Reachability properties of constrained linear systems

This paper presents reachability results for a linear control system and arbitrary terminal pointsp inR ⁿ with controls constrained within a compact setU containing the origin. The well-known results forp=0 are then a special case of our results.Geometric properties of the reachable set are presented and include: general containment properties which describe conditions that guarantee the inclusion of a reachable set in another reachable set; classification of the set of all pointsp that ensure the equivalence of two (different) reachable sets and the properties of this set.The topological properties of the reachable set to a pointp depends on the control setU, the final pointp, and the location of the spectrum of the system in the complex plane. We characterize the geometric properties of the reachable set when the spectrum lies in the closed right-half plane, the open left-half plane, or a combination of both.This work was supported in part by NSF Grant ECS-86-09586.     

Controllability of delay systems with restrained controls

Using a geometric growth condition, we characterize both the function space and Euclidean controllability of a nonlinear delay system which has a compact and convex control set. This extends analogous results for ordinary differential systems, and it yields conditions under which perturbed nonlinear delay controllable systems are controllable.This research was supported by the National Aeronautics and Space Administration under Contract No. NSG 1445Now on leave of absence as Professor, Department of Mathematics, University of Jos, Jos, Nigeria.     

Controllability of the perturbed adiabatic oscillator

In this paper, we consider the global controllability with bounded controls of the perturbed adiabatic oscillator. More explicitly, we study the equationÿ+(1+g(t))y=u, whereg(t) is a suitably small perturbation term andu is the control function, which is restricted either by norm or by range. We give sufficient conditions for global null controllability, under various conditions ong(t), whenu lies in a unit ball or is constrained to assume only positive values.     

Controllability questions for nonlinear systems in abstract spaces

In abstract spaces, we consider certain constrained controllability and approximate controllability properties of a nonlinear system that can be deduced from various controllability properties of its associated linear system. Several examples involving partial differential operators and functional delay operators are given to illustrate the theory.The research of the second author was supported in part by NSF Grant DMS-85-088651 and by the University of Tennessee Science Alliance Award.     

Controller synthesis for constrained discrete-time switched positive linear systems

This paper investigates the controller synthesis for a class of discrete-time switched linear systems with bounds on the controls and the states. First, the synthesis of state-feedback controllers guaranteeing positivity and stability of the closed-loop system is studied for sign-restricted inputs. Also, the results can be extended to asymmetrically bounded controls and constrained states. All the derived conditions are shown as linear programming framework. In addition, a cost function is proposed to maximize the length of the box constraints on the initial state. Finally, several numerical examples illustrate the validity of the developed results.     

Linear control systems: Controllability with constrained controls

We study the null controllability with bounded controls of an ordinary control process in ⁿ . To prove controllability conditions for the nonautonomous case, we need to generalize the definition of characteristic exponents of a linear system. Some theorems on characteristic exponents are stated.This work follows the program of the Gruppo Nazionale per l'Analisi Funzionale e le Sue Applicazioni, Consiglio Nazionale delle Ricerche, Rome, Italy.     

Extremality,Controllability, and Abundant Subsets of Generalized Control Systems

The generalized control system that we consider in this paper is a collection of vector fields, which are measurable in the time variable and Lipschitzian in the state variable. For such system, we define the concept of an abundant subset. Our definition follows the definition of an abundant set of control functions introduced by Warga. We prove a controllability–extremality theorem for generalized control systems, which says, in essence, that either a given trajectory satisfies a type of maximum principle or a neighborhood of the endpoint of the trajectory can be covered by trajectories of an abundant subset. We apply the theorem to a control system in the classical formulation and obtain a controllability–extremality result, which is stronger, in some respects, than all previous results of this type. Finally, we apply the theorem to differential inclusions and obtain, as an easy corollary, a Pontryagin-type maximum principle for nonconvex inclusions.     

Controllability for parabolic equations with uniformly bounded nonlinear terms

We consider a control system for a parabolic equation in a Banach space with uniformly bounded nonlinear termF,

An Example on the Controllability of Perturbed Linear Control Systems

We show by an example that small perturbations with respect to the convergence in measure of the coefficients A and B may affect the complete controllability of the linear control system , where x is the state and u is the control. This answers a question raised in Ref. 1.     

