Dissipativity-based analysis of stochastic model predictive control

• Lars Gruene

In this talk I will explain how the recently developed dissipativity-based qualitative analysis of stochastic optimal control problems helps in analysing stochastic model predictive control (MPC) schemes. I will on the one hand present stability and near-optimal performance results for problems with suitable stage costs. On the other hand, I will explain why the stochastic MPC closed loop may not be near-optimal when the stage cost is defined by certain risk measures but also when chance constraints are imposed.

System-Theoretic Analysis of Dynamic Generalized Nash Equilibria - Turnpikes and Dissipativity

• Sophie Hall (ETH Zürich)

Generalized Nash equilibria are used in multi-agent control applications to model strategic interactions between agents that are coupled in the cost, dynamics, and constraints, and provide the foundations for game-theoretic MPC (Receding Horizon Games). We study properties of finite-horizon dynamic GNE trajectories from a system-theoretic perspective. We show how strict dissipativity generates the turnpike phenomenon in GNE solutions. Moreover, we establish a converse turnpike result, i.e., the implication from turnpike to strict dissipativity. We derive conditions under which the steady-state GNE is the optimal operating point and, using a game value function, we give a local characterization of the geometry of storage functions. Finally, we design linear terminal penalties that ensure dynamic GNE trajectories applied in open-loop converge to and remain at the steady-state GNE. These connections provide the foundation for future system-theoretic analysis of GNEs similar to those existing in optimal control as well as for recursive feasibility and closed-loop stability results of game-theoretic MPC.

Energy conversion in port-Hamiltonian systems – An optimal control approach

• Guanru Pan (Hamburg University of Technology)

This talk investigates the asymptotic energy conversion efficiency of two-port port-Hamiltonian systems operating under supplied power limits and storage constraints. The extracted energy at the output port is defined as $ E(T) \doteq -\int_0^T y_2^\top(t)u_2(t)\,dt, $ and the corresponding normalized efficiency is $\eta_E(T) \doteq \frac{E(T)}{T\bar p}.$ Lower and upper bounds are derived for the asymptotic optimal efficiency $\limsup_{T\to\infty}\eta_E^\star(T)$ by investigating optimal cyclic operation. The framework is further specialized to port-Hamiltonian systems with a block-diagonal structure, in which the supply and extraction ports are geometrically decoupled. For this class of systems, we derive a Carnot-type upper bound on the achievable asymptotic efficiency, $ \limsup_{T\to\infty}\eta_E^\star(T) \le \eta_{\mathrm{Carnot}}= 1-\frac{\underline H_{x_1}}{\bar H_{x_1}},$ which generalizes the classical thermodynamic efficiency bound for heat engines. Numerical illustrations are provided for both linear and nonlinear systems. [1] van der Schaft, A. and Jeltsema, D. (2021). On energy conversion in port-hamiltonian systems. In 2021 60th IEEE Conference on Decision and Control (CDC), 2421–2427. doi:10.1109/CDC45484.2021.9683292. [2] van der Schaft, A. and Jeltsema, D. (2022). Limits to energy conversion. IEEE Transactions on Automatic Control, 67(1), 532–538. doi:10.1109/TAC.2021.3075652. [3] Philipp, F.M., Schaller, M., Worthmann, K., Faulwasser, T., and Maschke, B. (2024). Optimal control of port-Hamiltonian systems: energy, entropy, and exergy. Systems & Control Letters, 194, 105942. doi: 10.1016/j.sysconle.2024.105942

Realisations of matrix valued rational functions

• Michał Wojtylak (Faculty of Mathematics and Computer Science, Jagiellonian Univesity, Kraków, Poland)

