Controllability in Dimensions One and Two of Sobolev-Type Equations in Banach Spaces

We establish conditions necessary for -controllability in dimension one of first-order singular linear differential equation in Banach spaces. This result generalizes similar results for regular equations. For this class of equations, we show that the notion of -controllability in dimension two is more natural, and moreover, analogous necessary conditions are sufficient in the case of dimension two. Using an abstract approach, we derive sufficient conditions for the -controllability in dimension two of the Cauchy--Dirichlet problem for the Barenblatt--Zheltov--Kochina equation.     

Controllability onto a subspace of regular differential-algebraic systems with numerous delays

We study the H-controllability and complete H-controllability problems for differential-algebraic systems with numerous delays in the control. The original system can be reduced by special transformations to a delay-free system, which is analyzed with the use of results obtained earlier by the authors.     

Controllability of linear systems with positive controls: Geometric considerations

The problem of 0-controllability of linear systems with control restraint set Ω such that 0 ? int Ω is investigated.     

Controllability problems for the string equation

For the string equation controlled by boundary conditions, we establish necessary and sufficient conditions for 0-and ε-controllability. The controls that solve such problems are found in explicit form. Moreover, using the Markov trigonometric moment problem, we construct bangbang controls that solve the problem of ε-controllability. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 939–952, July, 2007.     

Nonsmooth Mapping of Linear Control Systems

This paper deals with linear control systems of a special form. The main goal of the paper is to find the exact analytic solution of the time-optimal control problem for an arbitrary linear control system with constant coefficients using the analytic solution of this problem for the canonical system. For this aim, we construct a certain nonsmooth mapping between the 0-controllability sets of the given systems. In other words, by means of this mapping, we investigate the equivalence of the systems with the same qualitative behavior in a neighborhood of the stationary point.     

F-controllability and observability of linear retarded systems

The paper gives new results onF-controllability of linear retarded functional differential equations. The concept ofF-controllability, weaker than the approximate controllability in the state space, has been introduced by this author in 1976. In the present paper new conditions are presented which, in the case of systems with one delay, become easily verifiable criteria ofF-controllability. A duality betweenF-controllability and observability is discussed, and several examples are given.This research has been supported in part from NSERC of Canada Grant A-9240 and in part from Grant FCAC 1978/79 and 1979/80 of the Quebec Ministry of Education.     

一类弹性系统的精确L2-能控性

该文研究带有压电驱动器的Rayleigh梁系统的精确能控性.先用算子半群方法和提升结果[9]建立了Rayleigh梁方程解的正则性;再用Hilbert唯一性方法结合Diophantine逼近理论的某些结果得到了系统的精确L2-能控性.     

The <Emphasis Type="Italic">R</Emphasis>-observability and <Emphasis Type="Italic">R</Emphasis>-controllability of linear differential-algebraic systems

We study the R-controllability (the controllability within the attainability set) and the R-observability of time-varying linear differential-algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “equivalent”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct the corresponding “equivalent form” for the conjugate DAE. We obtain conditions for the R-controllability and R-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.     

复杂微分网络的指数稳定性

研究树形弦网络在速度反馈控制下的指数稳定性及其控制器的有效性.用半群理论证明速度反馈控制下的闭环系统是适定的. 通过对算子谱的渐近分析, 得到在一定条件下, 系统的谱分布在平行于虚轴的带域中,并证明存在一列根向量构成Hilbert状态空间一个加括号的Riesz基, 从而系统满足谱确定增长条件.利用Riesz基性质和谱分布, 给出系统的指数稳定性结果. 提出控制器有效性的概念, 给出网络不同节点处控制器的有效性比较, 得到使树形弦网络指数稳定所需控制器的最少个数及其放置位置.     

Robust Controllability of Linear Differential-Algebraic Equations with Unstructured Uncertainty

We consider the linear stationary systems of ordinary differential equations (ODEs) that are unsolvedwith respect to the derivative of the unknown vector-function and degenerate identically in the domain of definition. These systems are usually called differential-algebraic equations (DAEs). The measure of how a system of DAEs is unsolved with respect to the derivative is an integer which is called the index of the system of DAEs. The analysis is carried out under the assumption of existence of a structural form with separated differential and algebraic subsystems. We investigate the robust controllability of these systems (controllability in the conditions of uncertainty). The sufficient conditions for the robust complete and R-controllability of a system of DAEs with the indices 1 and 2 are obtained.     

Tangential Interpolation in Weighted Vector-valued H p Spaces, with Applications

In this paper, norm estimates are obtained for the problem of minimal-norm tangential interpolation by vector-valued analytic functions in weighted H^p spaces, expressed in terms of the Carleson constants of related scalar measures. Applications are given to the notion of p-controllability properties of linear semigroup systems and controllability by functions in certain Sobolev spaces.     

