Abstract: The Operational Calculus as a distinct discipline has a history which has exceeded a century. But its roots can be found in the works of Leibniz, Bernoulli, Lagrange, Laplace, Euler, Fourier, Cauchy and others. Its importance is determined by its utility in solving complex problems in many domains such as Calculus, Number Theory, Special Functions, Ordinary Differential Equations, Mathematical Physics, Heat Transfer, Electronics, Automatics, etc.
In Systems and Control Theory the frequency domain methods, based on Laplace transformation in the continuous-time case or on Z transformation in the discrete-time case, play a very important role in the study of the "classical" 1D systems.
In the last two decades the study of two-dimensional (2D) systems (and more generally, of n-dimensional systems) developed as a distinct branch of Systems Theory, due to its applications in various domains as image processing, seismology and geophysics, control of multipass processes etc.
The two-dimensional (2D) systems were obtained from classical 1D linear dynamical systems by generalizing from a single time variable to two (space) variables. Different state space models for 2D systems have been proposed by Roesser, Fornasini and Marchesini, Attasi, Eising and others.
A subclass of 2D systems is represented by systems which are continuous with respect to one variable and discrete with respect to another one. The continuous-discrete models have applications in many problems like the iterative learning control synthesis, repetitive processes or in engineering problems such as metal rolling.
In order to extend the frequency domain methods to these multiple hybrid systems one needs a generalization of the Laplace and Z transformation.
The aim of this paper is to give a complete analysis of a suitable hybrid Laplace-Z type transformation and to emphasize its applications in the study of multidimensional continuous-discrete systems or for solving multiple hybrid equations.
In section 2 the continuous-discrete original functions are defined and it is shown that their set is a complex commutative linear algebra with unity. A multiple hybrid Laplace-Z transformation is defined as a linear operator defined on this algebra and taking values in the set of multivariable functions which are analytic over a suitable domain.
In section 3 the main properties of the multiple hybrid Laplace-Z transformation are stated and proved, including linearity, homothety, two time-delay theorems, translation, differentiation and difference of the original, differentiation of the image, integration and sum of the original, integration of the image, convolution, product of originals, initial and final values.
Section 4 is devoted to the inversion problem. Some formulas and methods for determining the original are given.
This hybrid transformation is employed in Section 5 to obtain transfer matrices for different classes of 2D (and more generally (q,r)-D) continuous-discrete linear control systems of Roesser-type, Fornasini-Marchesini-type and Attasi type models, including descriptor and delayed systems.
The realization problem is studied in Section 6. Two canonical controllable and observable realizations are provided. An algorithm is proposed which determines a minimal realization for separable (q,r)-D multi-input-multi-output (MIMO) systems. This method generalizes to (q,r)-D systems the celebrated Ho-Kalman algorithm. The proposed algorithm can also be used for MIMO separable nD discrete-time linear systems or for MIMO nD systems described by a class of hyperbolic partial differential equations.

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 several 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.