A classical result due to Arov [1] states that a rational matrix function $F(z)$ (of size $k\times l$) that takes contractive values for $z\in {\mathbb D}={ z \in{\mathbb C} : |z|<1}$ admits a contractive finite-dimensional realization; i.e., there exists a contractive matrix $ \left[ \begin{array}{cc} A & B \cr C & D \end{array} \right]\in \mathbb C^{d+k,d+l} $ such that $F(z) = D + zC (I~-~zA)^{-1} B ,\quad z\in {\mathbb D}.$ Since then, several other realizations have been introduced. In this talk we restrict attention to the case where $\left[ \begin{array}{cc} A & B \cr C & D \end{array} \right]$ is a matrix rather than an infinite dimensional operator. In particular, we present the annulus realization [2], the half-plane realization extending [3], and a multihalfplane realization. Finally, we discuss the practical implications of these results in connection with MOR and Padé approximation. [1] D. Z. Arov. Passive linear steady-state dynamical systems. Sibirsk. Mat. Zh., 20(2):211–228, 457, 1979. [2] R. Baran, P. Pikul, H. Woerdeman, M. Wojtylak. Contractive realization theory for the annulus and other intersections of discs on the Riemann sphere. Journal of Functional Analysis, 111346 (2026). [3] K. Cherifi, H. Gernandt, D. Hinsen. The difference between port-Hamiltonian, passive and positive real descriptor systems. Mathematics of Control, Signals, and Systems, 36(2), 451-482 (2024).

Closures of strict equivalence orbits of dissipative Hamiltonian matrix pencils

• Maria Dronka (Jagiellonian University)

The dissipative Hamiltonian (dH) matrix pencils are pencils of the form $L(\lambda) = \lambda E - (J-R)Q,$ where $J^* = -J$, $E^*Q = Q^*E \geq 0$, $R^* = R \geq 0$, and $\lambda E - Q$ is regular. Matrix pencils strictly equivalent to dH pencils were characterised in [2]. In this talk, we investigate the orbit structure of dH matrix pencils, in the setting proposed by Pokrzywa [3]. In particular, we determine, in terms of the Kronecker canonical form, when the closure of an orbit of a dH pencil contains only matrix pencils that are strictly equivalent to dH pencils. We also discuss the special case $Q = I$. Furthermore, we characterise matrix pencils that are strictly equivalent to possibly singular and possibly non-square pencils of the form $P(\lambda) = \lambda E - Q$ with $E^*Q = Q^*E \geq 0$, extending results from [1]. **References** [1] C. Mehl, V. Mehrmann, M. Wojtylak, Linear algebra properties of dissipative Hamiltonian descriptor systems, *SIAM J. Matrix Anal. Appl.*, 39 (3), 1489-1519, 2018. [2] C. Mehl, V. Mehrmann, M. Wojtylak, Matrix pencils with coefficients that have positive semidefinite Hermitian parts, *SIAM J. Matrix Anal. Appl.*, 43 (3), 1186-1212, 2022. [3] A. Pokrzywa, On perturbations and the equivalence orbit of a matrix pencil, *Linear Algebra Appl.*, 82, 99-121, 1986.

Mean-square dissipativity preserving stochastic numerical methods

• Raffaele D'AMBROSIO (University of L'Aquila)

This talk addresses recent developments in the design and analysis of structure-preserving numerical methods for selected classes of stochastic differential equations and stochastic partial differential equations, endowed with intrinsic invariance properties. Particular attention is devoted to the numerical preservation mean-square dissipativity, by means of stochastic $\theta$-methods. We show that, in this framework, the conservation of mean-square dissipativity emerges through conditional stability properties of the numerical method under consideration. Theoretical results are complemented by numerical experiments on selected test problems. The talk is based on recent papers in collaboration with H. Biscevic (Gran Sasso Science Insitute), E. Buckwar (Johannes Kepler University of Linz), S. Di Giovacchino (University of L'Aquila). $$ $$

A theory of energy-based learning in passive networks

• Henk van Waarde (University of Groningen)

Energy-based learning is a biologically plausible alternative to the widely used backpropagation method for training artificial neural networks. It considers models governed by an energy function and learns by shaping this function such that its minima coincide with observed data. This paradigm is particularly promising for training analog circuits in an energy-efficient manner, since learning can be implemented through relatively simple, local parameter updates. Despite its potential, a general theoretical understanding of energy-based learning is still largely missing. This talk presents steps toward such a theory. After a brief introduction, we focus on nonlinear resistive circuits and propose energy-based learning methods inspired by the literature on contrastive learning. Specifically, we define a contrastive function that compares two energy levels: the *free* energy, where the circuit’s output potentials are unconstrained, and the *clamped* energy, where the output potentials are fixed to match the target data. The learning objective is to minimize this contrastive function. As our main results, we introduce energy-based learning algorithms, and we establish conditions under which these algorithms converge. We will also discuss how the framework can be extended to more general classes of dynamical networks composed of passive subsystems.

