Recent zbMATH articles in MSC 93B05https://zbmath.org/atom/cc/93B052024-11-01T15:51:55.949586ZWerkzeugOn the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domainhttps://zbmath.org/1544.350972024-11-01T15:51:55.949586Z"Dekhkonov, Farrukh"https://zbmath.org/authors/?q=ai:dekhkonov.farrukh-n"Li, Wenke"https://zbmath.org/authors/?q=ai:li.wenkeSummary: In this paper, we consider a boundary control problem associated with a fourth order parabolic type equation in a bounded two-dimensional domain. The solution with the control function on the border of the considered domain is given. The constraints on the control are determined to ensure that the average value of the solution within the considered domain attains a given value. The initial-boundary problem is solved by the Fourier method, and the control problem under consideration is analyzed with the Volterra integral equation. The existence of admissible control is proved by the Laplace transform method.Approximate controllability for some retarded integrodifferential inclusions using a version of the Leray-Schauder fixed point theorem for the multivalued mapshttps://zbmath.org/1544.450132024-11-01T15:51:55.949586Z"El Matloub, Jaouad"https://zbmath.org/authors/?q=ai:el-matloub.jaouad"Ezzinbi, Khalil"https://zbmath.org/authors/?q=ai:ezzinbi.khalilSummary: The main goal of this work is to investigate the approximate controllability for a class of non-autonomous delayed integrodifferential inclusions with unbounded delay. Our technique starts with the search for the optimal control for a linear quadratic regulator problem. The existence of such an optimal control aids to establish sufficient conditions insuring our inclusion problem's approximate controllability. The findings we acquired represent a generalization and extension of previous results on this topic. Finally, we present an example to illustrate the abstract theory.A mathematical model of the visual MacKay effecthttps://zbmath.org/1544.912512024-11-01T15:51:55.949586Z"Tamekue, Cyprien"https://zbmath.org/authors/?q=ai:tamekue.cyprien"Prandi, Dario"https://zbmath.org/authors/?q=ai:prandi.dario"Chitour, Yacine"https://zbmath.org/authors/?q=ai:chitour.yacineSummary: This paper investigates the intricate connection between visual perception and the mathematical modeling of neural activity in the primary visual cortex (V1). The focus is on modeling the visual MacKay effect
[D. M. MacKay, \textit{Nature,} 180 (1957), pp. 849-850].
While bifurcation theory has been a prominent mathematical approach for addressing issues in neuroscience, especially in describing spontaneous pattern formations in V1 due to parameter changes, it faces challenges in scenarios with localized sensory inputs. This is evident, for instance, in MacKay's psychophysical experiments, where the redundancy of visual stimuli information results in irregular shapes, making bifurcation theory and multiscale analysis less effective. To address this, we follow a mathematical viewpoint based on the input-output controllability of an Amari-type neural fields model. In this framework, we consider sensory input as a control function, a cortical representation via the retino-cortical map of the visual stimulus that captures its distinct features. This includes highly localized information in the center of MacKay's funnel pattern ``MacKay rays''. From a control theory point of view, the Amari-type equation's exact controllability property is discussed for linear and nonlinear response functions. For the visual MacKay effect modeling, we adjust the parameter representing intra-neuron connectivity to ensure that cortical activity exponentially stabilizes to the stationary state in the absence of sensory input. Then, we perform quantitative and qualitative studies to demonstrate that they capture all the essential features of the induced after-image reported by MacKay.On the external estimation of reachable and null-controllable limit sets for linear discrete-time systems with a summary constraint on the scalar controlhttps://zbmath.org/1544.930432024-11-01T15:51:55.949586Z"Ibragimov, D. N."https://zbmath.org/authors/?q=ai:ibragimov.danis-nSummary: The problem of constructing reachable and null-controllable sets for stationary linear discrete-time systems with a summary constraint on the scalar control is considered. For the case of quadratic constraints and a diagonalizable matrix of the system, these sets are built explicitly in the form of ellipsoids. In the general case, the limit reachable and null-controllable sets are represented as fixed points of a contraction mapping in the metric space of compact sets. On the basis of the method of simple iteration, a convergent procedure for constructing their external estimates with an indication of the a priori approximation error is proposed. Examples are given.Study