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.     

On the 0-local controllability of a linear control system

The problem of the 0-local controllability of a linear control system when the origin does not belong to the constraint set is considered. Sufficient conditions for the 0-local controllability are presented here, extending a previous theorem by Saperstone and connecting this property with the distance of the constraint set from the origin.     

Complete controllability

A linear autonomous control system in ⁿ is said to be completely controllable iff there existsT>0 such that eachx ⁿ can be steered to anyy ⁿ in timeT. This paper presents a geometric characterization of this property in the case in which there are constraints on the values which the control maps can assume. A necessary and sufficient condition to get instant controllability (i.e., complete controllability for anyT>0) is also derived. This condition generalizes the well-known Kalman condition to the constrained case.     

Feedback stabilization for perturbed linear systems *

In this paper, we consider the questions of stabilization of perturbed (or uncertain) linear systems in Hilbert space. Perturbations of the system operator represent the uncertainty in the modelling process and could be bounded or even unbounded. Sufficient conditions are presented that guarantee stabilizability of the perturbed system given that the nominal (unperturbed) system is stabilizable. In particular it is shown that for certain class of perturbations weak and strong stabilizability properties are preserved for the same state feedback control. Two numerical examples are given to illustrate our theory.     

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.     

Invariance of asymptotic stability of perturbed linear systems on Hilbert space

The questions of stabilizability of structurally perturbed or uncertain linear systems in Hilbert space of the form are considered. The operatorA is assumed to be the infinitesimal generator of aC ₀-semigroup of contractionsT(t),t0, in a Hilbert spaceX;B is a bounded linear operator from another Hilbert spaceU toX; and {P(r),r } is a family of bounded or unbounded perturbations ofA inX, where is an arbitrary set, not necessarily carrying any topology. Sufficient conditions are presented that guarantee controllability and stabilizability of the perturbed system, given that the unperturbed system has similar properties. In particular, it is shown that, for certain classes of perturbations, weak and strong stabilizability properties are preserved for the same state feedback operator.This work was supported in part by the Natural Science and Engineering Research Council of Canada under Grant No. A7109.     

Approximate controllability for trajectories of semilinear control systems

We treat an abstract semilinear control system and study the controllability problem for its trajectories. Assuming a range condition of the control action operator and an inequality condition on the system parameters, we can show that the reachable trajectory set of the semilinear system is equivalent to that of its corresponding linear system.The author wishes to express his deep appreciation to Prof. T. I. Seidman for his many helpful suggestions and to Prof. W. Takahashi for many stimulating conversations.     

Controllability properties of constrained linear systems

In this paper, we present results on constrained controllability for linear control systems. The controls are constrained to take values in a compact set containing the origin. We use the results on reachability properties discussed in Ref. 1.We prove that controllability of an arbitrary pointp inR ⁿ is equivalent to an inclusion property of the reachable sets at certain positive times. We also develop geometric properties ofG, the set of all nonnegative times at whichp is controllable, and ofC, the set of all controllable points. We characterize the setC for the given system and provide additional spectrum-dependent structure.We show that, for the given linear system, several notions of constrained controllability of the pointp are the same, and thus the setC is open. We also provide a necessary condition for small-time (differential or local) constrained controllability ofp.This work was supported in part by NSF Grant ECS-86-09586.     

Small-time local controllability and continuity of the optimal time function for linear systems

For a finite-dimensional linear system, in which the control is restricted to belong to a completely arbitrary setJ, we give a simple necessary and sufficient condition for small-time local controllability from a pointp. The condition is equivalent to a characterization of the property that the Bellman function for the corresponding minimum-time optimal control problem is continuous atp.This work was partially supported by the National Science Foundation, Grant No. MCS-78-02442.     

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.     

Uniformly valid feedback expansions for optimal control of singularly perturbed dynamic systems

This paper shows how to construct a feedback control law for a class of singularly perturbed autonomous optimization problems. The control law is expressed as a single power series in the small parameter representing the ratio of the two effective time scales of the problem. The present approach avoids the need of expansion matching. The method is applied to a constant-speed interception problem. Comparison of numerical results with the exact solution shows an excellent agreement.Dedicated to G. Leitmann     

Linear controllability by piecewise constant controls with assigned switching times

For a linear time-dependent control process inR ⁿ, we prove that the complete controllability by means of the space of all the admissible controls is equivalent to the complete controllability by means of a suitablen-dimensional space of piecewise constant controls with at mostn preassigned switching times. An analogous result is also established for more general controllability problems.     

