Hans Zwart

University of Twente | UT · Department of Applied Mathematics

In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there are constraints coming from the underlying physics, but many standard PDE models can be seen as an adHDAE on an...

We consider differential operators A that can be represented by means of a so-called closure relation in terms of a simpler operator Aext\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \...

DC networks play an important role within the ongoing energy transition. In this context, simulations of designed and existing networks and their corresponding assets are a core tool to get insights and form a support to decision-making. Hereby, these simulations of DC networks are executed in the time domain. Due to the involved high frequencies a...

A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a nonzero energy flow through the boundaries. In this paper, we propose a novel framework for...

A class of linear hyperbolic partial differential equations, sometimes called networks of waves, is considered. For this class of systems, necessary and sufficient conditions are formulated for the operator dynamics to generate a C_0-group. This property turns out to be equivalent to the Riesz-spectral nature of that operator. In that case, its spe...

n this paper, we present port-Hamiltonian formulations of the incompressible Euler equations with a free surface governed by surface tension and gravity forces, modelling e.g. capillary and gravity waves and the evolution of droplets in air. Three sets of variables are considered, namely (v,Σ), (η,ϕ∂,Σ) and (ω,ϕ∂,Σ), with v the velocity, η the sole...

Linear-Quadratic optimal controls are computed for a class of boundary controlled, boundary observed hyperbolic infinite-dimensional systems, which may be viewed as networks of waves. The main results of this manuscript consist in converting the infinite-dimensional continuous-time systems into infinite-dimensional discrete-time systems for which t...

In this paper we consider BIBO stability of systems described by infinite-dimensional linear state-space representations, filling the so far unattended gap of a formal definition and characterization of BIBO stability in this general case. Furthermore, we provide several sufficient conditions guaranteeing BIBO stability of a particular system and d...

A class of linear hyperbolic partial differential equations, sometimes called networks of waves, is considered. For this class of systems, necessary and sufficient conditions are formulated on the system matrices for the operator dynamics to be a Riesz-spectral operator. In that case, its spectrum is computed explicitly, together with the correspon...

We provide an introduction to infinite-dimensional port-Hamiltonian systems. As this research field is quite rich, we restrict ourselves to the class of infinite-dimensional linear port-Hamiltonian systems on a one-dimensional spatial domainand we will focus on topics such as Dirac structures, well-posedness, stability and stabilizability, Riesz-ba...

In this paper, we present port-Hamiltonian formulations of the incompressible Euler equations with a free surface governed by surface tension and gravity forces, modelling e.g. capillary and gravity waves and the evolution of droplets in air. Three sets of variables are considered, namely $(v,\Sigma)$, $(\eta,\phi_{\partial},\Sigma)$ and $(\omega,\...

In this paper we consider BIBO stability of systems described by infinite-dimensional linear state-space representations, filling the so far unattended gap of a formal definition and characterization of BIBO stability in this general case. Furthermore, we provide several sufficient conditions guaranteeing BIBO stability of a particular system and d...

We consider differential operators $A$ that can be represented by means of a so-called closure relation in terms of a simpler operator $A_{\operatorname{ext}}$ defined on a larger space. We analyze how the spectral properties of $A$ and $A_{\operatorname{ext}}$ are related and give sufficient conditions for exponential stability of the semigroup ge...

A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a non-zero energy flow through the boundaries. In this paper we propose a novel framework for...

We characterise asymptotic stability of port-Hamiltonian systems by means of matrix conditions using well-known resolvent criteria from $C_0$-semigroup theory. The idea of proof is based on a recent characterisation of exponential stability established in [Trostorff, Waurick, arXiv:2201.10367], which itself goes back to a structural observation con...

This paper deals with the asymptotic stabilization of a class of port-Hamiltonian (pH) 1-D Partial Differential Equations (PDE) with spatial varying parameters, interconnected with a class of linear Ordinary Differential Equations (ODE), with control input on the ODE. The class of considered ODE contains the effect of a proportional term, that can...

The latest generation wafer scanners use extreme ultraviolet light to project a pattern of electronic connections onto a silicon wafer. A significant part of the projection light is absorbed by the mirrors in the projection system. This causes the mirrors to heat up and expand, which leads to a significant reduction in the imaging quality of the wa...