approximate controllability and null controllability of neutral delay Hilfer fractional stochastic integrodifferential system with Rosenblatt processhttps://zbmath.org/1544.930442024-11-01T15:51:55.949586Z"Ahmed, Hamdy M."https://zbmath.org/authors/?q=ai:ahmed.hamdy-mSummary: In this paper, we investigate the sufficient conditions for approximate controllability and null controllability of neutral delay stochastic Hilfer fractional integrodifferential system with Rosenblatt process are established. The required results are obtained based on a fixed point technique, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.Insensitizing controls with \(N-1\) components for the \(N\)-dimensional Ladyzhenskaya-Smagorinsky systemhttps://zbmath.org/1544.930452024-11-01T15:51:55.949586Z"Barreira, João Carlos"https://zbmath.org/authors/?q=ai:barreira.joao-carlos"Límaco Ferrel, Juan"https://zbmath.org/authors/?q=ai:limaco-ferrel.juanSummary: This article deals with a Ladyzhenskaya-Smagorinsky type differential turbulence model with partially known initial data. We are interested in the existence of insensitive controls, with \(N - 1\) scalar controls in an arbitrary control domain, for the local \(L^2\) norm of the solution of model, that is, the goal is to find a control function, having one vanishing component, such that some functional of the state is locally insensitive to the perturbations of these initial data. In this system, we find local and nonlocal nonlinearities, with the usual transport terms and a turbulent viscosity, respectively. Such a problem can be reduced to a non-standard null controllability problem, of a nonlinear cascade system governed by an equation forward in time and one backward. We use a known null controllability result, with controls having one vanishing component, for a linear problem and prove some technical lemmas. The key point to establish the null controllability for the nonlinear cascade system is to use an inverse mapping theorem in infinite-dimensional spaces.On the accessibility and controllability of statistical linearization for stochastic control: algebraic rank conditions and their genericityhttps://zbmath.org/1544.930462024-11-01T15:51:55.949586Z"Bonalli, Riccardo"https://zbmath.org/authors/?q=ai:bonalli.riccardo"Leparoux, Clara"https://zbmath.org/authors/?q=ai:leparoux.clara"Hérissé, Bruno"https://zbmath.org/authors/?q=ai:herisse.bruno"Jean, Frédéric"https://zbmath.org/authors/?q=ai:jean.fredericSummary: Statistical linearization has recently seen a particular surge of interest as a numerically cheap method for robust control of stochastic differential equations. Although it has already been successfully applied to control complex stochastic systems, accessibility and controllability properties of statistical linearization, which are key to make the robust control problem well-posed, have not been investigated yet. In this paper, we bridge this gap by providing sufficient conditions for the accessibility and controllability of statistical linearization. Specifically, we establish simple sufficient algebraic conditions for the accessibility and controllability of statistical linearization, which involve the rank of the Lie algebra generated by the drift only. In addition, we show these latter algebraic conditions are essentially sharp, by means of a counterexample, and that they are generic with respect to the drift and the initial condition.A temporal segmentation algorithm for restoring the controllability of networked control systemshttps://zbmath.org/1544.930472024-11-01T15:51:55.949586Z"Cui, Yulong"https://zbmath.org/authors/?q=ai:cui.yulong"Wu, Mincheng"https://zbmath.org/authors/?q=ai:wu.mincheng"Shao, Cunqi"https://zbmath.org/authors/?q=ai:shao.cunqi"He, Shibo"https://zbmath.org/authors/?q=ai:he.shiboSummary: Restoring the controllability of networked control systems is a fundamental issue that needs to be settled, especially when malicious attacks or malfunctions occur. Previous studies tried to address such an issue by adding extra driver nodes or rewiring edges, which will inevitably change the original network structure. In this paper, an algorithm to restore the controllability of networked systems while preserving the integrity of the original network structure is proposed. Specifically, a static uncontrollable network will be transformed into a controllable temporal network by means of the cactus-based segmentation method. The original problem will be equivalent to the classical set cover problem, which is known to be NP-hard, if the least number of segmentations is considered. An approximation algorithm with polynomial time complexity is proposed and it is proved that the solution to the problem is two-optimal. Finally, simulations are carried out to verify the effectiveness and feasibility of the proposed algorithm.