Empirical neural scaling laws for learning port-Hamiltonian systems

• Karim Cherifi (FEMTO-ST, Supmicrotech, Besancon, France)

Physics-informed learning has emerged as a powerful paradigm for system identification, enabling data-driven models to capture complex nonlinear dynamics while respecting underlying physical structure. Building on our recent work on learning nonlinear port-Hamiltonian (pH) systems from input–state–output data, we investigate how model performance scales with available learning resources. We present a unified framework for identifying nonlinear pH systems using neural networks as structured function approximators. By embedding the port-Hamiltonian framework into the learning architecture, the proposed approach preserves passivity and energy-based properties while leveraging the universal approximation capabilities of modern neural networks. Incorporating prior knowledge about the underlying physical structure constrains the hypothesis space, improves data efficiency, and yields models that are more accurate, physically consistent, and reliable for long-term prediction than purely data-driven approaches. Then the scalability of such physics-informed models is studied through the lens of neural scaling laws, relating identification loss to data, model size, and computational budget. While scaling behavior is well established in domains such as natural language processing, the quantitative relationship between learning resources and identification accuracy remains poorly understood in dynamical systems settings. We empirically verify neural scaling laws across a range of architectures, including standard input-affine models and physics-informed port-Hamiltonian representations. By training thousands of models and evaluating their performance using standardized metrics, we determine scaling relationships that quantify how improvements in resources translate to gains in accuracy. These scaling laws provide practical guidance for forecasting achievable accuracy, selecting architectures, and designing data acquisition strategies, connecting structure-preserving system identification with neural

Machine learning Method of Multiple Trajectories for time series data

• Victoria Rayskin (MNSU)

In this talk we discuss a machine learning method (Method of Multiple Trajectories [1]) for fitting time series data into a non-linear dynamical system. When restricted to a specific basin of attraction, the long-term forecast of the process can be associated with the tendency towards the attracting stationary point. Following M.W. Hirsch's definition of dissipativity (a system with a global attractor [2]) we define the dissipative restriction of the system to be the system on the restricted domain. Some dissipative properties can be discussed for this restricted system. Within the basin of dissipative restriction, our method allows to achieve high accuracy short-term predictions of the process development (compare to [3]). However, our method also provides us with several long-term trends that can be associated with various scenarios of the long-term process development. These scenarios are associated with different basins of attraction of the dynamical system, when the system is not restricted to a single basin. In our examples (in the fields of economics and epidemiology), the dynamics can be driven into different basins of attraction via random external effects. For this reason, our long-term forecasts consist of multiple scenarios. We want to use external forces for moving the dynamics into a desired basin of attraction. We are interested in extending the machine learning technique to find optimal control that allows the dynamics to be transferred from one basin to another. References: [1] V. Rayskin, Multivariate time series approximation by multiple trajectories of a dynamical system. Applications to internet traffic and COVID-19 data, American Institute of Physics Conference Proceedings, 2302 060011 (2020) [2] M.W. Hirsch, (1996). "Dynamical Systems." In P. Smolensky, M. C. Mozer, & D. E. Rumelhart (Eds.), Mathematical Perspectives on Neural Networks (pp. 271–324). Lawrence Erlbaum Associates. [3] E. Bradley, H. Kantz, Nonlinear time-series analysis revisited (2015), Chaos, 25(9)

Splitting schemes for time-integration of port-Hamiltonian systems

• Nicole Marheineke (Trier University)

In this talk, we discuss high-order commutator-based splitting methods for port-Hamiltonian systems, with a focus on preserving their intrinsic structure, in particular the dissipation inequality. Port-Hamiltonian systems provide a natural framework for modeling energy-conserving and dissipative processes, which is crucial for the accurate simulation of many physical systems. For port-Hamiltonian ordinary differential equations, we introduce an energy-associated decomposition that exploits the system’s energy properties. In addition, we present a port-based splitting as well as subsystem-tailored decompositions that reduce the effective dimension and thereby improve computational efficiency. We also address challenges arising from nonlinear dynamics and discuss possible extensions to systems with algebraic constraints, including coupled systems. The proposed approaches are validated through theoretical analysis and numerical experiments.

