Approximate controllability of second‐order distributed systems

In this paper, we study the approximate controllability of control systems with state and control in Banach spaces and described by a second‐order semilinear abstract differential equation. We compare the approximate controllability of the system with the approximate controllability of an associated discrete system. Copyright © 2013 John Wiley & Sons, Ltd.     

Novel Controllability Conditions for a Class of Singularly-Perturbed Systems with Small State Delays

A singularly-perturbed linear time-dependent controlled system with small pointwise and distributed delays in the state variable is considered. Two simpler parameter-free systems, the slow and fast ones, can be associated with the original system. It was established in the literature that the Euclidean space controllability of the original system, valid for all sufficiently small values of the parameter of singular perturbations, follows from the controllability properties of the slow and fast systems. It also was established that such a connection between the controllability properties of the original system and the slow and fast systems is correct, in general, only in one direction. Namely, the controllability of the slow and fast systems provides the controllability of the original system, while the controllability of the original system not always yields the controllability of both the slow and fast systems. In this paper, we consider the original system such that the respective fast system is uncontrollable, meaning that the previously established controllability conditions are not applicable to this original system. In this case, novel parameter-free sufficient conditions for the Euclidean space controllability of the original system, robust with respect to the small parameter of singular perturbations, are derived. Illustrative examples are presented.     

Controllability of nonlinear fractional order integrodifferential system with input delay

This paper addressed the controllability of nonlinear fractional order integrodifferential systems with input delay. Firstly, the Caputo fractional derivatives and the Mittag‐Leffler functions are employed. Thereafter, we establish a set of sufficient and necessary conditions for the controllability of the linear fractional system. Furtherly, controllability conditions of the nonlinear integrodifferential fractional order system with input delay are acquired by utilizing Arzela‐Ascoli theorem and Schauder's fixed‐point theorem. Finally, an example is presented to demonstrate our main results.     

Remarks on the controllability of fractional differential equations

In this paper, we study the approximate controllability of a linear fractional differential equation by using the method of regularization of Tikhonov. New concepts and results about controllability are established. Then, under the condition of the positivity of the controllability operator, we obtain that the linear system can be steered to an arbitrary small neighbourhood of the fractional integral of the state at final time.     

Controllability of nonlinear Hilfer fractional Langevin dynamical system

The main objective of this article is to prove the controllability result of a nonlinear Hilfer fractional Langevin dynamical system in an abstract-weighted space. This paper initiates with the persistence of the control formula for the Hilfer fractional Langevin dynamical system. Furthermore, As ensue is extended to the controllability result of Hilfer fractional Langevin dynamical system by the use of Krasnoselskii fixed point theorem. Ultimately, an example is furnished to demonstrate our outcomes.     

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.     

On the controllability of the Vlasov-Poisson system

In this paper, we study the controllability of the Vlasov-Poisson system in a periodic domain, by means of an interior control located in an spatial subdomain.The first result proves the local exact zero controllability in the two-dimensional torus between two small acceptable distribution functions, with an arbitrary control zone.A second result establishes the global exact controllability in arbitrary dimension, provided the control zone satisfies the condition that it contains a hyperplane of the torus.     

不变流形方法与非线性控制系统的能控制性

在Kovalev方法基础上运用不变流形研究非线性系统的能控制性问题,得出了一类仿射非线性系统能控的必要条件,讨论了必要条件的实现问题,研究了带有两个陀螺的刚体运动,证明了它满足能控性的必要条件.     

Boundary controllability for the quasi-linear wave equations coupled in parallel

We study the boundary exact controllability for a system of two quasi-linear wave equations coupled in parallel with springs and viscous terms. We prove the locally exact controllability around superposition equilibria under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the coupled quasi-linear system moves from a superposition equilibrium in one location to a superposition equilibrium in another location. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and superposition equilibria of the system.     

Some results on controllability of a nonlinear degenerate parabolic system by bilinear control

In this paper, we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control. Based on the shrinking property of the solutions, we prove that the system is not globally approximately controllable. Furthermore, we give an approximate null controllability result. We also prove that the system is not globally exactly null controllable by a comparison principle.     

Exact controllability of a fluid–rigid body system

This paper is devoted to the controllability of a 2D fluid–structure system. The fluid is viscous and incompressible and its motion is modelled by the Navier–Stokes equations whereas the structure is a rigid ball which satisfies Newton's laws. We prove the local null controllability for the velocities of the fluid and of the rigid body and the exact controllability for the position of the rigid body. An important part of the proof relies on a new Carleman inequality for an auxiliary linear system coupling the Stokes equations with some ordinary differential equations.     

