Abstract: The Geometric Approach is an important trend in Systems and Control Theory developed to achieve a better and deaper investigation of the structural properties of linear dynamical systems and to provide efficient and elegant solutions of problems of controller synthesis such as decoupling and pole-assignment problems or the compensator and  regulator synthesis for linear time-invariant multivariable systems.
The Geometric approach provides a very clear concept of minimality and explicit geometric conditions for controllability, observability, constructibility, pole assignability, tracking or regulation etc. The fundaments of this approach are some  invariant subspaces with respect to a linear transformation.
In 1969 Basile and Marro introduced and studied the basic geometric tools named by them controlled and conditioned invariant subspaces which were applied to disturbance rejection or unknown-input observability. In 1970 Wonham and Morse applied a maximal controlled invariant method to decoupling and noninteracting control problems and later on Wonham's book imposed the name of  "(A,B)-invariant" instead of  "(A,B)-controlled invariant". Basile and Marro opened new prospects to many applications (disturbance rejection, noninteraction etc.) by the robust controlled invariant and the emphasis of the duality. Other properties analysed using geometric tools are  invertibility, functional controllability or unknown input observability The LQ problem was also studied in a geometric framework by Silverman, Hautus and Willems. J.C. Willems also developed the theory of almost controlled and almost conditioned invariant subspaces used in high-gain feedback problems. Further contributions are due to numerous researchers among which Anderson, Akashi, Bhattacharyya, Kucera, Malabre, Molinari, Pearson, Francis and Schumacher. The range of the applications of the Geometric approach was extended to various areas, including, for instance, Markovian representations (Lindquist, Picci  and Ruckebusch) or modeling and estimation of linear stochastic systems).
The past three decades have seen a continually growing interest in the theory of two-dimensional (2D) or, more generally, multidimensional (nD) systems, which become a distinct and important branch of the systems theory. The reasons for the interests in this domain are on one side the richness in potential application fields and on the other side the richness and significance of the theoretical approaches. The application fields include circuits, control and signal processing, image processing, computer tomography, gravity and magnetic field mapping, seismology, control of multipass processes, etc.
A quite new field of the $n$D systems theory is represented by the 2D hybrid models, whose state equation is of differential-difference type. These hybrid models have applications in various areas such as linear repetitive processes, pollution modelling, long-wall coal cutting and metal rolling or in iterative learning control synthesis.
In the present paper, a class of multidimensional hybrid systems described by differential-difference state equations is studied from the point of view of the geometric approach. The state space representation of these systems is given: a  multidimensional hybrid systems is given by a set   where    and     are the    continuous-time and discrete-time drift matrices,    is the    input-state matrix,   is the    state-output matrix and    is the    input-output matrix.  The formulas of the state as well as of the input-output map are obtained. The considered systems represent extensions to multidimensional hybrid continuous-discrete models of Attasi's 2D discrete-time systems.
The notions of controllable and reachable states are defined, as well as the completely controllable and the completely reachable systems. states. A suitable reachability Gramian is constructed for time-invariant multidimensional systems and it is used to obtain conditions for the phase transfer and criteria of reachability. The controllability matrix is constructed and it is used to characterize the space of the reachable states as the minimal   -invariant subspace which contains the columns of the matrix B and to obtain necessary and sufficient conditions of reachability for multidimensional hybrid systems.
An algorithm is provided which compute the minimal   -invariant subspace,  (i.e. the subspace of the controllable states of the system   ) and which extends the 1D algorithm from.
By duality, similar results are obtained for the observability of the time-invariant multidimensional systems and an algorithm is proposed which compute the maximal   -invariant subspace.

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, Director 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 110 published papers in refereed journals or conference proceedings and author or co-author of 15 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..