Analysis of longitudinal vibration in a nanorod is an important subject in science and engineering due to its vast application in nanotechnology. This paper introduces a port-Hamiltonian formulation for the longitudinal vibrations in a nanorod, which shows that this model is essentially hyperbolic. Furthermore, it investigates the spectral properti...

This letter presents a method to estimate the space-dependent transport coefficients (diffusion, convection, reaction, and source/sink) for a generic scalar transport model, e.g., heat or mass. As the problem is solved in the frequency domain, the complex valued state as a function of the spatial variable is estimated using Gaussian process regress...

This work proposes a structure-preserving model reduction method for linear, time-invariant port-Hamiltonian systems. We show that a low order system of the same type can be constructed which interpolates the original transfer function in a given set of frequencies.

Introduction On-scene detection of acute coronary occlusion (ACO) during ongoing ventricular fibrillation (VF) may facilitate patient-tailored triage and treatment during cardiac arrest. Experimental studies have demonstrated the diagnostic potential of the amplitude spectrum area (AMSA) of the VF-waveform to detect myocardial infarction (MI). In f...

The location of the spectrum and the Riesz basis property of well-posed homogeneous infinite-dimensional linear port-Hamiltonian systems on a 1D spatial domain are studied. It is shown that the Riesz basis property is equivalent to the fact that system operator generates a strongly continuous group. Moreover, in this situation the spectrum consists...

Background: Closed loop bi-hormonal artificial pancreas systems, such as the artificial pancreas (AP™) developed by Inreda Diabetic B.V., control blood glucose levels of type 1 diabetes mellitus patients via closed loop regulation. As the AP™ currently does not classify postures and movements to estimate metabolic energy consumption to correct hor...

The well-known method of images relates the solution of the heat equation on Rn (typically n = 2 or n = 3) to the solution of the heat equation on certain spatial subdomains Ω of Rn. By reformulating the method of images in terms of a convolution kernel, two novel extensions are obtained in this paper. First, the method of images is extended from t...

In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order $N \in \N$ on a bounded $1$-dimensional spatial domain $(a,b)$. In order to achieve stabilization, we couple the system to a dynamic boundary controller, that is, a controller that acts on the system only via the boundary points $a,b$ of the...

The class of port-Hamiltonian systems incorporates many physical models, such as mechanical systems in the finite-dimensional case and wave and beam equations in the infinite-dimensional case. In this paper we study a subclass of linear first order port-Hamiltonian systems. These systems are exactly observable when the energy is not dissipated inte...

This paper contributes with a new formal method of spatial discretization of a class of nonlinear distributed parameter systems that allow a port-Hamiltonian representation over a one dimensional manifold. A specific finite dimensional port-Hamiltonian element is defined that enables a structure preserving discretization of the infinite dimensional...

Two-phase flows are frequently modelled and simulated using the Two-Fluid Model (TFM) and the Drift Flux Model (DFM). This paper proposes Stokes–Dirac structures with respect to which port-Hamiltonian representations for such two-phase flow models can be obtained. We introduce a non-quadratic candidate Hamiltonian function and present dissipative H...

In this paper, we provide an overview of some control synthesis methodologies for boundary control systems (BCS) in port-Hamiltonian form. At first, it is shown how to design a state-feedback control action able to shape the energy function to move its minimum at the desired equilibrium, and how to achieve asymptotic stability via damping injection...

In this paper we consider the stabilization problem of a beam clamped on a moving inertia actuated by an external torque and force. The beam is modelled as a distributed parameter port-Hamiltonian system (PDEs), while the inertia as a finite dimensional port-Hamiltonian system (ODEs). The control inputs correspond to a torque applied by a rotating...

In this paper we study approximations to the infinite-horizon quadratic optimal control problem for linear systems that may be only asymptotically stabilizable. For linear systems, this issue only arises with infinite-dimensional systems. We provide sufficient conditions which guarantee when approximations to the optimal feedback result in the cost...

The location of the spectrum and the Riesz basis property of well-posed homogeneous infinite-dimensional linear port-Hamiltonian systems on a 1D spatial domain are studied. It is shown that the Riesz basis property is equivalent to the fact that system operator generates a strongly continuous group. Moreover, in this situation the spectrum consists...

