Felix Schwenninger

University of Twente | UT · Department of Applied Mathematics

In this paper, we establish several results related to Crouzeix's conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of r...

We study a nonlinear, non-autonomous feedback controller applied to boundary control systems. Our aim is to track a given reference signal with prescribed performance. Existence and uniqueness of solutions to the resulting closed-loop system is proved by using nonlinear operator theory. We apply our results to both hyperbolic and parabolic equation...

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...

Crouzeix and Palencia recently showed that the numerical range of a Hilbert-space operator is a $(1+\sqrt2)$-spectral set for the operator. One of the principal ingredients of their proof can be formulated as an abstract functional-analysis lemma. We give a new short proof of the lemma and show that, in the context of this lemma, the constant $(1+\...

Input-to-state stability (ISS) for systems described by partial differential equations has seen intensified research activity recently, and in particular the class of boundary control systems, for which truly infinite-dimensional effects enter the situation. This note reviews input-to-state stability for parabolic equations with respect to general...

We study the question of when a distributed port-Hamiltonian system is bounded-input bounded-output (BIBO) stable. Exploiting the particular structure of the transfer function of these systems, we derive several sufficient conditions for BIBO stability.

We investigate when the algebraic numerical range is a $C$-spectral set in a Banach algebra. While providing several counterexamples based on classical ideas as well as combinatorial Banach spaces, we discuss positive results for matrix algebras and provide an absolute constant in the case of complex $2\times2$-matrices with the induced $1$-norm. F...

We characterise quantitative semi-uniform stability for $C_0$-semigroups arising from port-Hamiltonian systems, complementing recent works on exponential and strong stability. With the result, we present a simple universal example class of port-Hamiltonian $C_0$-semigroups exhibiting arbitrary decay rates slower than $t^{-1/2}$. The latter is based...

We thoroughly analyse the double-layer potential's role in approaches to spectral sets in the spirit of Delyon--Delyon, Crouzeix and Crouzeix--Palencia. While the potential is well-studied, we aim to clarify on several of its aspects in light of these references. In particular, we illustrate how the associated integral operators can be used to char...

In this paper we consider BIBO stability of infinite-dimensional linear state-space systems and the related notion of $L^1$-to-$L^1$ input-output stability (abbreviated LILO). We show that in the case of finite-dimensional input and output spaces, both are equivalent and preserved under duality transformations. In the general case, neither of these...

We extend classical duality results by Weiss on admissible operators to settings where the dual semigroup lacks strong continuity. This is possible using the sun-dual framework, which is not immediate from the duality of the input and output maps. This extension enables the testing of admissibility for a broader range of examples, in particular for...

We prove bounds for a class of homomorphisms arising in the study of spectral sets, by involving extremal functions and vectors. These are used to recover three celebrated results on spectral constants by Crouzeix–Palencia, Okubo–Ando and von Neumann in a unified way and to refine a recent result by Crouzeix–Greenbaum.

The sun dual space corresponding to a strongly continuous semigroup is a known concept when dealing with dual semigroups, which are in general only weak $$^*$$ ∗ -continuous. In this paper we develop a corresponding theory for bi-continuous semigroups under mild assumptions on the involved locally convex topologies. We also discuss sun reflexivity...

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...

Properly flying a walkalong glider is a challenging control problem, involving internal and external influences. We approach this problem from both a model-driven and data-driven viewpoint. We construct a mathematical model, based on well-established theory, which is used to derive both optimal paddle angle and distance between the paddle and the g...

The approximation properties of infinitely wide shallow neural networks heavily depend on the choice of the activation function. To understand this influence, we study embeddings between Barron spaces with different activation functions. These embeddings are proven by providing push-forward maps on the measures $\mu$ used to represent functions $f$...

We extend two characterizations of admissible operators with respect to Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{L}^p$$\end{document} to more general O...

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 derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. We show that input-to-state stability of a linear system does not imply existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded. If, however, the semigroup is similar to a contraction semigr...

Input-to-state stability estimates with respect to small initial conditions and input functions for infinite-dimensional systems with bilinear feedback are shown. We apply the obtained results to controlled versions of a viscous Burger equation with Dirichlet boundary conditions, a Schr\"odinger equation, a Navier--Stokes equation and a semilinear...

This note deals with Bounded-Input-Bounded-Output (BIBO) stability for semilinear infinite-dimensional dynamical systems allowing for boundary control and boundary observation. We give sufficient conditions that guarantee BIBO stability based on Lipschitz conditions with respect to interpolation spaces. Our results can be applied to guarantee feasi...

