arXiv subjects

Manuel Schaller

Publications and source records attributed to Manuel Schaller.

At least 19 recordsLinked to original sources

arXiv2026

Abstract second-order boundary control systems

Many physical systems, such as vibrating elastic bodies, acoustic media and electromagnetic fields, are governed by second-order evolution equations $\ddot x+D\dot x+Sx=0$ and interact with their environment through their boundary. The classical analysis via the operator matrix $\mathcal{A}=\left[\begin{smallmatrix}0&I\\-S&-D\end{smallmatrix}\right]$ requires $S$ to be self-adjoint, which typically fixes a boundary condition a priori and thus excludes boundary control. We therefore use $S$ only to define the energy space and govern the dynamics by a maximal operator $S_{\max}\supset S$ whose domain carries no boundary condition. Based on a factorization $S=A^*A$, we construct a boundary triplet for the associated operator matrix and define boundary inputs and outputs of Robin type, parametrized by an operator $P$ on the boundary space. We show that the resulting system is impedance passive and internally well-posed if $P$ is a contraction, and that for $D=0$ this covers all boundary conditions under which the free dynamics generate a contraction semigroup. Moreover, we show that the position-momentum formulation (e.g., $\left[\begin{smallmatrix}0&I\\Δ&0\end{smallmatrix}\right]$ for the wave equation) and the strain-momentum formulation (e.g., $\left[\begin{smallmatrix}0&\nabla\\\operatorname{div}&0\end{smallmatrix}\right]$) describe the same open system, including its boundary inputs and outputs. We illustrate our results by an $n$-dimensional wave equation and Maxwell's equations.

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arXiv2026

Dual companions and improved regularity of Lagrange multipliers in optimal control

We investigate optimal control problems whose state equation takes the form $\mathcal{A}x-\mathcal{B}u=b$, where $\mathcal{A}:\mathcal{X}\to\mathcal{P}'$ is an isomorphism from a Banach state space $\mathcal{X}$ onto the dual of a reflexive Banach space $\mathcal{P}$. The adjoint equation involves the Banach space adjoint $\mathcal{A}':\mathcal{P}\to\mathcal{X}'$, so that, in general, state and multiplier are of different regularity. For particular classes of partial differential equations, the typically weaker regularity of the multiplier is improved a posteriori by bootstrapping. In this work, we provide an abstract foundation for this methodology on the operator level. More precisely, we show that whenever the state cost extends to a differentiable function on a Hilbert space $\mathcal{H}$ in which $\mathcal{X}$ is densely embedded, as for $L^2$-tracking costs on Sobolev-type state spaces, the multiplier automatically lies in a subspace of $\mathcal{P}$, the dual companion, on which $\mathcal{A}'$ restricts to an isomorphism onto $\mathcal{H}'$. This yields an optimality system with balanced regularity, where state and multiplier solve equations of the same type. In applications, the dual companion can be identified by integration by parts in time or space. We illustrate the framework by linear elliptic and parabolic equations, Cauchy problems governed by general $C_0$-semigroups, and a wave equation with accumulated-state observation.

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arXiv2026

Extended dynamic mode decomposition with Fourier dictionaries: Error bounds and fast implementation

The Koopman operator has gained considerable attention in dynamical systems due to its capability to provide a linear viewpoint for nonlinear systems using data-driven methods such as extended dynamic mode decomposition (EDMD). In this work, we propose an EDMD-variant with a Fourier dictionary on the $d$-dimensional torus, where the data are sampled on an equispaced tensor grid. We show that the EDMD regression problem admits a unique closed-form solution and coincides with trigonometric interpolation of the Koopman image. This identification has two consequences. First, using Koopman invariance of Sobolev spaces, we transfer approximation-theoretic results for trigonometric interpolation to derive $L^p$-error bounds, $2 \le p \le \infty$, on approximations of the Koopman operator. In particular, the established bounds are of optimal order with explicit constants. Second, the EDMD matrix never has to be assembled, since its action reduces to (nonequispaced) fast Fourier transforms such that the EDMD-approximation can be evaluated matrix-free with quasi-linear cost in the dictionary size. We illustrate the results for the Kuramoto model of coupled oscillators and for the chaotic ABC flow with dictionaries of up to $10^7$ modes.