In this note we reflect on the work and life of Ruth Curtain. This article is strongly based on the article (Zwart, 2019), which was written in Dutch.

As the title indicates, we study the stability of strongly continuous semigroups. We distinguish between exponential, strong, and weak stability. We show that the stability cannot be concluded from the spectrum of the infinitesimal generator, i.e., the spectrum determined growth assumption does not need to hold. For our classes of systems; spatiall...

This chapter studies the abstract differential equation \(\dot{z}(t) = A z(t) + f(t)\) with the initial condition \(z(0)=z_0\), where we assume that A is the infinitesimal generator of a strongly continuous semigroup. We distinguish between different notions of solutions, such as classical and mild (or weak) solution. Furthermore, we study the asym...

For the state linear system as introduced in Chap. 6, two input-output maps are introduced. The first is in time domain, and writes the output y as a convolution of the input with the impulse response. The second one is the transfer function, which is in frequency domain. Instead of introducing the transfer function via the Laplace transform, we do...

In the previous chapters our focus was on systems in which the control action is within the spatial domain. In this chapter we show that systems in which the control action is at the boundary of the spatial domain can be rewritten into one with an in-domain control action. However, this comes with the price that we introduce a derivative in the inp...

The quadratic optimal problem is studied for our class of state linear systems. This is done on a finite- and on the infinite-time interval. The solution on the infinite horizon case is related to the smallest non-negative solution of the algebraic Riccati equation. Depending on the stability properties of the open loop system, the optimal feedback...

In this chapter the system differential equation \(\dot{z}(t) = A z(t) + B u(t)\), \(y(t) = C z(t) + D u(t)\) is introduced. For this state linear system the concepts of controllability and observability are defined, and it shown that there are different generalisations of their finite-dimensional counterparts. Using the characterisation of invaria...

Strongly continuous semigroups and their infinitesimal generators are treated in detail in this chapter. The chapter contains the classical concepts and results on these topics, such as the Hille-Yosida Theorem and the contraction- and dual semigroups. Although the theory can be extended to Banach spaces, we restrict ourselves to separable Hilbert...

The general concept of strongly continuous semigroup was introduced in the previous chapter. In this chapter we focus on examples. This is done by studying three main classes, namely spatially invariant operators, Riesz-spectral operators, and delay equations. Next to existence of the strongly continuous semigroup, we study for these classes the re...

In the previous chapter we treated linear systems. In this chapter we study the semilinear differential equation \(\dot{z}(t) = A z(t) + f(z(t))\), with A the infinitesimal generator of strongly continuous semigroup. We do this for two cases, the first one in which we assume that f is Lipschitz continuous on the state space, and the second one, whe...

One of the most important concepts of systems theory is that of stabilizability and its dual concept detectability. We characterise when a system with finitely many inputs is stabilizable. Additionally, we present tests for the stabilizability/detectability of spatially invariant, Riesz-spectral, and delay systems. Similar as for finite-dimensional...

Infinite-dimensional systems is a well established area of research with an ever increasing number of applications. Given this trend, there is a need for an introductory text treating system and control theory for this class of systems in detail. This textbook is suitable for courses focusing on the various aspects of infinite-dimensional state spa...

In this letter, advection-diffusion equations with constant coefficients on infinite 1-D and 2-D spatial domains are considered. Suitable sensor and/or actuator locations are determined for which high-gain and low-gain proportional feedback can effectively reduce the influence of a disturbance at a point of interest. These locations are characteriz...

In this paper we give sufficient conditions on a semi-linear differential equation ensuring the pre-compactness of its solution. The result is illustrated by two examples of vibrating strings in a network with a static damper.

For high-performance distributed parameter motion systems, the dynamics introduced by structural flexibilities need to be considered. Especially at the low frequency region, where most of the energy of the commonly used reference setpoint is concentrated. The contribution of non-rigid body modes at low frequencies is called the compliance function...

An important step in the production of integrated circuits is the projection of the pattern of electronic connections on a silicon wafer. The light used to project the pattern moves over the wafer and induces a local temperature increase. The resulting thermal expansion of the wafer leads to a significant reduction in the imaging quality of next-ge...

Analysis of nonlocal axial vibration in a nanorod is a crucial subject in science and engineering because of its wide applications in nanoelectromechanical systems. The aim of this paper is to show how these vibrations can be modelled within the framework of port-Hamiltonian systems. It turns out that two port-Hamiltonian descriptions in physical v...

