Plenary Lecture

Plenary Lecture

Boundary Value 1D and nD Linear Control Systems

Professor Valeriu Prepelita
University Politehnica of Bucharest
Department of Mathematics-Informatics I
Splaiul Independentei 313, 060042 Bucharest

Abstract: Linear one-dimensional (1D) acausal systems, i.e. systems with boundary conditions have been introduced in state space representation by A.J. Krener [12], [13], in connection with the modeling of boundary value regulation. M.B. Adams, A.S. Willsky and B.C. Levy [1], [2] have obtained important results in the linear estimation of stochastic processes governed by time varying systems with boundary conditions.
T. Kailath has tackled this topic in an input-output approach in a series of papers on linear estimation theory [11]. H. Bart, I. Gohberg and M.A. Kaashoek have come to linear systems with boundary conditions motivated by the analysis of Wiener-Hopf integral equation and related convolution equations [4], [5]. The theory of systems with boundary conditions has been developed by I. Gohberg and M.A. Kaashoek in a series of papers [7], [8], [9], [10]. They have brought this theory to the level of the classical theory of the causal linear systems. For instance, the characterization of the classes of irreducible and minimal systems has been obtained and it has been emphasized that these classes and the class of controllable and observable systems are different (whereas in the case of causal systems they coincide).
In the same time, in the framework of Systems Theory, different state space models of two-dimensional 2D systems has been proposed by Roesser [19], Fornasini and Marchesini [6], Attasi [3] and others. The study of 2D (and nD) systems has known an important development in the last three decades due to their significant applications in various areas as image processing, seismology, geophysics or computer tomography. The above mentioned papers and the subsequent ones have studied the causal systems, i.e.
systems whose states and outputs are determined by the inputs and the initial states.
In this paper we present the extension of these results (see [14]-[18]) to multidimensional (nD, n
2) boundary-value systems, by introducing a class of systems which represents the continuous-time time-varying counterpart of Attasi's discrete-time 2D model [3]. The state-space representation of the considered nD boundary-value systems is given, including well-posed boundary conditions. The formulas of the state and of the input-output map of the nD boundary-value systems are obtained, by means of a suitable variation-of-parameters formula.
Generalized nD separable kernels are associated to these systems. The realization problem is discussed for nD separable kernels and necessary and sufficient conditions for minimality are presented.
The adjoints of nD boundary-value systems are introduced and the input-output maps of the adjoint systems are derived. Two inner products are defined and they are used to obtain the relationship between the input-output operators of the boundary-value systems and their adjoints.

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.

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