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The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices. [ABSTRACT FROM AUTHOR]

This paper is concerned with the turnpike property for a class of stochastic linear-quadratic (LQ, for short) optimal control problems with periodic coefficients. The stability and stabilizability of the control system are studied, followed by the discussion of the existence and uniqueness of periodic solutions. A deterministic periodic LQ problem is introduced and solved, whose optimal pair, together with a pair of correction processes, serves as the turnpike limit of the stochastic problem. It is shown that the turnpike limit is periodic in the distribution sense. In the special case of constant coefficients, the turnpike limit turns out to have stationary distributions, with the expectation being the solution to a static optimization problem. [ABSTRACT FROM AUTHOR]

This paper concerns with the stabilizability for a quasilinear Klein–Gordon–Schrödinger system with variable coefficients in dimensionless form. The stabilizability of quaslinear Klein–Gordon-Wave system with the Kelvin–Voigt damping has been considered by Liu–Yan–Zhang (SIAM J Control Optim 61:1651–1678, 2023). Our main contribution is to find a suitable linear feedback control law such that the quasilinear Klein–Gordon–Schrödinger system is exponentially stable under certain smallness conditions. [ABSTRACT FROM AUTHOR]

Stabilization of the inverted pendulum by fractional-order proportional-derivative (PD) feedback with two delays is investigated. This feedback law is obtained as a combination of PD feedback with two delays and fractional-order PD feedback with a single delay. Different types of stabilizability boundaries and the corresponding geometric and multiplicity conditions are determined using the D-subdivision method. The stabilizable region is depicted in the plane of the delay parameters for given fractional derivative orders. Several special cases and the concept of delay detuning are also discussed. It is shown that the admissible delay can be slightly increased compared to the integer-order PD feedback by introducing a fractional-order feedback term. [ABSTRACT FROM AUTHOR]

In the first part of this article, we study feedback stabilization of a parabolic coupled system by using localized interior controls. The system is feedback stabilizable with exponential decay $ -\omega<0 $ for any $ \omega>0 $. A stabilizing control is found in feedback form by solving a suitable algebraic Riccati equation. In the second part, a conforming finite element method is employed to approximate the continuous system by a finite dimensional discrete system. The approximated system is also feedback stabilizable (uniformly) with exponential decay $ -\omega+\epsilon $, for any $ \epsilon>0 $ and the feedback control is obtained by solving a discrete algebraic Riccati equation. The error estimate of stabilized solutions as well as stabilizing feedback controls are obtained. We validate the theoretical results by numerical implementations. [ABSTRACT FROM AUTHOR]

We study abstract sufficient criteria for cost-uniform open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates are only needed for a single spectral parameter and, in particular, their constants do not depend on the growth rate w.r.t. this parameter. Our result unifies and generalizes earlier results obtained in the context of Hilbert spaces. As an application we consider fractional powers of elliptic differential operators with constant coefficients in L p (R d) for p ∈ [ 1 , ∞) and thick control sets. [ABSTRACT FROM AUTHOR]

We seek the solution of the coupled continuous-time algebraic Riccati equations, arising in the optimal control of Markovian jump linear systems. Newton's method is applied to construct the solution, under a mild and natural stabilizability assumption, leading to some coupled Lyapunov equations. Iterative methods of O (n 3) computational complexity for the coupled Lyapunov equations and the corresponding Newton's methods for the coupled continuous-time algebraic Riccati equations are analyzed. Illustrative examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, some new approaches for stability and stabilizability determination as well as state feedback stabilization controllers of linear time-invariant (LTI) interval systems are proposed. The presented stability conditions are less conservative than those of Kharitonov's theorem, and Gerschgorin's disc theorem methods. Moreover, some of the proposed stability, stabilizability, and feedback stabilization control methods for LTI interval systems are proved to be sufficient and necessary conditions. Compared with some traditional stability analysis and feedback stabilization design methods for LTI interval systems, these new approaches have lower computational complexity because of a special form of parameter vertex matrices developed in this work. Some numerical and practical examples are given to demonstrate the effectiveness and advantages of the proposed methods. • The proposed stability conditions are less conservative compared to traditional methods. • One of the state feedback stabilization control methods is proven to be both a sufficient and necessary condition. • The computational complexity of the proposed strategies is substantially reduced, courtesy of the KCV matrix. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem for a mean-field linear stochastic differential equation with a quadratic functional in the infinite time horizon. Under suitable conditions, including the stabilizability, the (strong) exponential, integral, and mean-square turnpike properties for the optimal pair are established. The keys are to correctly formulate the corresponding static optimization problem and find the equations determining the correction processes. These have revealed the main feature of the stochastic problems which are significantly different from the deterministic version of the theory. [ABSTRACT FROM AUTHOR]

