arXiv subjects

Michael Schönlein

Publications and source records attributed to Michael Schönlein.

11 recordsLinked to original sources

arXiv2026

A mathematical model for depression and resilience

A basic dynamical model for (clinical) depression is presented to describe the time evolution of two coupled states: a resilience level and a depression symptom. The resilience level can also be interpreted in terms of the memory of past symptoms. The model consists of a system of two coupled first order differential equations without free parameters that qualitatively captures different courses of illness, without the overhead of a derivation from neuroscientific first principles. A comprehensive mathematical analysis of the model is provided, including equilibria, stability properties, and monotonicity. The model can reproduce chronic, delayed, recovery, and resilience scenarios that are prominent in the literature, as well as scenarios such as improvement from pre-existing conditions, burnout from low-grade adversity, or isolated and recurrent depressive episodes. Supplementary to the manuscript are computational tools to replicate the figures and, without prior mathematical or programming skills, to experiment with the model interactively in the web browser (at https://rsmodel.org/).

math.DS↗

arXiv2026

Ensemble Feedback Methods for Families of Linear Systems

We consider feedback stabilization for one-parameter families of finite-dimensional linear systems over compact parameter sets in the complex field. Classical ensemble feedback induces compact control operators and therefore cannot modify the essential spectrum of the associated multiplication operators describing the free motion of the system. This precludes stabilization in many infinite-dimensional settings. To address this issue, multiplication feedback operators are introduced. For systems with constant Hermite indices, an analogue of Heymann's lemma is proved, as well as a pole placement theorem, and stabilization results. The relation of pointwise and ensemble controllability is investigated. For systems with nonconstant Hermite indices, corresponding results are obtained under additional assumptions on the structure of the parameter set.

math.OC↗

arXiv2023

Polynomial methods to construct inputs for uniformly ensemble reachable linear systems

This paper is concerned with linear parameter-dependent systems and considers the notion uniform ensemble reachability. The focus of this work is on constructive methods to compute suitable parameter-independent open-loop inputs for such systems. In contrast to necessary and sufficient conditions for ensemble reachability, computational methods have to distinguish between continuous-time and discrete-time systems. Based on recently derived sufficient conditions and techniques from complex approximation we present two algorithms for discrete-time single-input linear systems. Moreover, we illustrate that one method can also be applied to certain continuous-time single-input systems.

math.OC↗

arXiv2023

A Short Note on Output Controllability

In this paper we consider output controllability for linear time-invariant systems. In a recent paper by Danhane, Loh{é}ac and Jungers it has been pointed out that although output controllability is a classical notion in control theory, only a Kalman-type characterization was available in the literature. In their work that authors consider time-varying linear systems. In this short note we provide a Hautus-type characterization of output controllability for time-invariant linear systems which is more comprehensive and reduces to state controllability in case the output matrix is the identity matrix.

math.OC↗

arXiv2021

Computation of open-loop inputs for uniformly ensemble controllable systems

This paper presents computational methods for families of linear systems depending on a parameter. Such a family is called ensemble controllable if for any family of parameter-dependent target states and any neighborhood of it there is a parameter-independent input steering the origin into the neighborhood. Assuming that a family of systems is ensemble controllable we present methods to construct suitable open-loop input functions. Our approach to solve this infinite-dimensional task is based on a combination of methods from the theory of linear integral equations and finite-dimensional control theory.

math.OC↗

arXiv2021

Feedback equivalence and uniform ensemble reachability

This paper considers feedback methods for ensemble reachability of parameter-dependent linear systems $(A(θ),B(θ))$, where the parameter $θ$ is varying over a compact Jordan arc in the complex plane. Recently, pointwise testable sufficient conditions for uniform ensemble reachability have been developed. Beside the necessity of pointwise reachablility these conditions put restrictions on the spectra of the matrices $A(θ)$ and the Hermite indices of the pair $(A(θ),B(θ))$. In this paper we show that these conditions can be ensured by applying a suitable feedback transformation if the pair $(A(θ),B(θ))$ is pointwise reachable and it Kronecker indices are independent from the parameter.

