Stability analysis of infinite dimensional discrete and continuous time linear systems

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics are symmetric and uniformly elliptical and by using the properties of the Mittag–Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.

Convergence and convergence rate of the balanced realization truncations for infinite-dimensional discrete-time systems

In this study we consider the robust control of an inclined cable modelled using partial differential equations and subjected to external disturbances. This paper focuses on the construction of a standard linear infinite dimensional state space system and an H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> feedback control with full observation of the state.

This paper deals with the reduced order controller design for infinite dimensional port Hamiltonian systems (IDPHS). Firstly, a structure preserving and passive LQG control design equivalent to Control by Interconnection is proposed. Based on this LQG controller, a structure preserving reduction method is used to approximate both the closed loop IDPHS and the LQG controller. This closed loop reduction guarantees that the reduced order controller will ensure acceptable closed loop performances on the infinite dimensional system. The proposed methods is applied to the control of a vibro-acoustic system.

We establish left and right canonical factorizations of Hilbert-space operator-valued functions G ( z ) that are analytic on neighborhoods of the complex unit circle $${\mathbb {T}}$$ T and the origin 0 and that have the form $$G(z)=I+F(z)$$ G ( z ) = I + F ( z ) with F ( z ) taking strictly contractive values on $${\mathbb {T}}$$ T . Such functions can be realized as transfer functions of infinite dimensional dichotomous discrete-time linear systems, and we employ the strict bounded real lemma for this class of operators, together with associated Kreĭn space theory, to derive explicit formulas for the left and right canonical factorizations.

The theory of balanced realizations is essentially based on co-ordinating controllability and observability properties of a finite dimensional linear asymptotically stable system. By analyzing more general quadratic measures on the external behaviour of a dynamical system, we may obtain state space representations in which other quantities are internally balanced. For the class of L 2-trajectories of finite dimensional linear time invariant systems, we formalize the notion of a balanced state space with respect to these type of quadratic measures. A particular choice of quadratic measures is studied in more detail and it is shown that ‘classical’ balanced realizations are obtained as a special case.KeywordsBalanced realizationsfinite dimensional linear time invariant systemsstate space methodsmodel approximation

On Lipschitz conditions of infinite dimensional systems

In this paper we attempt to describe a circle of ideas which makes possible a unified exposition of a large part of linear algezbra, operator theory and both finite and infinite dimensional multivariable linear systems. As the title suggests the unifying conzcepts will be those of functional models, module theory and various factorizations of polynomial and analytic matrix valued functions. The full exposition of the ideas presented here will be the theme of a forthcoming monograph [20].

We consider the infinite dimensional linear control system described by the population dynamics model of Lotka-McKendrick with spatial diffusion. Considering control functions localized with respect to the spatial variable but active for all ages, we prove that the whole population can be steered to zero in any positive time. The main novelty we bring is that, unlike the existing results in the literature, we can also control the population of ages very close to 0. Another novelty brought in is the employed methodology: as far as we know, the present work is the first one remarking that the null controllability of the considered system can be obtained by using the Lebeau-Robbiano strategy, originally developed for the null-controllability of the heat equation.

Social network models aim to capture the complex structure of social connections. They are a framework for the design of control algorithms that take into account relationships, interactions, and communications between social actors. Based on three formation mechanisms — random attachment, triad formation, and network response — our work characterizes the dynamics of the degree distributions of social networks. In particular, we show that the complementary cumulative in- and out-degree distributions of highly clustered, reciprocal networks can be approximated by infinite dimensional time-varying linear systems. Furthermore, we determine the invariance of both limit distributions and the stability properties of the average degree.

This paper provides a general theory for characterizing and constructing a decoherence-free (DF) subsystem for an infinite dimensional linear open quantum system. The main idea is that, based on the Heisenberg picture of the dynamics rather than the commonly-taken Schrodinger picture, the notions of controllability and observability in control theory are employed to characterize a DF subsystem. A particularly useful result is a general if and only if condition for a linear system to have a DF component; this condition is used to demonstrate how to actually construct a DF dynamics in some specific examples. It is also shown that, as in the finite dimensional case, we are able to do coherent manipulation and preservation of a state of a DF subsystem.

A general perturbation theory of the Kolmogorov-Arnold-Moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems. The key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures. This approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems.

This paper describes the classification problem of continuous families of infinite dimensional linear systems. We are concerned with a family of controllable and observable linear systems $(F(s),G(s),H(s))$ in Banach spaces, which depend continuously on a parameter s in a Hausdorff topological space S. Our problem is to classify such families of systems by isomorphism relation and parametrize them by “moduli.” It is known that there exist “moduli” for finite dimensional controllable and observable systems. We show the result fails in the infinite dimensional case, and derive some conditions under which there exist “moduli,” i.e., a fine moduli theorem for infinite dimensional systems. The theory of Banach bundles plays an important role in our study.

A class of infinite dimensional linear systems over the binary field is introduced. Necessary and sufficient conditions for realizability by such systems are obtained. In particular, their input/output relations can be described by certain regular sets. An algebraic method is introduced to compute the generating functions of an example. Also bounds are placed on the complexity of the formal languages which can be recognised by these systems.

This paper is concerned with constructing observers for infinite dimensional linear systems characterized by semigroups on Hilbert spaces. Two types of observers (identity-type and general-type) are considered. Sufficient conditions are given for both types to exist for infinite dimensional systems. The basic tools of our approach make use of the properties of invariant and reducing subspaces of Hilbert spaces and the canonical decomposition of contraction semigroups.