We consider a control system describing the interaction of water waves with a partially immersed rigid body constraint to move only in the vertical direction. The fluid is modeled by the shallow water equations. The control signal is a vertical force acting on the floating body. We first derive the full governing equations of the fluid-body system in a water tank and reformulate them as an initial boundary value problem of a first-order evolution system. We then linearize the equations around the equilibrium state and we study its well-posedness. Finally we focus on the reachability and stabilizability of the linear system. Our main result asserts that, provided that the floating body is situated in the middle of the tank, any symmetric waves with appropriate regularity can be obtained from the equilibrium state by an appropriate control force. This implies, in particular, that we can project this system on the subspace of states with appropriate symmetry properties to obtain a reduced system which is approximately controllable and strongly stabilizable. Note that, in general, this system is not controllable (even approximately). [ABSTRACT FROM AUTHOR]

We design a new feedback law to stabilize the linear infinite-dimensional control system, where the state operator generates a $ C_0 $-group, and the control operator is unbounded. Our feedback law is based on the integration of a mutated Gramian operator-valued function. In the structure of the aforementioned mutated Gramian operator, we utilize the weak observability inequality in [21,13] and borrow some idea used to construct generalized Gramian operators in [11,23,24]. Unlike most related works where the exact controllability is required, we only assume the above-mentioned weak observability inequality, which is equivalent to the stabilizability of the system. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic linear-quadratic (LQ, for short) optimal control problem with jumps in an infinite horizon is studied, where the state system is a controlled linear stochastic differential equation containing affine term driven by a one-dimensional Brownian motion and a Poisson stochastic martingale measure, and the cost functional with respect to the state process and control process is quadratic and contains cross terms. Firstly, in order to ensure the well-posedness of our stochastic optimal control of infinite horizon with jumps, the $ L^2 $-stabilizability of our control system with jump is introduced. Secondly, it is proved that the $ L^2 $-stabilizability of our control system with jump is equivalent to the non-emptiness of the admissible control set for all initial state and is also equivalent to the existence of a positive solution to some integral algebraic Riccati equation (ARE, for short). Thirdly, the equivalence of the open-loop and closed-loop solvability of our infinite horizon optimal control problem with jumps is systematically studied. The corresponding equivalence is established by the existence of a $ stabilizing\ solution $ of the associated generalized algebraic Riccati equation, which is different from the finite horizon case. Moreover, any open-loop optimal control for the initial state $ x $ admiting a closed-loop representation is obatined.