{\copyright} 2021 The Authors. \textit{IET Control Theory \& Applications} published by John Wiley \& Sons Ltd on behalf of The Institution of Engineering and TechnologyOne-input linear control systems on the homogeneous spaces of the Heisenberg group -- the singular casehttps://zbmath.org/1544.930482024-11-01T15:51:55.949586Z"Da Silva, Adriano"https://zbmath.org/authors/?q=ai:silva.adriano-da"Duman, Okan"https://zbmath.org/authors/?q=ai:duman.okan"Kizil, Eyüp"https://zbmath.org/authors/?q=ai:kizil.eyupSummary: We have classified in [7] all linear control systems and studied their controllability and control sets on the homogeneous spaces \(L\setminus\mathbb{H}\) of the 3-dimensional Heisenberg group \(\mathbb{H}\) through its closed subgroups \(L\).
In this paper, we characterize the controllability and control sets of the induced linear control systems in the homogeneous spaces left. In particular, we focus on the singularity of the induced drift vector fields. Quite a technical analysis with many cases reveals the control sets. We give some geometric illustrations.Boundary null controllability of convection-diffusion equations with constraints on the state and application to the identification of boundary pollution parametershttps://zbmath.org/1544.930492024-11-01T15:51:55.949586Z"Fournier, Arnaud"https://zbmath.org/authors/?q=ai:fournier.arnaud"Larrouy, James"https://zbmath.org/authors/?q=ai:larrouy.jamesSummary: We study a null controllability problem for a convection-diffusion equation with the control contained in Fourier boundary condition to identify \(M\) parameters of pollution. The results are achieved by means of an observability inequality derived from new Carleman estimates for the boundary null controllability and the construction of a boundary sentinel to identify the parameters.Necessary conditions for local controllability of a particular class of systems with two scalar controlshttps://zbmath.org/1544.930502024-11-01T15:51:55.949586Z"Giraldi, Laetitia"https://zbmath.org/authors/?q=ai:giraldi.laetitia"Lissy, Pierre"https://zbmath.org/authors/?q=ai:lissy.pierre"Moreau, Clément"https://zbmath.org/authors/?q=ai:moreau.clement"Pomet, Jean-Baptiste"https://zbmath.org/authors/?q=ai:pomet.jean-baptisteSummary: We consider control-afflne systems with two scalar controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions for local controllability around this equilibrium, involving the iterated Lie brackets of the system vector fields, with controls that are either bounded, small in \(\mathrm{L}^\infty\) or small in \(\mathrm{W}^{1, \infty}\). These results are illustrated with several examples.The null controllability of transmission wave-Schrödinger system with a boundary controlhttps://zbmath.org/1544.930512024-11-01T15:51:55.949586Z"Guo, Ya-Ping"https://zbmath.org/authors/?q=ai:guo.yaping"Wang, Jun-Min"https://zbmath.org/authors/?q=ai:wang.junmin"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.32"Zhao, Dong-Xia"https://zbmath.org/authors/?q=ai:zhao.dongxiaSummary: This paper is devoted to investigate the null controllability of the transmission wave-Schrödinger system with only one boundary control. The domain of the system consists of two bounded intervals, where the wave and Schrödinger equations evolve, respectively. Two kinds of transmission conditions are considered: one is the simple continuous transmission, and by using the HUM method, the null controllability of the system is derived in the Hilbert space when a boundary control is added only on the wave equation. The other case is that the Schrödinger state is associated with the velocity of the wave, and we establish the null controllability of the system. It is found that the second space has more regularity.Mild solution and finite-approximate controllability of higher-order fractional integrodifferential equations with nonlocal conditionshttps://zbmath.org/1544.930522024-11-01T15:51:55.949586Z"Haq, Abdul"https://zbmath.org/authors/?q=ai:haq.abdul"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.3|ahmad.bashir.2|ahmad.bashir.1|ahmad.bashir.4|ahmad-khuda-bakhsh.bashirSummary: This article investigates the finite-approximate controllability properties for semi-linear integrodifferential systems involving higher-order fractional derivatives in Riemann-Liouville sense excluding Lipschitz assumptions of nonlinear operators. We discuss the existence of mild solutions by utilizing the Schaefer fixed point principle and compactness condition on the fractional resolvent. Then we show that one can steer the system in an arbitrary neighbourhood of any given target state simultaneously obeying the finitely many constraints. Lastly, an illustrative example is presented to validate the proposed methodology.Approximate controllability of neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert spacehttps://zbmath.org/1544.930532024-11-01T15:51:55.949586Z"Jeet, Kamal"https://zbmath.org/authors/?q=ai:jeet.kamal"Kumar, Ankit"https://zbmath.org/authors/?q=ai:kumar.ankit"Vats, Ramesh Kumar"https://zbmath.org/authors/?q=ai:vats.ramesh-kumarSummary: In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results.Controllability of higher order linear systems with multiple delays in controlhttps://zbmath.org/1544.930542024-11-01T15:51:55.949586Z"Klamka, Jerzy"https://zbmath.org/authors/?q=ai:klamka.jerzySummary: In the present chapter finite-dimensional dynamical control systems described by linear higher-order ordinary differential state equations with multiple point delays in control are considered. Using algebraic methods, necessary and sufficient conditions for relative controllability in a given time interval for linear dynamical system with multiple point delays in control are 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. Proof of the main result is based on necessary and sufficient controllability condition for linear systems without delays in 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.
