**
Plenary
Lecture**

Linear Control Systems Over Spaces of Regulated Functions

**Professor Valeriu Prepelita**

University Politehnica of Bucharest

Department of Mathematics I

Splaiul Independentei 313, Bucharest

ROMANIA

E-mail: vprepelita@mathem.pub.ro

**Abstract:** A model of generalized linear
control systems is considered, which is represented by matrices with
elements functions of bounded variation and controls over the space of
regulated functions (i.e. functions which possesses finite one-sided limits
on a given interval).

The Perron-Stieltjes integral with respect to the set of regulated functions
(which include the set of functions of bounded variation) was defined in
[12]. This integral is equivalent to the Kurzweil integral (see [2], [8] and
[9]).

In this paper, using the results of M.Tvrdy ([10], [11]) concerning the
properties of the Perron-Stieltjes integral with respect to regulated
functions and the differential equation in this space, the formulas of the
states and of the general response of the control systems are obtained. This
allows us to extend in this framework the concepts of reachability and
observability (see for instance [1] and [6]). These fundamental concepts are
analysed by means of two suitable controllability and observability Gramians.
The duality between the concepts of controllability and observability is
emphasized as well as Kalman's canonical form. The spaces of reachable and
observable states are described.

The minimal energy transfer is studied and the optimal control is provided.
In the case of completely observable systems a formula is obtained which
recovers the initial state from the exterior data. It is emphasized that
these systems are generalizations of the classical linear systems described
by differential equations with controls. The considered approach seems to be
the most general framework in which the linear control systems can be
studied.

Linear boundary value (acausal) systems are studied in the same framework
[3]. Semiseparable kernels are associated to acausal systems with well-posed
boundary conditions. Minimal realizations of semiseparable kernels are
characterized as well as the irreducibility of the acausal systems. Adjoint
systems are defined and an input-output operator is provided.

A Peano-Baker type formula is obtained for the calculus of the fundamental
matrix of the generalized linear differential systems [4].

This study can be continued in many directions such as stability, positivity,
multidimensional generalized systems [5], 2D generalized
differential-difference systems [7], linear quadratic optimal control etc.

**Brief Biography of the Speaker:**

Valeriu Prepelita graduated from the Faculty of Mathematics-Mechanics of the
University of Bucharest in 1964. He obtained Ph.D. in Mathematics at the
University of Bucharest in 1974. He is currently Professor at the Faculty of
Applied Sciences, the University Politehnica of Bucharest, Head of the
Department Mathematics-Informatics. His research and teaching activities
have covered a large area of domains such as Systems Theory and Control,
Multidimensional Systems, Functions of a Complex Variables, Linear and
Multilinear Algebra, Special Functions, Ordinary Differential Equations,
Partial Differential Equations, Operational Calculus, Probability Theory and
Stochastic Processes, Operational Research, Mathematical Programming,
Mathematics of Finance.

Professor Valeriu Prepelita is author of more than 100 published papers in
refereed journals or conference proceedings and author or co-author of 12
books. He has participated in many national and international grants. He is
member of the Editorial Board of some journals, member in the Organizing
Committee and the Scientific Committee of some international conferences,
keynote lecturer or chairman of some sections of these conferences. He is a
reviewer for five international journals. He received the Award for
Distinguished Didactic and Scientific Activity of the Ministry of Education
and Instruction of Romania.