For linear control systems, the usual state feedback stabilizability has two components: one is a continuous observation mode (i.e., to observe solutions continuously in time), and the other is a class of feedback laws (which is usually the space of all of the linear and bounded operators from a state space to a control space). This paper studies the stabilizability for abstract linear control systems, with a discrete observation mode (i.e., to observe solutions discretely in time) and two different classes of feedback laws. We first characterize these types of stabilizabilities via some weak observability inequalities for the dual systems. Then, we use these characterizations to reveal the connections between these types of stabilizabilities and those with continuous observation mode. Finally, we show some applications of the aforementioned weak observability inequalities. [ABSTRACT FROM AUTHOR]

This paper deals with global stability dynamics for the Klein–Gordon–Zakharov system in R 2 . We first establish that this system admits a family of linear mode unstable explicit quasi-periodic wave solutions. Next, we prove that the Kelvin–Voigt damping can help to stabilize those linear mode unstable explicit quasi-periodic wave solutions for the Klein–Gordon–Zakharov system in the Sobolev space H s + 1 (R 2) × H s + 1 (R 2) × H s + 1 (R 2) for any s ⩾ 1. Moreover, the Kelvin–Voigt damped Klein–Gordon–Zakharov system admits a unique Sobolev regular solution exponentially convergent to some special solutions (including quasi-periodic wave solutions) of it. Our result can be extended to the n-dimension dissipative Klein–Gordon–Zakharov system for any n ⩾ 1. [ABSTRACT FROM AUTHOR]

This paper concerns with the stabilizability for a quasilinear Klein-Gordon-wave system with variable coefficients in Rn. The stabilizability of linear wave-type equations with the Kelvin-Voigt damping has been considered by Liu-Zhang [SIAM J. Control Optim., 54 (2016), pp. 1859-1871] for an elastic string system, and by Yu-Han [SIAM J. Control Optim., 59 (2021), pp. 1973-1988] for the linear wave system in a cuboidal domain, respectively. In this paper, we establish that there exists a linear feedback control law such that the quasilinear Klein-Gordon-wave system is exponentially stable under certain smallness conditions, where the feedback control is a strong Kelvin-Voigt damping. [ABSTRACT FROM AUTHOR]

This paper studies the stabilizability problem with respect to an arbitrary stability domain for strict closed convex processes. The main result states necessary and sufficient conditions in terms of the eigenstructure of the dual processes under different assumptions depending on whether the convex process of interest is additive or not. The results presented in this paper are stronger than those existing in the literature in two respects: (i) they are valid for arbitrary stability domains and (ii) they guarantee existence of Bohl-type stable trajectories. We also illustrate the main result by means of examples for both nonadditive and additive processes. [ABSTRACT FROM AUTHOR]

Necessary conditions for asymptotic stability and stabilizability of subsets for dynamical and control systems are obtained. The main necessary condition is homotopical and is in turn used to obtain a homological one. A certain extension is ruled out. Questions are posed. [ABSTRACT FROM AUTHOR]

The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is considered in the unrestricted set of system matrices (E , A , B) ∈ R ℓ × × R ℓ × n × R ℓ × m , while for relative genericity we allow the restricted set Σ ℓ , n , m ≤ r : = { (E , A , B) ∈ R ℓ × n × R ℓ × n × R ℓ × m | rk R E ≤ r } , where r ∈ N . Our main results are characterizations of generic controllability and generic stabilizability in Σ ℓ , n , m ≤ r in terms of the numbers ℓ , n , m , r . [ABSTRACT FROM AUTHOR]

The stabilizing switching signal design of discrete-time linear compartmental switched systems (DT-LCSSs) has been heretofore unsolved. It has been proven that a DT-LCSS is stabilizable if and only if it is stabilizable by a periodic switching signal. However, it still needs to be determined whether the period of a stabilizing switching signal can be confined within a bound. Moreover, the existing design method for stabilizing periodic switching signals requires the diagonal entries of system matrices of all subsystems to be strictly positive. In this study, we propose a novel approach to solve this problem completely. We construct a discrete-time Markov chain for a given DT-LCSS, termed the associated Markov chain, and prove the equivalence of stability and stabilizability between the DT-LCSS and the associated Markov chain. Based on this, verifiable necessary and sufficient conditions for stability and stabilizability are derived. Especially, the period of a stabilizing switching signal for an n -dimensional DT-LCSS can always be chosen within the bound n 2 − n + 1. We propose a state-independent stabilizing switching signal design method for general stabilizable DT-LCSSs. We also prove the equivalence between stabilizability by state-independent switching laws and stabilizability by state-dependent switching laws. A state-dependent global stabilizing switching signal design method is also proposed. Additionally, the proposed results are applied to the consensus analysis of discrete-time leader–follower multi-agent systems with switching communication digraphs. The effectiveness of the theoretical results is demonstrated by examples. [ABSTRACT FROM AUTHOR]