For the entire collection see [Zbl 1522.93010].A note on the family of synchronizations for a coupled system of wave equationshttps://zbmath.org/1544.930552024-11-01T15:51:55.949586Z"Li, Tatsien"https://zbmath.org/authors/?q=ai:li.tatsien"Rao, Bopeng"https://zbmath.org/authors/?q=ai:rao.bopengSummary: We show that a coupled system of wave equations can be exactly synchronized by \(p\)-groups with respect to different groupings under the same control matrix.Null controllability of an abstract Riesz-spectral boundary control systemshttps://zbmath.org/1544.930562024-11-01T15:51:55.949586Z"Lourini, Abdellah"https://zbmath.org/authors/?q=ai:lourini.abdellah"El Azzouzi, Mohamed"https://zbmath.org/authors/?q=ai:el-azzouzi.mohamed"Laabissi, Mohamed"https://zbmath.org/authors/?q=ai:laabissi.mohamedSummary: This paper addresses the null controllability of an abstract boundary control systems in Hilbert spaces where the system operator is of Riesz type. Consequently, this document establishes a criterion for null controllability in such systems based on initial data, utilizing the moment problem. Furthermore, this criterion is formulated by employing a null controllability criterion that is applicable to a corresponding linear system with internal control. Finally, we apply our approach to the heat equation and the Mullins equation, demonstrating the practicality of our methodology.Controllability and control for discrete periodic systems based on fully-actuated system approachhttps://zbmath.org/1544.930572024-11-01T15:51:55.949586Z"Lv, Lingling"https://zbmath.org/authors/?q=ai:lv.lingling"Li, Zikai"https://zbmath.org/authors/?q=ai:li.zikai"Chang, Rui"https://zbmath.org/authors/?q=ai:chang.rui"Liu, Xinyang"https://zbmath.org/authors/?q=ai:liu.xinyangSummary: The problem of controllability and control of linear discrete-periodic systems is investigated in this paper. The high-order fully-actuated models for linear discrete periodic time-varying systems are constructed, and a controllability criterion based on fully-actuated system models is proposed. On this basis, stabilization as the fundamental issue is studied and periodic state-feedback control laws are designed via fully-actuated systems approach and parametric design, which converted the original problem into the pole assignment problem for linear constant systems. Finally, a numerical example is given to verify the validity and feasibility of the proposed method.Strong structural controllability based on leader-follower frameworkhttps://zbmath.org/1544.930582024-11-01T15:51:55.949586Z"Qi, Wei"https://zbmath.org/authors/?q=ai:qi.wei"Ji, Zhijian"https://zbmath.org/authors/?q=ai:ji.zhijian"Liu, Yungang"https://zbmath.org/authors/?q=ai:liu.yungang"Lin, Chong"https://zbmath.org/authors/?q=ai:lin.chongSummary: In this paper, the strong structural controllability of the leader-follower framework is discussed. Firstly, the authors analyze different edge augmentation methods to preserve the strong structural controllability of the path-bud topology. The following four cases are considered: Adding edges from the path to the bud; adding edges from the bud to the path; adding the reverse or forward edges to the path or bud; and adding both the reverse and forward edges to the path or bud. Then sufficient conditions are derived for the strong structural controllability of the new topologies which are generated by adding different edges. In addition, it is proved that \(\mathrm{rank}[A\,\,\,B]=n\) is a necessary condition for the strong structural controllability. Finally, three examples are given to verify the effectiveness of the main results.Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditionshttps://zbmath.org/1544.930592024-11-01T15:51:55.949586Z"Rissel, Manuel"https://zbmath.org/authors/?q=ai:rissel.manuel"Wang, Ya-Guang"https://zbmath.org/authors/?q=ai:wang.yaguang|wang.ya-guangSummary: We study the small-time global approximate controllability for incompressible magnetohydrodynamic (MHD) flows in smoothly bounded two- or three-dimensional domains. The controls act on arbitrary nonempty open portions of each connected boundary component, while linearly coupled Navier slip-with-friction conditions are imposed along the uncontrolled parts of the boundary. Some choices for the friction coefficients give rise to interacting velocity and magnetic field boundary layers. We obtain sufficient dissipation properties of these layers by a detailed analysis of the corresponding asymptotic expansions. For certain friction coefficients, or if the obtained controls are not compatible with the induction equation, an additional pressure-like term appears. We show that such a term does not exist for problems defined in planar simply-connected domains and various choices of Navier slip-with-friction boundary conditions.Controllability and stability of non-instantaneous impulsive stochastic multiple delays systemhttps://zbmath.org/1544.930602024-11-01T15:51:55.949586Z"Sathiyaraj, T."https://zbmath.org/authors/?q=ai:sathiyaraj.t"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrongSummary: This paper gives the controllability and Ulam-Hyers-Rassias (U-H-R) stability results for non-instantaneous impulsive stochastic multiple delays system with nonpermutable variable coefficients. The solution for nonlinear non-instantaneous impulsive stochastic systems is presented without the assumption of commutative property on delayed matrix coefficients. The kernel function of the solution operator is defined by sum of noncommutative products of delayed matrix constant coefficients. Sufficient conditions for controllability of linear and nonlinear non-instantaneous impulsive stochastic multiple delays system are established by using the Mönch fixed-point theorem under the proof that the corresponding linear system is controllable. Thereafter, U-H-R stability result is proved. Finally, the theoretical results are verified by a numerical example.Approximate controllability of semilinear fractional control systems of order \(\alpha \in (1, 2]\)https://zbmath.org/1544.930612024-11-01T15:51:55.949586Z"Shukla, Anurag"https://zbmath.org/authors/?q=ai:shukla.anurag"Sukavanam, N."https://zbmath.org/authors/?q=ai:sukavanam.nagarajan"Pandey, D. N."https://zbmath.org/authors/?q=ai:pandey.dwijendra-narainSummary: The objective of this paper is to present some sufficient conditions for approximate controllability of semilinear delay control systems of fractional order \(\alpha \in (1, 2]\). The results are obtained by the theory of strongly continuous \(\alpha\)-order cosine family and sequential approach under the natural assumption that the linear system is approximate controllable. At the end, an example is given to illustrate the theory.
For the entire collection see [Zbl 1466.93006].Structure-preserving interval-limited balanced truncation reduced models for port-Hamiltonian systemshttps://zbmath.org/1544.930622024-11-01T15:51:55.949586Z"Xu, Kangli"https://zbmath.org/authors/?q=ai:xu.kangli"Jiang, Yaolin"https://zbmath.org/authors/?q=ai:jiang.yaolinSummary: In this study, the authors propose structure-preserving balanced truncation methods of port-Hamiltonian systems over the frequency and time intervals. First, the port-Hamiltonian system is reduced based on the frequency-interval controllability and observability Gramians. In order to reduce the computational cost, the frequency-interval cross Gramian is utilised to yield the frequency-interval balanced truncation method. For the symmetric port-Hamiltonian systems, the authors prove that these two methods can generate equivalent reduced systems. Additionally, they are also devoted in exploring two structure-preserving balanced truncation methods over time intervals, where one is based on the time interval controllability and observability Gramians and the other is based on the time interval cross Gramian. All the resulting reduced systems have port-Hamiltonian structure, and as a consequence, they are passive. Two numerical examples are simulated to demonstrate the efficiency of the proposed methods.
