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 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.     

On a criterion for the complete controllability of nonautonomous linear systems

We describe the controllability sets of linear nonautonomous systems ẋ = A(t)x + B(t)u, x ∈ ℝ ⁿ , u ∈ U ⊆ ℝ ^m , with entire matrix functions A(t) and B(t) and with a linear set U of control constraints. We derive a criterion for the complete controllability of these linear systems in terms of derivatives of the entire matrix functions A(t) and B(t) at zero. This complete controllability criterion is compared with the Kalman and Krasovskii criteria.     

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 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.     

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.

     

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.     

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.     

On the stability properties of perturbed linear nonstationary systems

Several results are presented that relate the stability properties of a perturbed linear nonstationary system ?(t) = (A(t) + B(t)) x(t) to those of an unperturbed linear system ?(t) = A(t) x(t). Similarly, the stability properties of the discrete system x_{k + 1} = (A_k + B_k) x_k are related to those of x_{k + 1} = A_kx_k.     

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.     

Constraint controllability for linear control systems

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

On weak and strong reachability and controllability of infinite-dimensional linear systems

The notions of reachability and controllability generalize to infinite-dimensional systems in two different ways. We show that the strong notions are equivalent to finite-time reachability and controllability. For discrete systems in Hilbert space, we get simple relations generalizing the Kalman conditions. In the case of a continuous system in Hilbert space, weak reachability is equivalent to the weak reachability of a related discrete system via the Cayley transform.This research was partially supported by the Batsheva de Rothschild Fund for the Advancement of Science and Technology.     

Constrained stochastic controllability of infinite-dimensional linear systems

The main purpose of this article is to investigate the problem of (ε, δ)-stochastic controllability for linear systems of evolution type in infinite-dimensional spaces, wherein the controls are subjected to norm-bounded constrained sets. Some basic prerequisites of infinite-dimensional measures, in particular, Gaussian distributed type, are discussed. Corresponding to this measure, various properties of (ε, δ)-stochastic attainable sets in Hilbert spaces are studied. Necessary and sufficient conditions for (ε, δ)-stochastic controllability with respect to Hilbert space valued linear systems are obtained. Relationships with the deterministic counterpoint are noted. Pursuit game problems are also considered. Examples on systems governed by stochastic linear partial differential equations and stochastic differential delay equations are given for illustration.     

Quasidifferentiability and observability of linear nonstationary systems

We suggest a method for studying the observability of linear nonstationary ordinary differential systems on the basis of the quasidifferentiability of the output variables with respect to some lower-triangular matrix. This approach permits one to weaken the smoothness requirement on the coefficients in the statement of observability criteria.     

On linear nonstationary control processes

For a linear nonstationary optimal control problem, the number of switches in a time-optimal piecewise constant control is estimated above in the case where the control set U is a convex polytope and a genericity condition holds at all points of the time interval under consideration.