We prove bounds for a class of unital homomorphisms arising in the study of spectral sets, by involving extremal functions and vectors. These are used to recover three celebrated results on spectral constants by Crouzeix--Palencia, Okubo--Ando and von Neumann in a unified way and to refine a recent result by Crouzeix--Greenbaum.

Reproducing Kernel Hilbert spaces (RKHS) have been a very successful tool in various areas of machine learning. Recently, Barron spaces have been used to prove bounds on the generalisation error for neural networks. Unfortunately, Barron spaces cannot be understood in terms of RKHS due to the strong nonlinear coupling of the weights. This can be so...

Dissipation of energy — as well as its sibling the increase of entropy — are fundamental facts inherent to any physical system. The concept of dissipativity has been extended to a more general system theoretic setting via port-Hamiltonian systems and this framework is a driver of innovations in many of areas of science and technology. The particula...

We study integral input-to-state stability of bilinear systems with unbounded control operators and derive natural sufficient conditions. The results are applied to a bilinearly controlled Fokker–Planck equation.

The sun dual space corresponding to a strongly continuous semigroup is a known concept when dealing with dual semigroups, which are in general only weak$^*$-continuous. In this paper we develop a corresponding theory for bi-continuous semigroups under mild assumptions on the involved locally convex topologies. We also discuss sun reflexivity and Fa...

We investigate the well-posedness of the radiative transfer equation with polarization and varying refractive index. The well-posedness analysis includes non-homogeneous boundary value problems on bounded spatial domains, which requires the analysis of suitable trace spaces. Additionally, we discuss positivity, Hermiticity, and norm-preservation of...

In contrast to classical strongly continuous semigroups, the study of bi-continuous semigroups comes with some freedom in the properties of the associated locally convex topology. This paper aims to give minimal assumptions in order to recover typical features like tightness and equicontinuity with respect to the mixed topology as well as to carefu...

We study tracking control for a nonlinear moving water tank system modeled by the linearized Saint-Venant equations, where the output is given by the position of the tank and the control input is the force acting on it. For a given reference signal, the objective is that the tracking error evolves within a pre-specified performance funnel. Exploiti...

In this work we continue developments on the $p$-Weiss conjecture, which characterizes $\mathrm{L}^p$-admissibility in terms of a resolvent condition for a special class of Orlicz spaces. This extends previous derived characterizations due to Le Merdy ($p=2$) and Haak ($p \geq 1$), under the assumption of bounded analytic semigroups.

In contrast to classical strongly continuous semigroups, the study of bi-continuous semigroups comes with some freedom in the properties of the associated locally convex topology. This paper aims to give minimal assumptions in order to recover typical features like tightness and equicontinuity with respect to the mixed topology as well as to carefu...

New results on the boundedness of Laplace-Carleson embeddings on $L^\infty$ and Orlicz spaces are proved. These findings are crucial for characterizing admissibility of control operators for linear diagonal semigroup systems in a variety of contexts. A particular focus is laid on essentially bounded inputs.

The following corrections should be made to this article.

Crouzeix’s conjecture asserts that, for any polynomial f and any square matrix A, the operator norm of f(A) satisfies the estimate $$\begin{aligned} \Vert f(A)\Vert \le 2\,\sup \{|f(z)|:\ z \in W(A)\}, \end{aligned}$$ (1) where \(W(A):=\{\langle Ax,x\rangle : \Vert x\Vert =1\}\) denotes the numerical range of A. This would then also hold for all fu...

We investigate input-to-state stability (ISS) of infinite-dimensional collocated control systems subject to saturated feedback. Here, the unsaturated closed loop is dissipative and uniformly globally asymptotically stable. Under an additional assumption on the linear system, we show ISS for the saturated one. We discuss the sharpness of the conditi...

By Baillon's result, it is known that maximal regularity with respect to the space of continuous functions is rare; it implies that either the involved semigroup generator is a bounded operator or the considered space contains $c_{0}$. We show that the latter alternative can be excluded under a refined condition resembling maximal regularity with r...

We consider output trajectory tracking for a class of uncertain nonlinear systems whose internal dynamics may be modelled by infinite-dimensional systems which are bounded-input, bounded-output stable. We describe under which conditions these systems belong to an abstract class for which funnel control is known to be feasible. As an illustrative ex...

