Date of final exam: 26/03/1999

E-mail: p.vettori@dei.unipd.it

Tutor: Prof.  E. Fornasini, Università di Padova

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Delay-differential systems in the behavioral approach   ___________________________________________________________________________________________________________

Advisor:

Prof.  E. Fornasini, Università di Padova

Co-advisor:

Prof.  S. Zampieri, Università di Padova

Summary of the thesis

Delay-differential systems are a particular class of linear time invariant systems that can be modeled with differential equations that depend also on past values of the variables. This kind of functional equations arise in many practical applications: chemical reactors, remote control systems with communication constraints and even some distributed systems governed by partial differential equations that, as in the case of a flexible rod, may be reduced to delay-difference equations.
Infinite dimensional linear systems theory is the main tool used to deal with delay-differential systems; anyway this class has been treated also with other techniques such as those developed by the theory of linear systems over commutative rings.
Assuming, as in the latter case, a rather algebraic point of view, this thesis investigates delay-differential and more general systems that are defined by convolutional equations: it is based principally on the "behavioral approach", first proposed by Jan C. Willems in the eighties, but also compares its definitions and results with those arising within the module theoretic approach by Michel Fliess. As explained in the thesis, these two approaches are in a certain sense, "dual" of each other: in M. Fliess' approach systems are modules of operators while in the behavioral approach the main object in the definition of a dynamical system is the "behavior" which is the space of admissible trajectories.
The thesis is mainly concerned with a fundamental structural property of dynamical systems: controllability. In particular it extends some recently published results about the one delay case (i.e. every delay is an integer multiple of the same quantity) to the multidelay case (there are more noncommensurate delays) showing which relations occur among spectral controllability (a generalization of the well-known PBH test), different types of algebraic controllability as proposed by M. Fliess and various conditions that are equivalent to controllability for classical differential "behaviors".
 

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