Approximate controllability of second-order distributed implicit functional systems

In this paper we study the approximate controllability of control systems with states and controls in Hilbert spaces, and described by a second-order semilinear abstract functional differential equation with infinite delay. Initially we establish a characterization for the approximate controllability of a second-order abstract linear system and, in the last section, we compare the approximate controllability of a semilinear abstract functional system with the approximate controllability of the associated linear system.     

On the complete controllability of hybrid differential-difference systems

For time-independent hybrid differential-difference linear systems, we study the complete controllability problem, that is, the problem of complete quieting of such systems. We derive necessary and sufficient parametric conditions for strict complete controllability in various classes of admissible controls. In the case of simplest basic classes, we prove necessary and sufficient parametric conditions for the weak complete controllability and suggest a method for constructing the desired controls quieting the system by using methods of the theory of entire functions of finite degree. We discuss problems of estimating the duration of the transient process. As an example, we consider the strict complete controllability problem in various classes of functions for the case of a system of scalar differential-difference equations.     

Controllability of a Reaction-Diffusion System Describing Predator–Prey Model

This article establishes the controllability to the trajectories of a reaction-diffusion-advection system describing predator–prey model by using distributed controls acting on a single equation with the no-flux boundary conditions. We first prove the exact null controllability of an associated linearized problem by establishing an observability estimate, derived from a global Carleman type inequality, for the adjoint system. The proof of the nonlinear problem relies on the suitable regularity of the control and Kakutani's fixed point theorem.     

Null controllability in a fluid-solid structure model

We consider a system coupling the Stokes equations in a two-dimensional domain with a structure equation which is a system of ordinary differential equations corresponding to a finite dimensional approximation of equations modeling deformations of an elastic body or vibrations of a rigid body. For that system we establish a null controllability result for localized distributed controls acting only in the fluid equations and there is no control in the solid part. This controllability result follows from a Carleman inequality that we prove for the adjoint system.     

Controllability of Some Nonlinear Systems in Hilbert Spaces

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.     

Optimal Actuator Placement for Dynamical Systems

Vibrations occur in many industrial areas and produce undesirable side effects. To avoid respectively suppress these effects actuators can be attached to the structure. Actuators can transfer forces into the structure, which can be used to control the system behavior of the structure in order to achieve, for example, vibration absorption, noise reduction and/or exact positioning. The appropriate positioning of actuators is of significant importance for the controllability of the structure. We present a method for determining the optimal actuator placement. We consider applications, where the dynamics of a system can be described by a finite dimensional, controllable and observable state-space model. The concept of controllability by Kálmán forms the basis for the definition of controllability measures. In this context, the controllability gramian plays an important role. In our approach actuators are placed in such a way that the selected controllability measure becomes maximal. This leads to an optimization problem with binary and continuous variables and linear matrix inequalities. The numerical results show the optimal actuator placement for a two dimensional flexible clamped plate. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

APPROXIMATE CONTROLLABILITY OF INFINITE DIMENSIONAL BILINEAR SYSTEMS

1.IntroductionRecentlymanyresultshavebeenobtainedfordistributedbilinearsystemsl3'7'8].In[3]thecontrollabilityofabilinearsystem%=Aw(t) p(t)Bw(t)wasstudied,whereAistheindnitesimalgeneratorofaCosemigroupofboundedlinearoperatorsT(t)onaBanachspaceX.B:X-- 5isaboundedlinearoperator,andpELI([0,T];R)isacontrol.TheconditionsforelementsofEtobeaccessiblefromagiveninitialstatecoweregiven.Itisclearthattheystudiedthebilinearsysteminaspecialcase.In[8]theystudiedthesysteminamorespecialcasebecausetheyass…     

半线性热方程精确内部能控性

主要采用了构造辅助系统u′ △u=0的方法，同时也利用了强单调算子 的满射性。在条件（H)的修改下，得到在有限时间内的精确可控性，研究半线性热方程系统ey/et=△y-f(x,t,y,倒△y) h(x,t)的精确内部能控性。     

弹性结构有限元控制系统

本文讨论了经有限元方法处理后的弹性结构系统的可控、可观测、镇定等问题.所得的结论与用分布参量系统模型所得的结论一致,但却便于用计算机计算且方法简单.在一、中研究了系统的可控与可观测的问题,给出了易于用计算机判别的条件.在二、中对于采用线性反馈镇定弹性体的问题进行了仔细的讨论,指出对弹性结构系统而言,若系统完全可控仅用位移反馈可以任意配置振动频率但却无法镇定系统,而仅用速度反馈虽可以进行镇定但镇定能力是有限的,对于在系统运动方程中包含刚体运动成分的情形也作了研究.在三、中对梁的控制问题用有限元进行了处理,指出直梁作为一个系统可以分解为拉压、扭转和两个方向弯曲这四个互不关联的子系统,它们的可控与可观测问题可以分别进行讨论.最后对折线型刚架的可控与可观测的问题也作了探讨.