{\copyright} 2021 The Authors. IET Control Theory \& Applications published by John Wiley \& Sons, Ltd. on behalf of The Institution of Engineering and TechnologyStabilisation of multi-agent systems over finite fields based on high-order fully actuated system approacheshttps://zbmath.org/1544.930632024-11-01T15:51:55.949586Z"Yang, Yunsi"https://zbmath.org/authors/?q=ai:yang.yunsi"Feng, Jun-e"https://zbmath.org/authors/?q=ai:feng.june"Jia, Lei"https://zbmath.org/authors/?q=ai:jia.leiSummary: In this paper, the controllability of multi-agent systems over finite fields is studied based on the high-order fully actuated (HOFA) system approaches. The concept of HOFA systems over finite fields is introduced. Necessary and sufficient conditions for the controllability of linear systems over finite fields are established. It is proved that a leader-follower multi-agent system over finite fields is controllable if and only if it can be transformed into an HOFA system. Then the stabilisation control protocol is proposed based on the full-actuation property, and the upper bound of settling time for the system can be determined. The uncontrollable leader-follower multi-agent system over finite fields can be transformed into a controllable subsystem and an autonomous subsystem via structural decomposition, the conditions for the system to be stabilisable are provided. Finally, an illustrative example is shown to verify the validity of the main results of this paper.On approximate solution of mobile (scanning) control problemshttps://zbmath.org/1544.932792024-11-01T15:51:55.949586Z"Khurshudyan, Asatur Zh."https://zbmath.org/authors/?q=ai:khurshudyan.asatur-zhSummary: We describe an approximate technique for solving the so-called mobile (scanning) control problems. The method is based on the Bubnov-Galerkin procedure and allows us to reduce the control problem, in which the unknown function is included in nonlinear manner, to a \textit{finite-dimensional} system of integral constraints of equality type. An efficient numerical scheme is described reducing the solution of the nonlinear system to a problem of nonlinear programming. The proposed method is described thoroughly for nonlinear equations with \textit{linear} boundary conditions. Two particular problems of heating by a moving source and vibration damping by a moving absorber are considered. The system of necessary and sufficient conditions for controllability are obtained in both cases. Main points of numerical implementations are discussed.
For the entire collection see [Zbl 1466.93006].The chain control set of discrete-time linear systems on the affine two-dimensional Lie grouphttps://zbmath.org/1544.934542024-11-01T15:51:55.949586Z"Cavalheiro, Thiago Matheus"https://zbmath.org/authors/?q=ai:cavalheiro.thiago-matheus"Cossich, João Augusto Navarro"https://zbmath.org/authors/?q=ai:cossich.joao-augusto-navarro"Santana, Alexandre José"https://zbmath.org/authors/?q=ai:santana.alexandre-joseLet \(G=\mathrm{Aff}\left( E,\mathbb{R}\right) \) be the two-dimensional Lie group, i.e.
\[
\mathrm{Aff}\left( E,\mathbb{R}\right) =\left\{ \left[ \begin{array}{cc} a & b \\
0 & A \end{array} \right] :a>0,b\in \mathbb{R}\right\}
\]
so that \(\mathrm{Aff}\left( E,\mathbb{R}\right) \cong \mathbb{R}^{+\ast }\times \mathbb{R}\) endowed with the semidirect product \(\left( a,b\right) .\left( c,d\right) =\left( ac,d+bc\right) .\) The distance of two elements \( g,h\in G\) is denoted as \(\delta \left( g,h\right) .\) (The notation of the paper is modified in this review to avoid some confusions.) The discrete-time system \(\Sigma \) is given by
\[
x_{k+1}=f_{u_{k}}\left( x_{k}\right) ,\quad u_{k}\in U,\quad x_{k}\in G\text{ for all integers }k
\]
where \(U\) is a compact neighborhood of \(0\) and \(f_{\upsilon }\) is a diffeomorphism for all \(\upsilon \in U.\) By induction, the solution at time \( k\) can be written
\[
x_{k}=\varphi \left( k,g,u\right) ,\quad x_{0}=g
\]
where \(u=\left( u_{i}\right) _{i\in \mathbb{Z}}\in \mathcal{U}:=\prod_{i\in \mathbb{Z}}U.\) The system \(\Sigma \) is called linear if \(f_{\upsilon }\left( g\right) =f_{\upsilon }\left( 0\right) f_{0}\left( g\right) ,\) which implies \(\varphi \left( k,g,u\right) =\varphi \left( k,e,u\right) f_{0}^{k}\left( u\right) .\)
Reachability and controllability are defined as usual and the set of all points \(h\) reachable from \(g\) is denoted as \(\mathcal{R}\left( g\right) .\) Furthermore, a subset \(D\neq \varnothing \) of \(G\) is called a \textit{control set} if (i) for all \(g\in D,\) \(\exists u\in \mathcal{U}:\varphi \left( k,g,u\right) \in D\) for all \(k\geq 0;\) (ii) for all \(g,\) \(D\) is included in the closure of \(\mathcal{R}\left( g\right) ;\) \(D\) is maximal with these properties.