math.OC↗

arXiv2020

Uniform and $L^q$-Ensemble Reachability of Parameter-dependent Linear Systems

In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs $\big( A(θ),B(θ) \big)$ depending on a parameter $θ\in \p$ that is varying over a compact subset $\p$ of the complex plane. In particular, we investigate the following control task: Find an open-loop control which is {\it independent} of the parameter $θ\in \p$ and steers a given family of initial states $x_0(θ)$ arbitrarily close to a desired family of terminal states $f(θ)$ in finite time. Here, the maps $θ\mapsto x_0(θ)$ and $θ\mapsto f(θ)$ are assumed to lie in a common appropriately chosen {Banach space $X_n(\p)$ of $\C^n$-valued functions}. If this task is solvable for all initial and terminal states, the pair $\big( A(θ),B(θ) \big)$ is called {(completely)} ensemble controllable with respect to $X_n(\p)$. Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family $A(θ)$. Based on thses findings as well as approximation theory and cyclicity conditions of multiplications operators, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the Banach spaces of continuous functions and $L^q$-functions. In the last section, results on {averaged} controllability (reachability) for linear families $\big( A(θ),B(θ),C(θ) \big)$ are presented.

math.OC↗

arXiv2015

Controllability of ensembles of linear dynamical systems

We investigate the task of controlling ensembles of initial and terminal state vectors of parameter-dependent linear systems by applying parameter-independent open loop controls. Necessary, as well as sufficient, conditions for ensemble controllability are established, using tools from complex approximation theory. For real analytic families of linear systems it is shown that ensemble controllability holds only for systems with at most two independent parameters. We apply the results to networks of linear systems and address the question of open-loop robust synchronization.

math.OC↗

arXiv2015

Asymptotic Stability and Smooth Lyapunov Functions for a Class of Abstract Dynamical Systems

This paper deals with asymptotic stability of a class of dynamical systems in terms of smooth Lyapunov pairs. We point out that well known converse Lyapunov results for differential inclusions cannot be applied to this class of dynamical systems. Following an abstract approach we put an assumption on the trajectories of the dynamical systems which demands for any trajectory the existence of a neighboring trajectory such that their difference grows linearly in time and distance of the starting points. Under this assumption, we prove the existence of a $C^\infty$-smooth Lyapunov pair. We also show that this assumption is satisfied by differential inclusions defined by Lipschitz continuous set-valued maps taking nonempty, compact and convex values.

math.DS↗

arXiv2012

A Lyapunov view on positive Harris recurrence of multiclass queueing networks

This paper addresses the question when the underlying Markov process of a multiclass queueing network is positive Harris recurrent. It is well-known that stability of the fluid limit model is a sufficient condition for this. Hence, stability of fluid (limit) models is of vital interest. Recently, it has been shown that if the fluid model satisfies certain properties, it is stable if and only if there exists a Lyapunov function. In this paper a new method is provided to conclude that the underlying Markov process is positive Harris recurrent if the fluid model is stable by using explicitly the Lyapunov function of the fluid model.

math.PR↗

arXiv2011

On converse Lyapunov theorems for fluid network models

We consider the class of closed generic fluid networks (GFN) models, which provides an abstract framework containing a wide variety of fluid networks. Within this framework a Lyapunov method for stability of GFN models was proposed by Ye and Chen. They proved that stability of a GFN model is equivalent to the existence of a functional on the set of paths that is decaying along paths. This result falls short of a converse Lyapunov theorem in that no state dependent Lyapunov function is constructed. In this paper we construct state-dependent Lyapunov functions in contrast to path-wise functionals. We first show by counterexamples that closed GFN models do not provide sufficient information that allow for a converse Lyapunov theorem. To resolve this problem we introduce the class of strict GFN models by forcing the closed GFN model to satisfy a concatenation and a semicontinuity condition of the set of paths in dependence of initial condition. For the class of strict GFN models we define a state-dependent Lyapunov and show that a converse Lyapunov theorem holds. Finally, it is shown that common fluid network models, like general work-conserving and priority fluid network models as well as certain linear Skorokhod problems define strict GFN models.

math.DS↗