In this paper, the fully actuated system (FAS) approaches for continuous-time systems with time-varying state delays are proposed. Two types of continuous-time high-order FAS models with time delays: single-order time-delay FAS models and multi-order time-delay FASs, are proposed. Particularly, the type of sub-FASs that do not completely but partially satisfy the full actuation is investigated, and the sets of feasible points are defined. When the system states are constrained to the feasible set, a controller can be easily constructed for the sub-FAS such that the closed-loop system is a constant linear system with an arbitrarily assignable eigenstructure. In addition, it is demonstrated that the feasibility constraint can be transformed into a constraint on the initial values of the system, which vanishes when the system is a (global) FAS. Based on the unique control characteristic of the type of FAS models, the concepts of controllability and stabilizability of general dynamical time-delay systems are also proposed. The effect of the proposed theories is illustrated with examples. [ABSTRACT FROM AUTHOR]

Positive reachability of a positive system on a time scale means that system trajectories starting from the origin can reach all the points of the nonnegative cone. Positive stabilization of such a system means feedback stabilization that preserves positivity of the system. It is shown that if a positive (linear or nonlinear) continuous-time system is positively reachable from the origin, then it is positively feedback stabilizable. On the other hand, such implication does not hold in general for systems on discrete time scales, even linear ones, though it holds for one-dimensional systems. These differences between continuous-time and discrete-time systems follow from different characterizations of positive reachability for these two classes of systems. [ABSTRACT FROM AUTHOR]

This paper analyzes the limiting behavior of stochastic linear-quadratic optimal control problems in finite time-horizon [0, T] as T → ∞. The so-called turnpike properties are established for such problems, under stabilizability condition which is weaker than the controllability, normally imposed in the similar problem for ordinary differential systems. In dealing with the turnpike problem, a crucial issue is to determine the corresponding static optimization problem. Intuitively mimicking the deterministic situations, it seems to be natural to include both the drift and the diffusion expressions of the state equation to be zero as constraints in the static optimization problem. However, this would lead us to a wrong direction. It is found that the correct static problem should contain the diffusion as a part of the objective function, which reveals a deep feature of the stochastic turnpike problem. [ABSTRACT FROM AUTHOR]

This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the N N -dimensional linear reaction-convection-diffusion equation with N ≥ 1 N ≥ 1 and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state. [ABSTRACT FROM AUTHOR]

In this article we consider the γ‐stabilization of nth‐order linear time‐invariant dynamical systems using multiplicity‐induced‐dominancy‐based controller design in the presence of delays in the input or the output channels. A sufficient condition is given for the dominancy of a real root with multiplicity at least n+1 and at least n using an integral factorization of the corresponding characteristic function. A necessary condition for γ‐stabilizability is analyzed utilizing the property that the derivative of a γ‐stable quasipolynomial is also γ‐stable under certain conditions. Sufficient and necessary conditions are given for systems with real‐rooted open‐loop characteristic function: the delay intervals are determined where the conditions for dominancy and γ‐stabilizability are satisfied. The efficiency of the proposed controller design is shown in the case of a multilink inverted pendulum. [ABSTRACT FROM AUTHOR]

In this article, we consider several coupled ode-parabolic systems. These systems are known to be not null controllable at any time by localized interior controls. We show that, these systems are not exponentially stabilizable with arbitrary decay rate. And consequently, we recover the known results that they are not null controllable at any time. [ABSTRACT FROM AUTHOR]