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arXiv2026

Optimal energy conversion -- A port-Hamiltonian approach

This paper studies optimal energy conversion in port-Hamiltonian systems. We analyze state- and control-constrained finite-horizon optimal control problems that maximize energy conversion efficiencies. Under suitable reachability conditions, we show that the asymptotic optimal efficiencies exist and coincide with the corresponding optimal cyclic efficiencies. For a class of port-Hamiltonian systems describing coupled storage-conversion components, we further derive intrinsic limits on achievable energy conversion and establish a generalized Carnot-type efficiency bound. Our results provide an optimal-control framework connecting asymptotic optimal operation, cyclic operation, and fundamental efficiency limits for port-Hamiltonian systems.

eess.SY↗

arXiv2026

ANOVA approximation for high-dimensional control systems

We study the data-driven approximation of control-affine systems, where the drift and each control vector field exhibit a low-dimensional structure. While the ANOVA (ANalysis Of VAriance) decomposition is tailored to low-dimensional structures, a straightforward implementation does not exploit control affinity, rendering the approximation computationally expensive. We propose a coupled least squares approach for given state-control samples, whose ansatz is affine linear in the control by construction and can be realized by a preprocessing of the ANOVA index sets. Based on a Marcinkiewicz--Zygmund inequality for the resulting ansatz space, we prove a probabilistic $L_2$-error bound with explicitly given constants. Separate error control on drift and input vector fields renders the approach flexible to, e.\,g., parameter changes. Finally, we numerically illustrate the identification of the coupling structure via global sensitivity indices on discretizations of a nonlinear advection-diffusion equation, the shallow water equations and the viscous Burgers equation.

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arXiv2026

Nonlinear controlled port-Hamiltonian systems: Existence of (optimal) solutions

We investigate the existence of solutions of reversible and irreversible port-Hamilto\-nian systems. To this end, we utilize the associated exergy, a function that is composed of the system's Hamiltonian and entropy, to prove global existence in time for bounded control functions. Then, we rigorously verify our existence conditions for two examples, the gas-piston system and a network of heat exchangers. Last, we explore model predictive control tailored to irreversible port-Hamiltonian systems by means of a numerical case study with a heat exchanger network.

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arXiv2026

Maximum-likelihood reprojections for reliable Koopman predictors of parametric nonlinear dynamical systems

Koopman-based methods leverage a nonlinear lifting to enable linear regression techniques. Consequently, system analysis, learning, and prediction are considered through the lens of this lifting resulting in a nonlinear manifold that is invariant under the action of the Koopman operator. Typically, this invariance is not preserved in data-driven approximations such as Extended Dynamic Mode Decomposition (EDMD). In this work, we systematically generalize the coordinate reprojection to render Koopman predictors more reliable and versatile. To this end, we propose consistency projectors to align the dynamics of the Koopman surrogate model with the geometry of the nonlinear manifold using concepts from differential geometry. Further, we establish a link to maximum likelihood estimates, which can be leveraged as an event-triggering mechanism to only reproject if really necessary. In addition, we extend our approach to parameter-affine systems enabling sensitivity analysis. In numerical examples ranging from a supercritical pitchfork bifurcation over the Duffing oscillator to the Lorenz system, we showcase that the reprojected Koopman surrogate reproduces the bifurcation behaviour of the underlying system, i.e., the emerging equilibria, their stability and the resulting attractors, whereas predictions without reprojection either converge to spurious equilibria or diverge. This provides the basis for data-driven bifurcation analysis and control applications.

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arXiv2026

Energy-optimal predictive control of discrete-time port-Hamiltonian systems: Closed-loop practical stability

We study a dissipativity-based model predictive control (MPC) scheme for energy-optimal constrained output stabilization of a discrete-time nonlinear SISO port-Hamiltonian system described by difference and differential representation. For the optimal control problem to be solved in each MPC step, we establish a measure turnpike behavior. The turnpike set in our work is obtained from the underlying port-Hamiltonian structure without assuming existence of steady-states or periodic orbits. In particular, unlike existing results which typically establish the turnpike property w.r.t. a controlled forward invariant set such as optimal steady-states or optimal periodic orbits, we do not require the turnpike set to be forward control invariant. For MPC, we establish recursive feasibility and show practical stability to the turnpike set w.r.t. the MPC closed loop leveraging both dissipativity and the port-Hamiltonian structure. Finally, we demonstrate our results using two numerical examples.