Regularity modulus of arbitrarily perturbed linear inequality systems

We aim to quantify the stability of systems of (possibly infinitely many) linear inequalities under arbitrary perturbations of the data. Our focus is on the Aubin property (also called pseudo-Lipschitz) of the solution set mapping, or, equivalently, on the metric regularity of its inverse mapping. The main goal is to determine the regularity modulus of the latter mapping exclusively in terms of the system's data. In our context, both, the right- and the left-hand side of the system are subject to possible perturbations. This fact entails notable differences with respect to previous developments in the framework of linear systems with perturbations of the right-hand side. In these previous studies, the feasible set mapping is sublinear (which is not our current case) and the well-known Radius Theorem constitutes a useful tool for determining the modulus. In our current setting we do not have an explicit expression for the radius of metric regularity, and we have to tackle the modulus directly. As an application we approach, under appropriate assumptions, the regularity modulus for a semi-infinite system associated with the Lagrangian dual of an ordinary nonlinear programming problem.     

The monotone method for controllability of the nonlinear evolution systems

In this paper,we study the controllability of the nonlinear evolution systems.We establish the controllability results by using the monotone operator theory.No compactness assumptions are imposed in the main results.We present an example to illustrate our results.     

Solvability,controllability, and observability of singular systems

The problems of solvability, controllability, and observability for the singular systemKx(t)=Ax(t)+Bu(t) are studied, whereK is a singular, square matrix andu(t) is a complex vector function sufficiently differentiable. The classical theories of matrix pencils are first related to the solvability of singular systems. Then, the concepts of reachability, controllability, and observability of regular systems are extended to singular systems. Finally, the set of reachable states is described. The proposed matrix conditions for testing the controllability and observability of singular systems are simple and always feasible.     

Global robust controllability of the triangular integro-differential Volterra systems

A solution of the global controllability problem for a class of nonlinear control systems of the Volterra integro-differential equations is presented. It is proven that there exists a family of continuous controls that solve the global controllability problem for this class. The constructed controls depend continuously on the initial and the terminal states. It makes possible to prove the global controllability of the uniformly bounded perturbations of these systems under the global Lipschitz condition for the unperturbed system with respect to the states and the controls.     

On output feedback control of singularly perturbed systems

We treat the problem of robustness of output feedback controllers with respect to singular perturbations. Given a singularly perturbed control system whose boundary layer system is exponentially stable and whose reduced order system is exponentially stabilizable via a (possibly dynamical) output feedback controller, we present a sufficient condition which ensures that the system obtained by applying the same controller to the original full order singularly perturbed control system is exponentially stable for sufficiently small values of the perturbation parameter. This condition, which is less restrictive than those previously given in the literature, is shown to be always satisfied when the singular perturbation is due to the presence of fast actuators and/or sensors. Furthermore, we show explicitly that, in the linear time-invariant case, if this condition is not satisfied then there exists an output feedback controller which stabilizes the reduced order system but destabilizes the full order system.     

On the characterization of the controllability property for linear control systems on nonnilpotent,solvable three-dimensional Lie groups

In this paper we show that a complete characterization of the controllability property for linear control system on three-dimensional solvable nonnilpotent Lie groups is possible by the LARC and the knowledge of the eigenvalues of the derivation associated with the drift of the system.     

Output controllability and optimal output control of positive fractional order linear discrete system with multiple delays in state, input and output

The article concerns output controllability and optimal output control of positive fractional order discrete linear systems with multiple delays in state, input and output. Necessary and sufficient conditions for output reachability (output controllability from zero initial conditions) and null output controllability (output controllability to zero final output) are given and proven. We also prove that the positive system is output controllable if it is output reachable and null output controllable with the output reachability index is equal or less than the null output controllability index. Sufficient conditions for the solvability of the optimal output control problem are given. Numerical examples are presented to illustrate the theoretical results.     

On perturbations of controllable implicit linear systems

Various notions of controllability for general (i.e. possiblynon-regular) implicit linear systems are considered. Some genericityproperties of the corresponding controllable systems are established.