F-controllability and observability of linear retarded systems

Abstract

Abstract. The paper gives new results on F-controllability of linear retarded functional differential equations. The concept of F-controllability, weaker than the approximate controllability in the state space, has been introduced by this author in 1976. In the present paper new conditions are presented which, in the case of systems with one delay, become easily verifiable criteria of F-controllability. A duality between F-controllability and observability is discussed, and several examples are given.

Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

“…Controllability in the full state space (approximate, functional controllability [15,16]) is quite a restrictive concept [16]. For instance, many delay systems do not satisfy one of the necessary conditions of approximate controllability, even though they often possess other good properties such as stabilizability and spectral controllability.…”

“…Therefore, one should search for a "smaller space" in which a controllability would be characterized by less restrictive conditions and would be related to stabilizability and spectral controllability. The concept of F-controllability, weaker than the approximate controllability in the state space, has been introduced (see [16] and the references therein). Works on the study of systems with delay on the basis of an approach of the space of minimal states also appeared (see [17] and the references therein).…”

“…The functional controllability of systems with delay according to the approach, based on the notion of time delay systems' state minimality [17], turns out to be equivalent to the complete controllability, which solves one of the most difficult problems of controllability for systems with delay: the problem of complete damping of such systems in a finite time (total quieting, controllability to null function, null controllability). The spectral criterion of complete controllability for a system with delay (without parameter) was proven in [12,16,18] (see also the references in [17]). In [12,15,19] and a number of other papers, these conditions were associated with parametric rank Kalman-type conditions.…”

“…However, the stiffness, as well as a possible high dimension of the SPS, can considerably complicate this application. Therefore, for example, with the direct use of rank controllability criteria [12,16,18], the controllability matrix of such systems has a large dimension and is ill-conditioned. Correct checking of the rank of such matrices can be complicated.…”

Algebraic criteria of approximate controllability of retarded systems

“…Let K t K t These examples provide some motivation for a controllability concept weaker than the approximate controllability in the state space X, namely the one given by relat~t (21). An additional motivation for studying this type of controllability is given [13]. Similarly as in [15], let G*: X ÷ X be the following linear operator Condition (i) has been previously known [15], but (ii) is new.…”