Controllability of linear systems on Heisenberg groups ℍ n

Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner stability, asymptotic stability, and BIBO (Bounded Input, Bounded Output) stability, are well understood for classical linear control systems in Euclidean spaces. This paper aims to thoroughly address the stability problem for a class of linear control systems defined on matrix Lie groups. This approach generalizes classical models corresponding to the latter when the group is Abelian and non-compact. It is important to note that this generalization leads to a very difficult control system, due to the complexity of the state space and the special dynamics resulting from the drift and control vectors. Several mathematical concepts help us understand and characterize stability in the classical case. We first show how to extend these algebraic, topological, and dynamical concepts from Euclidean space to a connected Lie group of matrices. Building on classical results, we identify a pathway that enables us to formulate conjectures about stability in this broader context. This problem is closely linked to the controllability and observability properties of the system. Fortunately, these properties are well established for both classes of linear systems, whether in Euclidean spaces or on Lie groups. We are confident that these conjectures can be proved in future work, initially for the class of nilpotent and solvable groups, and later for semi-simple groups. This will provide valuable insights that will facilitate, through Jouan’s Equivalence Theorem, the analysis of an important class of nonlinear control systems on manifolds beyond Lie groups. We provide an example involving a three-dimensional solvable Lie group of rigid motions in a plane to illustrate these conjectures.

A finitely generated subgroup${\rm\Gamma}$of a real Lie group$G$is said to be Diophantine if there is${\it\beta}>0$such that non-trivial elements in the word ball$B_{{\rm\Gamma}}(n)$centered at$1\in {\rm\Gamma}$never approach the identity of$G$closer than$|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group$G$is said to be Diophantine if for every$k\geqslant 1$a random$k$-tuple in$G$generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most$5$, or derived length at most$2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

This paper is concerned with the topological structure of solvmanifolds i.e. spaces on which solvable Lie groups of transformations act transitively. The special case of nilmanifolds in which the group of transformations is nilpotent has been studied by Malcev [5]. Examples of two dimensional solvmanifolds are the torus, Klein bottle, cylinder, mobius strip. The central results are THEOREM A. Two compact solvmanifolds having the same fundamental group are isomorphic. THEOREM B. Any solvmanifold is covered a finite number of times by the direct product of a compact solvmanifold and a euclidean space; moreover, the covering group is abelian. THEOREM C. If S is an algebraically subgroup (cf. ?9 for definition) of the solvable Lie group G, then G/S is homeomorphic to the direct product of a compact solvmanifold and a euclidean space. Theorem C applies for example, if S is a closed subgroup which is generated by its connected identity-component together with elements s1, s2, -such that all the eigenvalues of Adsi are positive (i = 1, 2, * ). The assertion in Theorem A, applied to nilmanifolds, was proved by Malcev [5]. The situation in that case is simplified by virtue of the fact that two nilpotent groups which operate transitively and effectively on spaces with same fundamental group are isomorphic-a fact which is not true in the case of general solvmanifolds. Accordingly, there is a natural homeomorphism between nilmanifolds with isomorphic fundamental groups but there is no natural homeomorphism in the case of solvmanifolds. In the absence of uniqueness, therefore, proof of the existence of the homeomorphisin in Theorem A is rather elusive. After a number of algebraic maneuvers, proof of Theorem A is reduced to comparing two fibre bundles over homeomorphic toroids-the fibres being the orbits of transformation groups which are isomorphic as abstract topological groups. The main topological lemma asserts the existence of an equivariant homeomorphism between the two bundles. The major algebraic fact which underlies our method is the theorem of ?5 on uniform subgroups of solvable groups. In Theorem B, the covering need not be univalent-as is illustrated by the mobius band. The sufficiency condition in Theorem C yields as a special case a result of Chevalley: the factor space of a solvable Lie group by a connected subgroup is a product of circles and straight lines. Theorem A yields more generally

In this paper, we continue with the results in \cite{Pg} and compute the group of quasi-isometries for a subclass of split solvable unimodular Lie groups. Consequently, we show that any finitely generated group quasi-isometric to a member of the subclass has to be polycyclic, and is virtually a lattice in an abelian-by-abelian solvable Lie group. We also give an example of a unimodular solvable Lie group that is not quasi-isometric to any finitely generated group, as well deduce some quasi-isometric rigidity results.

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

Let G be a connected Lie group with Lie algebra g. This review is devoted to studying the fundamental dynamic properties of elements in the normalizer NG of G. Through an algebraic characterization of NG, we analyze the different dynamics inside the normalizer. NG contains the well-known left-invariant vector fields and the linear and affine vector fields on G. In any case, we show the shape of the solutions of these ordinary differential equations on G. We give examples in low-dimensional Lie groups. It is worth saying that these dynamics generate the linear and bilinear control systems on Euclidean spaces and the invariant and linear control systems on Lie groups. Moreover, the Jouan Equivalence Theorem shows how to extend this theory to control systems on manifolds.