We study existence of solutions, and in particular well-posedness, for a class of inhomogeneous, nonlinear partial differential equations (PDE’s). The main idea is to use system theory to write the nonlinear PDE as a well-posed infinite-dimensional linear system interconnected with a static nonlinearity. By a simple example, it is shown that in gen...

The zero dynamics of infinite-dimensional systems can be difficult to characterize. The zero dynamics of boundary control systems are particularly problematic. In this paper the zero dynamics of port-Hamiltonian systems are studied. A complete characterization of the zero dynamics for port-Hamiltonian systems with invertible feedthrough as another...

Background: The amplitude spectrum area (AMSA) of the ventricular fibrillation (VF) waveform predicts shock success and clinical outcome after out-of-hospital cardiac arrest (OHCA). Recently, also AMSA-changes demonstrated prognostic value. Until now, most studies focused on early shocks, while many patients require prolonged resuscitations. We st...

Background Several models for educational simulation of labor and delivery were published in the literature and incorporated into a commercially available training simulator (CAE Healthcare Lucina). However, the engine of this simulator does not include a model for the clinically relevant indicators: uterine contraction amplitude and frequency, and...

The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also...

In this survey we use an operator theoretic approach to infinite‐dimensional systems theory. As this research field is quite rich, we restrict ourselves to the class of infinite‐dimensional linear port‐Hamiltonian systems and we will focus on topics such as well‐posedness, stability and stabilizability. We combine the abstract operator theoretic ap...

Old and recent experiments show that there is a direct response to the heating power of transport observed in modulated ECH experiments both in tokamaks and stellarators. This is most apparent for modulated experiments in the Large Helical Device (LHD) and in Wendelstein 7 advanced stellarator (W7-AS). In this paper we show that: (1) this power dep...

In this paper, techniques for optimal input design are used to optimize the waveforms of perturbative experiments in modern fusion devices. The main focus of this paper is to find the modulation frequency for which the accuracy of the estimated diffusion coefficient is maximal. Mathematically, this problem can be formulated as an optimization probl...

In this note, we are concerned with the stabilization of linear port-Hamiltonian systems on an interval (a, b) (for instance, vibrating strings or beams) in the presence of external disturbances. In order to achieve stabilization we couple the system to a nonlinear dynamic boundary controller whose output is allowed to be corrupted by an external d...

The heat flux is one of the key parameter used to quantify and understand transport in fusion devices. In this paper, a new method is introduced to calculate the heat flux including its confidence with high accuracy based on perturbed measurements such as the electron temperature. The new method is based on ideal filtering to optimally reduce the n...

Moving heat load problems appear in many manufacturing processes, such as lithography, welding, grinding, and additive manufacturing. The simulation of moving heat load problems by the finite-element method poses several numerical challenges, which may lead to time consuming computations. In this paper, we propose a 2D semi-analytic model in which...

In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order $N \in \mathbb{N}$ on a bounded $1$-dimensional spatial domain $(a,b)$. In order to achieve stabilization, we couple the system to a dynamic boundary controller, that is, a controller that acts on the system only via the boundary points $a,b...

Assuming only strong stabilizability, we construct the maximal solution of the algebraic Riccati equation as the strong limit of a Kleinman–Newton sequence of bounded nonnegative operators. As a corollary we obtain a comparison of the solutions of two algebraic Riccati equations associated with different cost functions. We show that the weaker stro...

The zero dynamics of infinite-dimensional systems can be difficult to characterize. The zero dynamics of boundary control systems are particularly problematic. In this paper the zero dynamics of port-Hamiltonian systems are studied. A complete characterization of the zero dynamics for a port-Hamiltonian systems with invertible feedthrough as anothe...

The conditions for existence of solutions and stability, asymptotic and exponential, of a large class of boundary controlled systems on a 1D spatial domain subject to nonlinear dynamic boundary actuation are given. The consideration of such class of control systems is motivated by the use of actuators and sensors with nonlinear behavior in many eng...

This article deals with characterizations of $L^\infty$-admissible operators for linear control systems. We present a result linking a version of the Weiss conjecture for $L^{\infty}$ to the boundedness of the $H^\infty$-calculus for analytic semigroup generators $A$. Moreover, we obtain admissibility with respect to Orlicz spaces, which has conseq...