We study control properties of a linearized fluid–structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to prove this, we prove a general result for this kind of systems that generalizes in particular the case without structure. The exponential stabilization of the system is obtained with a finite-dimensional feedback control acting only on the momentum equation on a subset of the fluid domain and up to some rate that depends on the coefficients of the system. We also show that as in the case without structure, the system is not exactly null-controllable in finite time. [ABSTRACT FROM AUTHOR]

We give a classification of linear nondissipative mechanical control system under mechanical change of coordinates and feedback. First, we consider a controllable case that is somehow a mechanical counterpart of Brunovský classification, then we extend the result to all linear nondissipative mechanical systems (not necessarily controllable) which leads to a mechanical canonical decomposition. The classification of Lagrangian systems is given afterwards. Next, we show an application of the classification results to the stability and stabilization problem and illustrate them with several examples. All presented results in this paper are expressed in terms of objects on the configuration space R n only, while the state-space of a mechanical control system is R n × R n consisting of configurations and velocities. [ABSTRACT FROM AUTHOR]

We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1–61. https://doi.org/10.1007/978-3-642-34928-7%5f1)), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats. [ABSTRACT FROM AUTHOR]

We extend the approach based on the linearization of triangular systems to new classes of non-linearizable control systems that are almost linearizable. This means that there exists a change of variables and control mapping all but one equations of the initial nonlinear system to a linear system. We show how this property can be used for solving the problem of constructive controllability, i.e., finding trajectories connecting two given points. Namely, we explicitly find a change of variables and control that maps n - 1 equations of the initial system to a linear system. For the remaining first-order nonlinear differential equation, which contains one unknown scalar parameter, the boundary value problem is considered. Once this one-dimensional problem is solved, a trajectory connecting two given points for the initial system is explicitly found. Moreover, we solve the stabilization problem for systems from the proposed classes under additional natural conditions. We give several examples to illustrate a constructive character of our approach. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of controllability and stabilizability of the perturbed linear time-varying systems defined in some suitable real Hilbert space. The aim of this paper is to show that any globally null-controllable system is completely stabilizability and conversely, under some additional condition the complete stabilizability implies global null-controllability [ABSTRACT FROM AUTHOR]

• The paper deals with design problem of LPV systems through directly exploring the primary design space. • The LPV system affinely depends on two groups of parameters namely time-varying system's parameters and constant design parameters. • The algorithm directly searches the primary design space to find a feasible solution that is capable to stabilize the model. • To find the appropriate controller, the design space is iteratively divided until reaching a feasible solution. • The algorithm is able to satisfy some convex or linear constraints on the model due to searching the design space. This paper deals with the stabilizability problem of liner parameter varying (LPV) systems. It is assumed that LPV models affinely depend on time-varying uncertain and time-invariant design parameters. The uncertain parameters, their time-derivations, and design parameters belong to polygonal convex spaces. The stabilizability problem of such systems is studied. Extending the stability conditions to stabilizability conditions generally causes nonlinearity issues due to the coupling between the Lyapunov and design variables. To cope with this issue, a design space exploration algorithm (DSEA) is proposed to accurately determine the design parameters with a feasibility performance similar to stability analysis approaches. DSEA removes the undesired parts of the design subspace that cannot stabilize the model. Then, it checks the corner points of the remaining subspaces to find a stabilizing point. This procedure continues until a stabilizing point is found or the whole design subspaces are detected to be undesirable. Three hundred random LPV systems are generated to compare the feasibility performance of DSEA with existing approaches. Also, the proposed approach is used to stabilize the LPV model of a microgrid consisting of several distributed generation units and energy storage systems. The simulation results show the superiority of DSEA over the existing approaches. [ABSTRACT FROM AUTHOR]