In this paper we consider several applications of bilinear stochastic models in which state estimation is an important problem. Bilinear stochastic models occur naturally in many communication problems, including noisy oscillators and phase-lock loops, in which the system evolves on the circle Sl. Similar models arise in the estimation of the position of an orbiting body (in which the state evolves on the 2-sphere S2) and in the estimation of the orientation of a rotating rigid body (which evolves on SO(3)).Three techniques for the solution of bilinear estimation problems are presented. First, finite dimensional optimal nonlinear estimators are presented for certain bilinear systems evolving on solvable and nilpotent Lie groups. Then the use of harmonic analysis for estimation problems evolving on spheres and other compact manifolds is investigated. Finally, an approximate estimation technique utilizing cumulants is discussed.KeywordsBilinear SystemNonlinear FilterBilinear EquationBrownian Motion ProcessVariable Structure SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Performance limits and deconstraints of linear feedback control systems disturbed by random noise are studied from the viewpoint of information theory. By using Bode integral formulas, information transmission in such systems is connected to unstable poles and nonminimum phase zeros of open-loop transfer functions. Performance limits of disturbance rejection are discussed by using such connection, and the conservation law of variety is revised based on a new extended definition of variety. Meanwhile, a necessary information condition which implies a new kind of design constraints of linear tracking problem is proposed with the concept of mutual information rate, and the law of requisite variety is derived from this condition.

A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is easy to decide whether a nilmanifold is formal, Kaehlerian or (Hard) Lefschetz. The study of solvmanifolds meets with noticeably greater obstacles than the study of nilmanifolds. Even the construction of solvmanifolds is considerably more difficult than is the case for nilmanifolds. The reason is that there is no simple criterion for the existence of a lattice in a connected and simply-connected solvable Lie group. We consider the question of existence of lattices in solvable Lie groups up to dimension six and examine whether the corresponding solvmanifolds are formal, symplectic, Kaehler or (Hard) Lefschetz.

The aim of this paper, is to study electrical cardiography problem. Thus, we construct the state space system of this model as mathematical model. Moreover, we present some definitions and results which is described some concepts of linear control system analysis related to this problem. More precisely, the sufficient conditions which characterize the observability notion of linear dynamical controlled system are presented and discussed. Finally, we prove that, the electrical cardiography model is completely observable system over finite time.

Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate relevant matrices and some relevant quantities such as the topological entropy. An application to stabilization via algebraic Riccati equations is included by viewing a control system as an autonomous system in an extended space of states and control inputs. Kernel methods are the main techniques used in this paper and the approach is illustrated via a series of numerical examples. The advantage of using kernel methods is that they allow to perform function approximation from data and, as illustrated in this paper, allow to approximate linear discrete-time autonomous and control systems from data.

We establish analogs of the three Bieberbach theorems for a lattice \(\Gamma\) in a semidirect product \(\mathsf{K}\rtimes\mathsf{K}\) where \(\mathsf{S}\) is a connected, simply connected solvable Lie group and \(\mathsf{K}\) is a compact subgroup of its automorphism group. We first prove that the action of \(\Gamma\) on \(\mathsf{S}\) is metrically equivalent to an action of \(\Gamma\) on a supersolvable Lie group. The latter is shown to be determined by \(\Gamma\) itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group \(\Gamma\) as a lattice in a semidirect product \(\mathsf{S}\rtimes\mathsf{F}\) with \(\mathsf{F}\) being a finite group whose order is bounded by a constant only depending on the dimension of \(\mathsf{S}\). This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup \(\Upsilon \subset \mathsf{K}\rtimes\mathsf{K}\) that the quotient \(\mathsf{S}/\Upsilon\) is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group.

In the present paper finite-dimensional dynamical control systems described by linear higherorder ordinary differential state equations with multiple point delays in control are considered. Using algebraic methods, necessary and sufficient condition for relative controllability in a given time interval for linear dynamical system with multiple point delays in control is formulated and proved. This condition is generalization to relative controllability case some previous results concerning controllability of linear dynamical systems without multiple point delays in the control. Simple numerical example, which illustrates theoretical result is also given. Finally, some remarks and comments on the existing results for controllability of dynamical systems with delays in control are also presented.

By complementarily fusing the robust stabilizability condition, the orthogonal-functions approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA), an integrative method is proposed in this paper to design the robust-stable and quadratic-optimal static output feedback controller such that i) the linear singular control system with structured parameter uncertainties is regular, impulse-free and asymptotically stable and ii) a quadratic finite-horizon integral performance index for the linear nominal singular control system can be minimized. Based on some essential properties of matrix measures, a new sufficient condition is presented for ensuring that the linear singular system with structured and quadratically-coupled structured parameter uncertainties is regular, impulse free and asymptotically stable. By using the OFA and the robust stabilizability condition, the dynamic-optimization problem for the robust-stable and quadratic-optimal static output feedback control design of the linear uncertain singular system is transformed into a static-constrained-optimization problem represented by the algebraic equations with constraint of robust stabilizability condition; thus greatly simplifying the robust-stable and quadratic-optimal static output feedback control design problem of the linear uncertain singular system. Then, for the static-constrained-optimization problem, the HTGA is employed to find the robust-stable and quadratic-optimal static output feedback controller of the linear uncertain singular control system. One design example of the robust-stable and quadratic-optimal static output feedback controller for a mass-spring-damper mechanical system with structured parameter uncertainties is given to demonstrate the applicability of the proposed integrative approach.

In the paper, we investigate three-dimensional solvable Lie groups from the point of view of the generalized Hermitian geometry. The corresponding three-dimensional solvable Lie algebras were firstly classified by G. M. Mubarakzyanov in 1963. Using the classification in somewhat different notations, we construct basic left-invariant metric f-structures of rank 2 on all three-dimensional solvable Lie groups equipped with the standard left-invariant Riemannian metric. It was proved that all the considered f-structures belong to one or several classes of generalized almost Hermitian structures. As a result, it gives the opportunity to present new examples of left-invariant Killling, nearly Kähler, generalized nearly Kähler and Hermitian f-structures on solvable Lie groups.