A new methodology to analyze non-linear components in perturbative transport experiments is introduced. The methodology has been experimentally validated in the Large Helical Device for the electron heat transport channel. Electron cyclotron resonance heating with different modulation frequencies by two gyrotrons has been used to directly quantify...

In 1968 Fattorini published in SIAM Journal on Control his well-known paper with the title Boundary Control Systems. In this paper he sets up the abstract theory for systems described by partial differential equations with a control at the boundary. The theory as presented is complete, meaning that it could be copied into textbooks and articles wit...

The stability of an undamped Euler Bernoulli beam connected to non-linear mass spring damper systems is addressed. It is shown that under mild assumptions on the local behaviour of the non-linear springs and dampers the solutions exist and the system is globally asymptotically stable.

Transient electron temperature measurements of a step power experiment at W7-AS are reassessed by direct comparison of the up- and downward responses of the electron temperature. The analysis shows that the response at some distance to the center behaves linearly and the model predicted responses based on a power-dependent diffusion coefficient tha...

Small inter-vehicle distances can increase traffic throughput on highways. Human drivers are not able to drive safely under such conditions. To this aim, cooperative adaptive cruise control (CACC) systems have been developed, which require vehicles to communicate with each other through a wireless communication network. By communicating control-rel...

In this short note we show that under some mild conditions on the space and the operators, an estimate for $\|Sf(A) - f(B)S\|$ needs only to be studied for invertible $S$ and $B$ equal to $A$. Thus estimates for a quasi-commutator can be derived from that for the commutator.

The implementation of lightweight high-performance motion systems in lithography and other applications imposes lower requirements on actuators, amplifiers, and cooling. However, the decreased stiffness of lightweight designs increases the effect of structural flexibilities especially when the point of interest is not at a fixed location. This is f...

The asymptotic stability of boundary controlled port-Hamiltonian systems defined on a 1D spatial domain interconnected to a class of non-linear boundary damping is addressed. It is shown that if the port-Hamiltonian system is approximately observable, then any boundary damping which behaves linear for small velocities asymptotically stabilizes the...

We design control laws for a class of dissipative infinite-dimensional systems using nonlinear boundary control action. The applications are to saturated control of SCOLE-type models.

Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated to a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, a...

We introduce LPMLE3, a new 1D approach to quantify vertical water flow components at streambeds using temperature data collected in different depths. LPMLE3 solves the partial differential equation for coupled water flow and heat transport in the frequency domain. Unlike other 1D approaches it does not assume a semi-infinite halfspace with the loca...

Single-degree-of-freedom (single-DOF) nonlinear mechanical systems under periodic excitation may possess multiple coexisting stable periodic solutions. Depending on the application, one of these stable periodic solutions is desired. In energy-harvesting applications, the large-amplitude periodic solutions are preferred, and in vibration reduction p...

In this article we introduce a technique that derives from the existence and uniqueness of solutions to a simple hyperbolic partial differential equation (p.d.e.) the existence and uniqueness of solutions to hyperbolic and parabolic p.d.e.’s. Among others, we show that starting with an impedance passive system associated to the undamped wave equati...

This paper is concerned with the energy shaping of 1-D linear boundary controlled port-Hamiltonian systems. The energy-Casimir method is first proposed to deal with power preserving systems. It is shown how to use finite dimensional dynamic boundary controllers and closed-loop structural invariants to partially shape the closed-loop energy function...

This volume collects a selected number of papers presented at the International Workshop on Operator Theory and its Applications (IWOTA) held in July 2014 at Vrije Universiteit in Amsterdam. Main developments in the broad area of operator theory are covered, with special emphasis on applications to science and engineering. The volume also presents...

Consider a network with linear dynamics on the edges, and observation and control in the nodes. Assume that on the edges there is no damping, and so the dynamics can be described by an infinite-dimensional, port-Hamiltonian system. For general infinite-dimensional systems, the zero dynamics can be difficult to characterize and are sometimes ill-pos...

In this discussion paper we present an idea of combining techniques known from systems theory with energy estimates to show existence for a class of non-linear partial differential equations (pde's). At the end of the paper a list of research questions with possible approaches is given.