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arXiv2026

A Generalized Scalar Auxiliary Variable Method for Structure-Preserving and Efficient Integration of Nonlinear port-Hamiltonian DAEs

We develop an energy-optimal generalized scalar auxiliary variable (EOP-GSAV) framework for nonlinear index-one port-Hamiltonian differential-algebraic equations (pH-DAEs). Exploiting the port-Hamiltonian structure, we separate the nonlinear effort, interconnection, and dissipation terms from a constant implicit core. The resulting BDF-1 and BDF-2 schemes require one linear solve per time step with a reusable factorization, while retaining discrete passivity and accurate tracking of the Hamiltonian. The schemes are compared with the implicit midpoint method equipped with full, modified, and frozen-Jacobian Newton iterations. Numerical experiments ranging from a strongly state-dependent nonlinear stress test to large-scale benchmarks demonstrate robust and competitive performance, with substantial efficiency gains in matched-accuracy regimes. A comparison with SUNDIALS IDA shows comparable work-precision behavior at equal order despite a non-specialized Python/SciPy implementation, while unrestricted variable-order adaptive IDA is faster in the high-accuracy regime.

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arXiv2026

A constraint dissolving inexact penalty method for optimization problems with geometric constraints

Optimization problems with geometric constraints have a broad range of applications, including machine learning, finance, and control. A powerful algorithmic tool to resolve these geometric constraints are constraint dissolving methods. To this end, we propose a framework for constraint dissolving mappings for nonconvex geometric constraints. Leveraging these, we develop a constraint dissolving inexact penalty method to solve optimization problems with general set-membership constraints and possibly nonconvex geometric constraints. We establish the convergence of the proposed algorithm and prove that every feasible accumulation point is Mordukhovich stationary. Notably, we rely only on mild asymptotic Mordukhovich regularity, which is significantly weaker than the constraint qualifications adopted in the existing literature on constraint dissolving methods. Numerical experiments addressing classical equality-, complementarity-, sparsity-, and low-rank constrained optimization problems demonstrate that the proposed method is competitive with the safeguarded augmented Lagrangian method in terms of solution quality and significantly outperforms the penalty decomposition method.

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arXiv2026

Optimization-based control by interconnection of nonlinear port-Hamiltonian systems

In this paper, we develop a control-by-interconnection approach for the stabilization of nonlinear port-Hamiltonian systems. Motivated by model predictive control, the controller is realized as a continuous-time primal-dual gradient flow associated with a finite-horizon, control-constrained optimal control problem. By exploiting its port-Hamiltonian structure, the optimization dynamics are interconnected with the nonlinear plant. Introducing a time-scale parameter for the optimization dynamics reveals a singularly perturbed closed-loop system, whose reduced dynamics correspond to the optimal finite-horizon feedback, while the boundary-layer dynamics capture convergence of the primal-dual variables to the optimality system. Using a composite Lyapunov function and singular-perturbation arguments, we prove asymptotic stability of the coupled plant-controller dynamics, which is local for sufficiently fast optimization dynamics and becomes global under additional assumptions. Numerical experiments for a nonlinear port-Hamiltonian oscillator illustrate the results.

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arXiv2026

Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators

Conditional expectation operators (CEOs) and their associated conditional mean embeddings (CMEs) play a central role across applied mathematics and machine learning, appearing in nonparametric regression, Bayesian inverse problems, and Koopman operator theory. A fundamental question is when a CEO maps a function space on $\mathcal{Y}$ into a prescribed function space on $\mathcal{X}$, particularly a reproducing kernel Hilbert space (RKHS). We show that such mapping properties are characterized by the regularity of the Radon--Nikodym density of the conditional law, and establish a simple, verifiable sufficient condition under which the CEO is bounded and Hilbert--Schmidt. For RKHSs norm-equivalent to Sobolev spaces, this condition reduces to Sobolev regularity of the conditional density. The result yields a direct route to validate CME representations and error bounds for Galerkin-type and CME-based estimators. We verify the regularity condition in three settings: nonparametric regression, Bayesian inverse problems, and Koopman operator theory for stochastic dynamical systems. We show in each case that classical regularity results on the underlying probabilistic model imply the required mapping properties. The resulting framework offers a unified perspective on conditional expectation operators across probability, operator theory, kernel methods, and stochastic dynamics.

math.DS↗

arXiv2026

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

Many numerical algorithms for optimal control leverage an elimination of the state via the control-to-state 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.