Precise bounds for the petersen time-varying real parameter uncertainty system stabilized via linear static controllers

Using a class of linear static controllers, we stabilize the Petersen open-loop two-dimensional linear system (Ref. 1), which consists of one time-varying uncertainty in the state matrixA and one timevarying uncertainty in the input matrixB. We show that the worst-case uncertainty strategy for the closed-loop system is a piecewise constant strategy of the angular state with three switches on the half-turn, –/2/2; it is unique with respect to a set of measure zero. Formulas are derived for the worst-case half-turn radius gainr _HT as a function of the parameters of the class of stabilizing linear static controllers. Using the class of scalar-quadratic Lyapunov functions, we show that a necessary and sufficient condition for the closed-loop system to be robustly stable against all time-varying admissible uncertainties is thatr _HT be less than unity. The bound on the time-varying real parameter uncertainties for the closed-loop system to be robustly stable is derived for the class of linear static feedback controllers. We obtain stabilizing linear static controllers such that the bound is as close to infinity as desired. The derived results are compared with numerical results obtained using commerical robust-control software.     

Construction of a Class of Finite H ∞ Controllers for Infinite-Dimensional Systems

We consider the suppression of forced oscillations in distributed systems of a hyperbolic type by finite-dimensional controllers using an H objective. The system is split into a finite-dimensional and an infinite-dimensional subsystems. The controller receives a signal from the output of both systems. The class of controllers is described in the form of a system of ordinary differential equations.     

Adaptive synchronization of new fractional‐order chaotic systems with fractional adaption laws based on risk analysis

In this paper, a new fractional‐order chaotic system and an adaptive synchronization of fractional‐order chaotic system are proposed. Parameters adaption laws are obtained to design adaptive controllers using Lyapunov stability theory of fractional‐order system. Finally, reliability of designed controllers and risk analysis of adaptive synchronization problem are formulated and, risk of using the proposed controllers in presences of external disturbances are demonstrated. Also, risk of controllers are reduced using an optimizing method. Numerical examples are used to verify the performance of the proposed controllers.     

Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems

Based on the Lyapunov stability and adaptive synchronization theory, optimization design of adaptive controllers and parameter observers with controllable gain coefficient are investigated in detail. The linear errors of corresponding variables and parameters are used to construct different appropriate positive Lyapunov functions V and the parameter observers and adaptive controllers are approached analytically by simplifying the differential inequality dV/dt?0. Particularly, an optional gain coefficient is selected in the parameter observers and positive Lyapunov function, which decides the transient period to identify the unknown parameters and reach synchronization. The scheme is used to study the synchronization of two non-identical hyperchaotic Rössler systems. The theoretical and numerical results confirm that the four unknown parameters in the drive system are estimated exactly and the two hyperchaotic systems reach complete synchronization when the controllers and parameter observers work on the driven system. To confirm the model independence of this scheme, an alternative hyperchaotic system is investigated, whereby the results confirm that the five unknown parameters are identified rapidly and exactly, and that the two hyperchaotic systems reach complete synchronization as well.     

CONTROLLING HYPERCHAOS IN A NEW HYPERCHAOTIC SYSTEM

In this paper,the control strategies for a new hyperchaotic system is investiga-ted. Several kinds of feedback controllers are constructed,such as the linear,speed,nonlinear doubly-periodic function feedback controllers. These controllers are used to prevent the new hyperchaos becoming an unstable equilibrium. Finally,numerical simulations are used to verify the effectiveness of the proposed controllers.     

Discrete model reference adaptive control of continuous-time linear multi-input multi-output systems via multirate sampled-data controllers

The use of multirate sampled-data controllers for linear multivariable time-invariant systems with unknown parameters is investigated. Such controllers contain periodically time-varying elements and a multirate sampling mechanism with different sampling periods at each system input. Their application to unknown continuous-time linear multi-input, multi-output systems results in a sampled closedloop system for which an arbitrary discrete-time transfer function matrix can be assigned, as is shown in the present paper. The contribution of the present paper is twofold: the use of multirate sampled-data controllers in the area of model reference adaptive control; and the application, for the first time, of periodically varying controllers for model reference adaptive control of multi-input, multi-output systems.The work described in this paper has been partialy funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research, and Technology and by the Heracles General Cement Company of Greece.     

Reduced-order feedback control of distributed parameter systems via singular perturbation methods

Implementable feedback control of distributed parameter systems must often be based on reduced-order models due to the infinite dimensional nature of the actual open-loop system. The behavior of controllers designed via reduced-order models obtained with singular-perturbation techniques is analyzed. When such controllers are used in the actual distributed parameter system, the closed-loop stability is in question. The results presented here provide bounds on the smallness of the singular perturbation parameter to ensure stable operation; such a priori bounds may be used to evaluate potential reduced-order controllers for distributed parameter systems.     

Internal Stabilization of a Mutualistic Reaction Diffusion System

We study the internal stabilization of steady-state solutions to a 2-species mutualistic reaction diffusion system via finite-dimensional feedback controllers. Our main idea is to use different internal controllers to stabilize different steady-state solutions. The controllers axe provided by considering LQ problems associated with the lineaxized systems at steady-state solutions.