostic_neural_2024">[Kostić et al., 2024b]</a> - have accelerated their
adoption, enabling scalable implementations for complex systems modeled by partial differential
equations (PDEs) or stochastic processes. Notably, operator methods provide a natural framework for
integrating physics-informed learning <a href="#giannakis2025physics">[Giannakis & Valva, 2025]</a>,
where domain knowledge
is embedded into observable functions, enhancing model interpretability and generalization.
</p>
<p>
We are hopeful that this invited session may bring together researchers with different perspectives to
strengthen joint research efforts at the intersection of classical control theory, machine learning, and
operator learning to position systems and control as indispensable also in machine learning-based
approaches.
</p>
</div>
</div>
</div>
</div>
</section>
<!-- Scope and Directions -->
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<h2 class="title is-3">Scope and Directions</h2>
<div class="content has-text-justified">
<p>
This invited track seeks to gather cutting-edge contributions at the intersection of operator theory,
machine learning, and control engineering, with a focus on operator-based system representation,
data-driven control, theoretical guarantees, and applications. By uniting researchers from dynamical
systems, optimization, and machine learning, this track aims to foster cross-disciplinary dialogue and
shape the next frontier of scalable, interpretable, and theoretically grounded control solutions.
</p>
<!-- Scope Figure -->
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<figure class="image">
<img src="static/images/scope.png" alt="Scope Overview Diagram"
style="max-width: 800px; margin: auto;">
<figcaption class="caption" style="margin-top: 1rem;">
Figure 2: Overview of the scope and main areas of focus for the invited track.
</figcaption>
</figure>
</div>
<h4 class="title is-5">Operator-based system representation</h4>
<p>
Methods that learn tractable representations of composition (Koopman) and transfer operators, as well as
related ones, such as their generators. Possible approaches include physics-informed basis, neural,
spectral, or Reproducing Kernel Hilbert Space (RKHS) techniques and algorithms that model or analyze
nonlinear dynamics through their operators.
</p>
<h4 class="title is-5">Data-driven control</h4>
<p>
Automated design of control laws for nonlinear dynamical systems is a notoriously challenging problem. A
promising direction is to connect ideas from classical control design with operator representations.
This includes risk-aware methods that integrate uncertainty quantification and criteria such as CVaR,
operator-theoretic formulations of Bellman's principle of optimality, and evolution operators enabling
stochastic control, PDE-constrained optimization, and predictive control.
</p>
<h4 class="title is-5">Theoretical guarantees</h4>
<p>
Contributions regarding the theoretical properties of the operators of interest in common hypothesis
spaces, as well as on algorithms in terms of statistical learning, error bounds, and convergence
analysis for operator learning in infinite-dimensional settings.
</p>
<h4 class="title is-5">Applications</h4>
<p>
We encourage reproducible case studies including strong baselines demonstrating advantages or failure
modes of state-of-the-art operator models for prediction, estimation, or control. Another possible
application is the analysis of nonlinear systems through spectral techniques to obtain dominant
timescales, frequencies, and modes. Possible applications include but are not limited to fluid dynamics,
robotics, power grids, and biomedical and biological systems.
</p>
</div>
</div>
</div>
</div>
</section>
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<h2 class="title">References</h2>
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<ol class="references">
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</div>
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<pre id="bibtex-code"><code>@article{houska2026operator,
title={Operator Methods in Systems and Control: Theory and Applications},
author={Hoischen, Nicolas, Max, Beier, Bevanda, Petar, Hirche, Sandra and Houska, Boris},
booktitle={Open invited track @ IFAC World Congress},
year={2026},
url={https://tum.itr.github.io/IFAC-2026-Operator-Methods-in-Systems-and-Control-Theory-and-Applications}
}</code></pre>
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