We investigate input-to-state stability of infnite-dimensional collocated control systems subject to saturated feedback where the unsaturated closed loop system is dissipative and uniformly globally asymptotically stable. We review recent results from the literature and explore limitations thereof.

We consider output trajectory tracking for a class of uncertain nonlinear systems whose internal dynamics may be modelled by infinite-dimensional systems which are bounded-input, bounded-output stable. We describe under which conditions these systems belong to an abstract class for which funnel control is known to be feasible. As an illustrative ex...

We study tracking control for a moving water tank system, which is modelled using the Saint-Venant equations. The output is given by the position of the tank and the control input is the force acting on it. For a given reference signal, the objective is to achieve that the tracking error evolves within a prespecified performance funnel. Exploiting...

We study input-to-state stability of bilinear control system with a possibly unbounded control operator. Sufficient conditions for integral input-to-state stability are given. The obtained results are applied to the bilinearly controlled Fokker-Planck equation.

We consider constant-coefficient differential-algebraic equations from an operator theoretic point of view. We show that the Kronecker form allows to determine the nullspace and range of the corresponding differential-algebraic operators. This yields simple matrix-theoretic characterizations of features like closed range and Fredholmness.

This paper deals with strong versions of input-to-state stability and integral input-to-state stability of infinite-dimensional linear systems with an unbounded input operator. We show that infinite-time admissibility with respect to inputs in an Orlicz space is a sufficient condition for a system to be strongly integral input-to-state stable but,...

This work contributes to the recently intensified study of input-to-state stability for infinite-dimensional systems. The focus is laid on the relation between input-to-state stability and integral input-to-state stability for linear systems with a possibly unbounded control operator. The main result is that for parabolic diagonal systems both noti...

In this work, the relation between input-to-state stability and integral input-to-state stability is studied for linear infinite-dimensional systems with an unbounded control operator. Although a special focus is laid on the case $L^{\infty}$, general function spaces are considered for the inputs. We show that integral input-to-state stability can...

We investigate the boundedness of the $H^\infty$-calculus by estimating the bound $b(\varepsilon)$ of the mapping $H^{\infty}\rightarrow \mathcal{B}(X)$: $f\mapsto f(A)T(\varepsilon)$ for $\varepsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\infty}$ is the space of bounded analytic functions on a domain strictly containi...

We show $H^{\infty}$-functional calculus estimates for Tadmor-Ritt operators (also known as Ritt operators), which generalize and improve results by Vitse. These estimates are in conformity with the best known power-bounds for Tadmor-Ritt operators in terms of the constant dependence. Furthermore, it is shown how discrete square function estimates...

For $\left(C(t)\right)_{t\in\mathbb R}$ being a cosine family on a unital normed algebra, we show that the estimate $\limsup_{t\to\infty^{+}}\|C(t) - I\| <2$ implies that $C(t)=I$ for all $t\in\mathbb R$. This generalizes the result that $\sup_{t\geq0}\|C(t)-I\|<2$ yields that $C(t)=I$ for all $t\geq0$. We also state the corresponding result for di...

For $\left(C(t)\right)_{t \geq 0}$ being a strongly continuous cosine family on a Banach space, we show that the estimate $\limsup_{t\to0^{+}}\|C(t) - I\| <2$ implies that $C(t)$ converges to $I$ in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of $\sup_{t\geq0}\|C(t)-I\|<2$...

In this short note we use ideas from systems theory to define a functional calculus for infinitesimal generators of strongly continuous semigroups on a Hilbert space. Among others, we show how this leads to new proofs of (known) results in functional calculus.

Assume that a block operator of the form $\left(\begin{smallmatrix}A_{1}\\A_{2}\quad 0\end{smallmatrix}\right)$, acting on the Banach space $X_{1}\times X_{2}$, generates a contraction $C_{0}$-semigroup. We show that the operator $A_{S}$ defined by $A_{S}x=A_{1}\left(\begin{smallmatrix}x\\SA_{2}x\end{smallmatrix}\right)$ with the natural domain gen...

In this paper we show that from the estimate $\sup_{t \geq 0}\|C(t) - \cos(at)I\| <1$ we can conclude that $C(t)$ equals $\cos(at) I$. Here $\left(C(t)\right)_{t \geq 0}$ is a strongly continuous cosine family on a Banach space.

We show that, given a reflexive Banach space and a generator of an exponentially stable C-0-semigroup, a weakly admissible operator g (A) can be defined for any g bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separabl...

We show that, given a Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable H...