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arXiv2026

Splitting Schemes for ODEs with Goal-Oriented Error Estimation

We present a hybrid a-priori/a-posteriori goal oriented error estimator for a combination of dynamic iteration-based solution of ordinary differential equations discretized by finite elements. Our novel error estimator combines estimates from classical dynamic iteration methods, usually used to enable splitting-based distributed simulation, and from the dual weighted residual method to be able to evaluate and balance both, the dynamic iteration error and the discretization error in desired quantities of interest. The obtained error estimators are used to conduct refinements of the computational mesh and as a stopping criterion for the dynamic iteration. In particular, we allow for an adaptive and flexible discretization of the time domain, where variables can be discretized differently to match both goal and solution requirements, e.g. in view of multiple time scales. We endow the scheme with efficient solvers from numerical linear algebra to ensure its applicability to complex problems. Numerical experiments compare the adaptive approach to a uniform refinement.

math.NA↗

arXiv2026

Koopman meets input-output data: Data-driven output-feedback control of nonlinear systems with closed-loop guarantees

Data-driven control of nonlinear systems from input-output measurements remains a fundamental challenge, as existing approaches with rigorous closed-loop guarantees predominantly require access to full state measurements. In this paper, we address this gap by proposing a data-driven output-feedback controller design method for nonlinear systems that provides provable closed-loop guarantees while operating solely on measured input-output data. Our approach combines Koopman operator theory with an extended state representation of the nonlinear system constructed from input-output trajectories. This allows us to obtain a bilinear surrogate model directly from data, on which robust state-feedback design methods can be applied. By exploiting the observability of the underlying nonlinear system, we establish exponential stability of the extended state, which in turn implies exponential convergence of the original system state to the origin. Finally, we validate our theoretical findings in numerical simulations.

eess.SY↗

arXiv2026

Coupling optimization algorithms and monotone control systems: Suboptimal model predictive control as an operator splitting scheme

We propose a framework for suboptimal model predictive control (MPC) based on the interconnection of monotone dynamical systems, such as port-Hamiltonian systems. In contrast to classical MPC formulations, where the optimizer is treated as an instantaneous mapping, we model both the plant and the optimizer as dynamical systems and couple them through a structured interconnection. This leads to a continuous-time closed-loop formulation governed by (quasi-)monotone operators. Within this setting, we establish well-posedness of the coupled optimizer-plant dynamics and provide a unified interpretation of suboptimal MPC schemes. In particular, we reveal a direct connection between iterative optimization algorithms and dynamical control systems theory by showing that standard suboptimal MPC algorithms can be understood as time discretizations of the underlying continuous-time dynamics via operator splitting methods.

math.OC↗

arXiv2026

Goal-Oriented Time Adaptivity for Linear Port-Hamiltonian Differential-Algebraic Equations of Index~1

Port-Hamiltonian systems provide a highly-structured framework for modeling of physical systems. By definition, they encode a balance equation relating energy changes to supplied and dissipated energy. Capturing this energy balance in discrete approximations is a fundamental challenge and often has been achieved by designing particular schemes such as discrete gradient methods. In this work, we propose an approach that controls the energy balance violation for port-Hamiltonian differential algebraic equations via time adaptivity using a posteriori grid refinement techniques based on the dual weighted residual method. In particular, we show how one may leverage the port-Hamiltonian structure to efficiently compute the error estimators using a dissipativity-exploiting block-Jacobi approximation. We illustrate the efficacy of the method by means of simulations of electrical circuit models.

math.NA↗

arXiv2026

Koopman for stochastic dynamics: error bounds for kernel extended dynamic mode decomposition

We prove $L^\infty$-error bounds for kernel extended dynamic mode decomposition (kEDMD) approximants of the Koopman operator for stochastic dynamical systems. To this end, we establish Koopman invariance of suitably chosen reproducing kernel Hilbert spaces and provide an in-depth analysis of the pointwise error in terms of the data points. The latter is split into two parts by showing that kEDMD for stochastic systems involves a kernel regression step leading to a deterministic error in the fill distance as well as Monte Carlo sampling to approximate unknown expected values yielding a probabilistic error in terms of the number of samples. We illustrate the derived bounds by means of Langevin-type stochastic differential equations involving a nonlinear double-well potential.

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