Complete controllability of perturbed linear control systems,revisited

LetL ¹(I, ^n,n )×M(I, ^n,m ) be the space of all pairs (A, B), whereA andB are measurable functions from a compact intervalI to ^n,n and ^n,m , respectively, andA is Lebesgue integrable. Also, let this space be endowed with the topology of theL ¹-norm with respect toA and the topology of convergence in measure with respect toB. Then, the set of all pairs (A, B), for which the corresponding linear control system

Controllability of nonlinear discrete systems without differentiability assumption

The paper gives another approach to studying controllability of nonlinear discrete systems, which is based on proving the support principle for some boundary problems involving locally Lipschitzian set-valued maps. The approach enables us to develop sufficient conditions for various types of local controllability of nonlinear non-stationary discrete systems without differentiability assumption. For linear discrete systems with constrained states and controls, necessary and sufficient conditions for local controllability are given.     

Local controllability of control systems with feedback

We prove sufficient conditions for the instantaneous local controllability of nonlinear (nonsmooth) control systems with feedback. We introduce for that purpose the high-order variations of the reachable map, which is related strongly to its shape. It is a direction in which the reachable map evolves. The cone of variations is convex. This allows one to prove the following theorem: if there exist variationsv ₁, ...,v _p such that int co{v ₁, ...,v _p }, then the system is small-time locally controllable at the point of equilibrium. We provide a short proof of the main result of Sussmann (Ref. 12) and extend it to differential inclusions.This work was supported in part by FCAR of Canada.     

Controllability of series connections of arbitrarily many linear systems

In this paper we study the controllability of series connections of arbitrarily many linear systems. As the main result, we completely determine the controllability and the possible controllability indices of a system obtained by a special series connection of arbitrarily many linear systems.     

On exact controllability and complete stabilizability for linear systems

This work is concerned with the relations between exact controllability and complete stabilizability for linear systems in Hilbert spaces. We give an affirmative answer to the open problem posed by Rabah and Karrakchou [R. Rabah, J. Karrakchou, Exact controllability and complete stabilizability for linear systems in Hilbert spaces, Appl. Math. Lett. 10 (1997) 35–40]. More precisely, if the $C_0$-semigroup $S\left(t\right)$ generated by $A$ is surjective and the pair $(A,B)$ with a bounded operator $B$ is completely stabilizable, then $(A,B)$ is exactly controllable without any additional condition.     

Controllability properties of nonlinear behaviors

This paper proposes a topological framework for the analysis of the time shift on behaviors. It is shown that controllability is not a property of the time shift, while chain controllability is. This also leads to a global decomposition of behaviors.

     

Estimating reachable sets for two-dimensional linear discrete systems

We are concerned with the problem of estimating the reachable set for a two-dimensional linear discrete-time system with bounded controls. Different approaches are adopted depending on whether the system matrix has real or complex eigenvalues. For the complex eigenvalue case, the quasiperiodic nature of minimum time trajectories is exploited in developing a simple, but often accurate, procedure. For the real eigenvalue case, over estimates of reachable sets can be trivially obtained using a decomposition method.The second author was supported by funds supplied by the John M. Bennett Faculty Fellowship, Trinity University, San Antonio, Texas.     

Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints

We present some necessary and sufficient conditions for null controllability for a class of general linear evolution equations on a Banach space with constraints on the control space. We also present a result on the existence of time-optimal controls and some partial results on the maximum principle. Some interesting insights that can be obtained from these results are discussed, and the paper is concluded with an application to a boundary control problem.This work was supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.The author is thankful to Professor L. Cesari for many helpful suggestions and also for calling his attention to the recent papers of Professor K. Narukawa.     

Controllability of semilinear systems

The most general results known about controllability of semilinear systems are found in Refs. 1 and 2. The authors of these papers used different approaches, and no connection between them has ever been pointed out. The purpose of this paper is to present a general theorem under which these results easily follow.This paper is taken from a chapter of the author's doctoral thesis written at University of California, Los Angeles, California.     

On the boundedness of outer polyhedral estimates for reachable sets of linear systems

New properties of outer polyhedral (parallelepipedal) estimates for reachable sets of linear differential systems are studied. For systems with a stable matrix, it is determined what the orientation matrices are for which the estimates possessing the generalized semigroup property are bounded/unbounded on an infinite time interval. In particular, criteria are found (formulated in terms of the eigenvalues of the system’s matrix and the properties of bounding sets) that guarantee for previously mentioned tangent estimates and estimates with a constant orientation matrix that either there are initial orientation matrices for which the corresponding estimate tubes are bounded or all these tubes are unbounded. For linear stationary systems, a system of ordinary differential equations and algebraic relations is derived that determines estimates with constant orientation matrices for reachable sets that have no generalized semigroup property but are tangent and also bounded if the matrix of the system is stable.     

Controllability and stabilizability of linear time‐varying distributed hereditary control systems

This paper is concerned with the controllability and stabilizability problem for control systems described by a time‐varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.