Single and double inverted pendulum systems subjected to delayed state feedback are analyzed in terms of stabilizability. The maximum (critical) delay that allows a stable closed-loop system is determined via the multiplicity-induced-dominancy property of the characteristic roots, that is the dominant (rightmost) roots are associated with higher multiplicity under certain conditions of the system parameters. Other methods such as tracking the changes of the D-curves with increasing delay and the Walton–Marshall method are also demonstrated for the example of the single pendulum. For the double inverted pendulum subjected to full state feedback, the number of control gains is four, and application of numerical methods requires therefore high computational effort (i.e. optimization in a four-dimensional space). It is shown that, with the multiplicity-induced-dominancy–based approach, the critical delay and the associated control gains can be determined directly using the characteristic equation and its derivatives. [ABSTRACT FROM AUTHOR]

We consider the parabolic type equation in R n : (0.1) (∂ t + H) y (t , x) = 0 , (t , x) ∈ (0 , ∞) × R n ; y (0 , x) ∈ L 2 (R n) , where H can be one of the following operators: (i) a shifted fractional Laplacian; (i i) a shifted Hermite operator; (i i i) the Schrödinger operator with some general potentials. We call a subset E ⊂ R n as a stabilizable set for (0.1) , if there is a linear bounded operator K on L 2 (R n) so that the semigroup { e − t (H − χ E K) } t ≥ 0 is exponentially stable. (Here, χ E denotes the characteristic function of E , which is treated as a linear operator on L 2 (R n).) This paper presents different geometric characterizations of the stabilizable sets for (0.1) with different H. In particular, when H is a shifted fractional Laplacian, E ⊂ R n is a stabilizable set for (0.1) if and only if E ⊂ R n is a thick set, while when H is a shifted Hermite operator, E ⊂ R n is a stabilizable set for (0.1) if and only if E ⊂ R n is a set of positive measure. Our results, together with the results on the observable sets for (0.1) obtained in [1,19,25,33] , reveal such phenomena: for some H , the class of stabilizable sets contains strictly the class of observable sets, while for some other H , the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for (0.1) where H is the Schrödinger operator with some general potentials. [ABSTRACT FROM AUTHOR]

The presented paper considers problems of asymptotic, exponential, and uniform exponential stabilizability of discrete time-varying systems. Relations between different types of stabilizability, controllability, and linear-quadratic problem and the Riccati equation for discrete time-varying linear systems are discussed. A few examples illustrating obtained results are presented. At the end, we conclude with a summary of the new results we proved. [ABSTRACT FROM AUTHOR]

For the computation of the lower spectral radius of a finite family of matrices that shares an invariant cone, two recent papers by Guglielmi and Protasov [8] and Guglielmi and Zennaro [9] make use of so-called antinorms. Antinorms are continuous, nonnegative, positively homogeneous and superadditive functions defined on the cone and turn out to be related to the lower spectral radius of the family in a similar way as norms are related to the joint spectral radius. In this paper, we revisit the theory of antinorms in a systematic way, filling in some theoretical holes, correcting a common mistake present in the literature and adding some new properties and results. In particular, we prove that, under suitable assumptions, the lower spectral radius is characterized by a Gelfand type limit computed on an antinorm. [ABSTRACT FROM AUTHOR]

This article studies internal control of the sixth order Boussinesq equation utt uxx+ βuxxxx uxxxxxx +(u²)xx = f posed on a periodic domain T with the internal control input f(·, t) acting on an arbitrarily small open subset of the domain T. It is shown that the system is locally exactly controllable and exponentially stabilizable in the classic Sobolev space Hs+3(T)Ã--Hs(T) for any s ≥ 0. [ABSTRACT FROM AUTHOR]

