On the complete controllability of linear nonautonomous systems

We suggest a geometric approach to the controllability of nonautonomous linear control systems of the form $\dot x = A(t)x + B(t)u$ , x ?? ? ⁿ , u ?? U ? ? ^m , with conical control constraint set U and continuous matrices A(t) and B(t). We derive two new complete controllability criteria, the first of which is reduced to the analysis of the arrangement of the cones ??^?1(t)B(t)U in the state space of the system [ $\dot \Phi (t) = \left. {A(t)\Phi } \right|(t)$ , ??(0) = E] and the second is based on the existence of appropriate controls bringing zero back to zero. We prove a theorem on the approximation of the control constraint set U by cones with finitely many generators lying inside the cone U with the preservation of the complete controllability property. We present a number of examples illustrating some peculiarities in the evolution of controllability sets of nonautonomous linear systems.     

On complete controllability of linear nonstationary systems

By use of the first integrals of the equations of unperturbed motion we obtain necessary and sufficient conditions for complete controllability of linear nonstationary dynamical systems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 160–163.     

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.     

An explicit criterion for finite-time stability of linear nonautonomous systems with delays

In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result.     

On the controllability of infinite-dimensional linear systems

Two types of controllability, namely weak controllability and strong controllability, are defined for general linear systems. Both types of controllability are useful in control theory. If the system is finite-dimensional, the two types of controllability are equivalent to each other and to the standard concept of complete state-controllability.     

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.     

On strong controllability of infinite-dimensional linear systems

If an infinite-dimensional linear system is strongly controllable (i.e., every state can be reached from any state in finite time), then it is strongly controllable in uniform finite time.     

On single input controllability for infinite dimensional linear systems

Given a controllable linear system {A, B} where A is a Volterra operator, there exists a vector b in the range of B such that {A, b} is controllable. The case where A is a convolution operator on L²(0, ∞) is discussed and an example is given where a controllable system is not replaceable by a single input controllable system.     

Almost complete controllability of linear autonomous difference-differential systems of neutral type

For linear autonomous systems of neutral type with absolutely continuous initial states, it is shown that the spectral condition

$$rank[\lambda E - D - e^{ - \lambda h} (D_1 + D_2 \lambda ),B] = n \forall \lambda \in \mathbb{C}$$

is necessary and sufficient for the controllability of “almost all” initial states to zero. Considerations have a constructive nature. The results are illustrated by an example.

     

Constraint controllability for linear control systems

Summary We investigate some controllability properties of linear control systems in finite or infinite dimensions.     

Spline approximations for linear nonautonomous delay systems

We develop a semi-discrete approximation framework for linear nonautonomous nonhomogeneous functional differential equations of retarded type. The approximation schemes are constructed and convergence results are obtained through the approximation of an associated abstract evolution operator. Computer implementation of the schemes is outlined and a spline-based method included in the framework is constructed. Extensions of the semi-discrete methods to schemes incorporating full discretization and difference equation approximation are also discussed. Numerical results for several examples demonstrating the feasibility of the schemes are presented.     

On constraint controllability of linear systems in Banach spaces

In this paper, we give an addendum to a result of Dolecki and Russell (Ref. 1) related to the duality relationship between observation and control for linear systems in Banach spaces. Our results relate the controllability of a system to the constraint controllability of that system and to the observability of an adjoint system. The main tool used here is an extension of the classical open mapping theorem.     

An asymptotic result and a stability criterion for linear nonautonomous delay difference equations

Linear delay difference equations with variable coefficients and constant delays are considered. By the use of an appropriate solution of the so called generalized characteristic equation, an asymptotic result is obtained and a stability criterion is established.     

Floquet theorem for linear implicit nonautonomous difference systems

The aim of this paper is to develop the Floquet theory for linear implicit difference systems (LIDS). It is proved that any index-1 LIDS can be transformed into its Kronecker normal form. Then the Floquet theorem on the representation of the fundamental matrix of index-1 periodic LIDS has been established. As an immediate consequence, the Lyapunov reduction theorem is proved. Some applications of the obtained results are discussed.     

Instant controllability of linear autonomous systems

In this paper, the complete controllability of linear autonomous control systems in any interval of time and the local controllability in any interval of time are investigated. Necessary and sufficient conditions are obtained for both types of controllability.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.