Linear delay-differential systems with commensurate delays: an algebraic approach.

*(English)*Zbl 0989.34001
Lecture Notes in Mathematics. 1770. Berlin: Springer. vi, 176 p. (2002).

This book deals with linear delay-differential equations (DDEs) with constant coefficients and commensurate point delays. The investigation of DDEs presented here is with respect to their general control-theoretic properties. An approach is developed, which shows that the dynamical systems described by DDEs can be successfully studied from the behavioral point of view. In this sense the key notion for specifying a system is the space of all possible trajectories of that system. The solutions under consideration are in the space of \( C^{\infty} \)-functions. From the algebraic point of view, the approach consists in obtaining a setting where a polynomial ring in two operators acts on a module of functions. This algebraic setting is developed in Chapter 2. The ring of operators consisting of point-delay-differential operators as well as certain distributed delays is denoted by \( \mathcal H \). The ring \( \mathcal H \) is investigated from a purely algebraic point of view in Chapter 3. The main result in this chapter is that, under unmodular transformations, matrices with entries in the ring \( \mathcal H \) behave like matrices over Euclidean domains. Moreover, a description of \( \mathcal H \) as a convolution algebra consisting of distributions with compact support is presented in the chapter. In Chapter 4 an algebraic characterization of systems of DDEs sharing the same solution space is made. Some of the basic concepts of system theory, defined purely in terms of trajectories, are characterized by algebraic properties of the associated equations. This includes the notions of controllability, input/output partitions and the investigation of interconnection of systems. In Chapter 5 explicit DDEs of first order and of related type are studied. Some applications of DDEs are considered as well.

Reviewer: Angela Slavova (Sofia)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

34Kxx | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

34H05 | Control problems involving ordinary differential equations |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

93B05 | Controllability |