The uncertainty principle is one of the most important features in modeling and solving linear time-invariant (LTI) systems. The neutrality phenomena of some factors in real models have been widely recognized by engineers and scientists. The convenience and flexibility of neutrosophic theory in the description and differentiation of uncertainty terms make it take advantage of modeling and designing of control systems. This paper deals with the controllability and stabilizability of LTI systems containing neutrosophic uncertainty in the sense of both indeterminacy parameters and functional relationships. We define some properties and operators between neutrosophic numbers via horizontal membership function of a relative-distance-measure variable. Results on exponential matrices of neutrosophic numbers are well-defined with the notion e tA deployed in a series of neutrosophic matrices. Moreover, we introduce the concepts of controllability and stabilizability of neutrosophic systems in the sense of Granular derivatives. Sufficient conditions to guarantee the controllability of neutrosophic LTI systems are established. Some numerical examples, related to RLC circuit and DC motor systems, are exhibited to illustrate the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]

The positivity, stability and stabilizability of switched discrete-time linear singular (SDLS) systems are studied. Our analysis builds on the recently introduced one-step-map for SDLS systems of index-1 and the stability of positive singular systems. We first provide the positive conditions for SDLS systems and the sufficient stability conditions for positive SDLS systems. After that we derive notions and characterizations for stabilizability of positive SDLS systems. Furthermore, we generalize the notion of joint spectral subradius of a finite set of matrix pairs, which allows us to fully characterize stabilizability. [ABSTRACT FROM AUTHOR]

In view of the problems of large consumption of communication and computing resources in the control process, this paper studies a fundamental property for a class of multi-agent systems under event-triggered strategy: the S-stabilizability of a group of multi-agent systems with general linear dynamics under weakly connected directed topology. The results indicate that the S-stabilizability can be described in some way that the stabilizability region and feedback gain can evaluate the performance of the protocol. Firstly, a new distributed event-triggered protocol is proposed. Under this protocol, a kind of hybrid static and dynamic event-triggered strategy are presented, respectively. In particular, by using Lyapunov stability theory and graph partition tool, it is proved that the proposed event-triggered control strategy can guarantee the closed-loop system achieve S-stabilizability effectively, if at least one vertex in each iSCC (independent strongly connected component) cell receives information from the leader, which reflects the ability of distributed control law. Further, we demonstrate that the stabilizability can be realized if the initial system matrix A is Hurwitz. Moreover, it is confirmed that the designed static event-triggered condition is a limit case of dynamic event condition and can guarantee Zeno-free behavior. Finally, the validity of the theoretical results is proved by numerical simulation. [ABSTRACT FROM AUTHOR]

We give a classification of linear nondissipative mechanical control system under mechanical change of coordinates and feedback. First, we consider a controllable case that is somehow a mechanical counterpart of Brunovský classification, then we extend the result to all linear nondissipative mechanical systems (not necessarily controllable) which leads to a mechanical canonical decomposition. The classification of Lagrangian systems is given afterwards. Next, we show an application of the classification results to the stability and stabilization problem and illustrate them with several examples. All presented results in this paper are expressed in terms of objects on the configuration space Rn only, while the state-space of a mechanical control system is Rn×Rn consisting of configurations and velocities.

Sensory uncertainties and imperfections in motor control play important roles in neural control and Bayesian approaches to neural encoding. However, it is difficult to estimate these uncertainties experimentally. Here, we show that magnitude of the uncertainties during the generation of motor control force can be measured for a virtual stick balancing task by varying the feedback delay, τ. It is shown that the shortest stick length that human subjects are able to balance is proportional to τ 2. The proportionality constant can be related to a combined effect of the sensory uncertainties and the error in the realization of the control force, based on a delayed proportional-derivative (PD) feedback model of the balancing task. The neural reaction delay of the human subjects was measured by standard reaction time tests and by visual blank-out tests. Experimental observations provide an estimate for the upper boundary of the average sensorimotor uncertainty associated either with angular position or with angular velocity. Comparison of balancing trials with 27 human subjects to the delayed PD model suggests that the average uncertainty in the control force associated purely with the angular position is at most 14% while that associated purely with the angular velocity is at most 40%. In the general case when both uncertainties are present, the calculations suggest that the allowed uncertainty in angular velocity will always be greater than that in angular position.