Long time convergence of Runge-Kutta methods for dissipative ordinary differential equations on infinite time intervals

• Chengming Huang (School of Mathematics and Statistics, Huazhong University of Science and Technology)

In this talk, we are concerned with the numerical solution of stiff ordinary differential equations by Runge-Kutta methods. A B-convergence result on infinite time intervals is provided for algebraically stable methods applied to strictly dissipative systems. As an application of this result, the $s$-stage Radau IIA methods are proved to be B-convergent of order $s$ on infinite time intervals, and the $s$-stage Radau IA and Lobatto IIIC methods are B-convergent of order $s-1$ on infinite time intervals. Compared to Theorem 15.3 in the monograph [1] by Hairer and Wanner, the error bounds obtained here are independent of the length of the integration interval and are applicable to infinite time intervals. [1] E. Hairer and G. Wanner, Solving ordinary differential equations. II. Stiff and differential-algebraic problems. Second edition. Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 1996.

Contractivity-preserving IMEX Runge-Kutta methods

• Simone Di Donato (University of L'Aquila)

This talk investigates the geometric numerical integration of dissipative systems using Implicit-Explicit Runge-Kutta (IMEX-RK) schemes. We analyze the ability of these methods to maintain the dissipative nature of the exact flow within the numerical solution. This theoretical analysis leads to the construction of some structure-preserving schemes, the effectiveness of which is checked through a selection of numerical experiments. This talk is based on a joint paper with D'Ambrosio Raffaele (University of L'Aquila), Jackiewicz Zdzislaw and Welfert Bruno(Arizona State University). $$ $$

Data-Driven Port-Hamiltonian Systems

• Thomas Beckers (Vanderbilt University)

Reliable models of dynamical systems are essential for tasks such as state estimation, prediction, and the implementation of safe control strategies. However, developing first-principles models for nonlinear systems is often time-consuming and requires significant expert knowledge. While machine learning offers an alternative, learned models frequently lack trustworthiness, generalizability, and physical consistency, making them ill-suited for safety-critical applications. In this talk, I will present our recent work on data-driven port-Hamiltonian systems (PHS) for compositional and physically consistent modeling of complex dynamics, including ODE and PDE systems. We leverage machine learning methods with built-in uncertainty quantification to learn unknown Hamiltonian functions directly from data. Unlike many physics-informed learning approaches that enforce physical constraints through penalty terms, our models are physically consistent by design. This structure naturally supports composability: physical consistency is preserved under interconnection. Finally, I will discuss how data-driven port-Hamiltonian systems enable robust and safe learning-based control, making them a promising foundation for trustworthy and scalable modeling of physical systems.

Structure-preserving model reduction of linear time-varying port-Hamiltonian systems

• Riccardo Morandin (OvGU Magdeburg)

Many physical processes can be naturally modeled using port-Hamiltonian (pH) systems, which are inherently passive and stable, and allow for structure-preserving interconnection, making them particularly suitable for the modeling of complex networks. Furthermore, many dedicated numerical methods have been developed to exploit and preserve the structure of pH systems, e.g. for space- and time-discretization, and model order reduction (MOR). In our work, we focus on the structure-preserving MOR of linear time-varying (LTV) pH systems. LTV systems appear quite naturally in many applications, e.g. in the linearization of nonlinear systems around non-stationary reference solutions, or when some of the system parameters are time-dependent. In this talk we introduce a general approach based on (Petrov)-Galerkin projection for the structure-preserving MOR of LTV-pH systems. This includes (but is not limited to) the extension of the effort constraint method to LTV-pH systems. Furthermore, we combine balancing and projection to obtain a reduced model that is guaranteed to be pH. We exhibit numerical experiments to validate our algorithms. This is joint work with Karim Cherifi (FEMTO-ST, SUPMICROTECH, Besançon, France).

H2-Optimal Passivation by Low-Rank Approximation

• Matthias Voigt (UniDistance Suisse)

We present a novel passivity enforcement (passivation) method for linear time-invariant systems based on the Kalman-Yakubovich-Popov (KYP) lemma and the closely related Lur'e equations. The passivation problem in our framework corresponds to finding a perturbation to a given non-passive system that renders the system passive while minimizing the $\mathcal{H}_2$-norm distance between the original non-passive and the resulting passive system. We show that this problem can be formulated as an unconstrained optimization problem whose objective function can be differentiated efficiently even in large-scale settings. To solve the resulting numerical optimization problem efficiently, we propose an initialization strategy based on modifying the feedthrough term and a restart strategy when it is likely that the optimization has converged to a non-global local minimum. Numerical examples illustrate the effectiveness of the proposed method.