Weakening this notion, given \(g,h\in G\) and a real number \(\varepsilon >0,\) an \(\left( \varepsilon ,k\right) \)-controllable chain between \(g\) and \(h\) is the set
\[
\mu _{\left( \varepsilon ,k\right) }=\left\{ \begin{array}{c} n\in \mathbb{N} \\
x_{0},x_{1},\dots,x_{n}\in G \\
u_{0},\dots,u_{n-1}\in \mathcal{U} \\
k_{0},\dots,k_{n-1}\geq k \end{array} \right\}
\]
where \(x_{0}=g,\) \(x_{n}=h\) and \(\delta \left( \varphi \left( k_{i},x_{i},u_{i}\right) ,x_{i+1}\right) <\varepsilon \) \(\left( i=0,\dots,n-1\right) .\) The pair \(\left( g,h\right) \) is called \textit{chain-controllable} if for all \(\left( \varepsilon ,k\right) \in \mathbb{R} ^{+\ast }\times \mathbb{N}\), there is an \(\left( \varepsilon ,k\right) \)-controllable chain between \(g\) and \(h.\) A subset \(E\neq \varnothing \) of \(G\) is called a \textit{chain-control set} if (i) for all \(g\in E,\) \(\exists u\in \mathcal{U}:\varphi \left( k,g,u\right) \in E\) for all \(k\in \mathbb{Z};\) (ii) every pair \(\left( g,h\right) \in E\times E\) is chain-controllable\(;\) (iii) \(E\) is maximal with these properties.
From a result obtained in [\textit{V. Ayala} and \textit{A. Da Silva}, J. Differ. Equations 268, No. 11, 6683--6701 (2020; Zbl 1440.93025)], it is shown that \(f_{u}\) is of the form
\[
f_{u}\left( a,b\right) =\left( \mathfrak{h}\left( u\right) a,\alpha \left( a-1\right) +\gamma b+\mathfrak{g}\left( u\right) a\right) ,\quad \gamma \neq 0
\]
where \(\mathfrak{h},\mathfrak{g}\) are smooth and \(\mathfrak{h}\left( 0\right) =1,\mathfrak{g}\left( 0\right) =0.\)
Assuming that \(\mathfrak{h}\equiv 1,\) control sets \(D_{a}\) \(\left( a\in \mathbb{R}^{+\ast }\right) \)\ with empty interior are determined when \( \left\vert \gamma \right\vert <1.\) In that case, \(E\) is the union of the \( D_{a}\) \(\left( a>0\right) ,\) hence unique. Furthermore, if \(\gamma =\pm 1,\) then the sets \(D_{a}=\left\{ a\right\} \times \mathbb{R}\) \(\left( a>0\right) \) are control sets and, if \(\gamma =-1,\) \(G\) is chain-controllable.
Reviewer: Henri Bourlès (Paris)Stabilizability of linear discrete time-varying systemshttps://zbmath.org/1544.936362024-11-01T15:51:55.949586Z"Babiarz, Artur"https://zbmath.org/authors/?q=ai:babiarz.artur"Czornik, Adam"https://zbmath.org/authors/?q=ai:czornik.adamSummary: For linear discrete time-varying systems we discuss the relation between stabilizability, controllability and finiteness of quadratic cost functional. The role of the existence of global and bounded solutions of the discrete time-varying Riccati equation for stabilizability is also explained.
For the entire collection see [Zbl 1522.93010].Kalman rank criterion for the controllability of fractional impulse controlled systemshttps://zbmath.org/1544.938052024-11-01T15:51:55.949586Z"Cai, Rui-Yang"https://zbmath.org/authors/?q=ai:cai.ruiyang"Zhou, Hua-Cheng"https://zbmath.org/authors/?q=ai:zhou.hua-cheng"Kou, Chun-Hai"https://zbmath.org/authors/?q=ai:kou.chunhaiSummary: This study aims to address the impulsive controllability of linear fractional impulse controlled systems. The authors establish the Kalman rank criterion to guarantee the impulsive controllability of the considered system. They also present the minimum number of impulsive controls for a realisation of such controllability. An example is finally presented to demonstrate the effectiveness of the authors' results.
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