Semi-dissipative systems and their optimal representation

• Volker Mehrmann (TU Berlin, Inst. f. Mathematik)

Different representations of asymptotically/exponentially stable evolution equations are studied. These arise from the solution of Lyapunov inequalities. We discuss the construction of optimal representations via different criteria: Field of values, optimal decay, maximal coercivity, distance to instability. We discuss finite dimensional problems and infinite dimensional problems with bounded operators. Furthermore the evolution equations are studied in continuous and discrete time. This summary of recent publications presents work with different coauthors: A. Arnold, C. Beattie, S. Egger, E. Nigsch, P. Van Dooren, H. Xu, H. Zwart

Energy Based Dissipativity and Stability Analysis of Nonlinear Coupled Dynamical Systems with Memory

• Johnson Audu (Prince Mohammed Bin Fahd University)

In this work, we investigate the dissipative structure of a class of nonlinear coupled suspension bridge systems governed by partial differential equations with memory effects. We develop an energy based analytical framework that combines multiplier techniques with Lyapunov functionals tailored to the underlying physical energy of the system. This approach allows us to characterize the dissipativity of the coupled system and to establish well-posedness and exponential stability under suitable assumptions on the memory kernel. The results provide a rigorous description of how memory induced dissipation influences the decay of energy and the stabilization of nonlinear distributed systems. The proposed analysis highlights the role of energy based methods in the modeling and stabilization of complex infrastructure systems governed by PDEs.

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

• Till Preuster (Technische Universität Chemnitz)

Model Predictive Control (MPC) provides a powerful optimization-based framework for feedback design, but its real-time deployment remains computationally demanding. Suboptimal MPC schemes address this issue by terminating the underlying optimal control solver prematurely, thereby trading optimality for computational tractability. In this talk, we develop a suboptimal MPC approach tailored to port-Hamiltonian systems that leverages dissipativity at multiple levels. Specifically, we exploit the dissipative structure of the optimality system induced by its primal–dual formulation, in conjunction with the intrinsic dissipativity of the port-Hamiltonian dynamics. This perspective enables a reformulation of the closed-loop scheme as a control-by-interconnection of two port-Hamiltonian systems. Building on this structural viewpoint, we establish well-posedness and convergence of the resulting algorithm in an infinite-dimensional (function space) setting. In particular, we demonstrate mesh-independent convergence, highlighting the robustness of the approach with respect to discretization.

Towards Polynomial Immersion of Port-Hamiltonian Systems

• Timm Faulwasser (Hamburg University of Technology)

Port-Hamiltonian (pH) systems provide a highly structured and physically motivated framework for modeling and controlling complex dynamical systems by explicitly capturing energy-based dynamics, dissipation, and interconnection geometry. However, many practical physical systems exhibit non-polynomial non-linearities that hinder the application of modern constructive control design methods, such as sum-of-squares (SOS) optimization, which typically require polynomial system representations. We address the fundamental problem of transforming non-polynomial port-Hamiltonian systems into equivalent polynomial representations without sacrificing their favorable structural properties. We explore a novel *lifted immersion* technique that combines conceptual ideas from system lifting and immersion to construct a higher-dimensional polynomial port-Hamiltonian system that preserves the essential features of the original system along system trajectories. Specifically, we prove that the proposed immersion preserves: (i) the internal interconnection geometry with identical rational Hamiltonian, (ii) the port structure and conjugated input-output relationship, and (iii) the dissipation and passivity properties with respect to the original Hamiltonian on an invariant embedded submanifold. The resulting polynomial pH system structure enables the design of stabilizing feedback laws by combining interconnection and damping assignment passivity-based control with sum-of-squares optimization techniques. We demonstrate the effectiveness of this approach through several illustrative examples, showing how polynomial immersions bridge the gap between the structural advantages of port-Hamiltonian systems and the computational benefits of polynomial control design methods.