On the existence of an optimal solution of the Mayer problem governed by 2D continuous counterpart of the Fornasini-Marchesini model Open Access Article

First Online: 23 October 2012 Received: 30 May 2012 Revised: 04 October 2012 Accepted: 10 October 2012 DOI :
10.1007/s11045-012-0207-2

Cite this article as: Bors, D. & Majewski, M. Multidim Syst Sign Process (2013) 24: 657. doi:10.1007/s11045-012-0207-2 Abstract In the paper the optimization problem described by some nonlinear hyperbolic equation being continuous counterpart of the Fornasini-Marchesini model is considered. A theorem on the existence of at least one solution to this hyperbolic PDE is proved and some properties of the set of all solutions are established. The existence of a solution to an optimization problem under appropriate assumptions is the main result of this paper. Some application of the obtained results to the process of gas filtration is also presented.

Keywords Mayer problem Continuous counterpart of the Fornasini-Marchesini model Existence of optimal solutions Download to read the full article text

References Cesari L. (1983) Optimization—theory and application. Springer, Berlin

Google Scholar Cheng H., Saito T., Matsushita S., Xu L. (2011) Realization of multidimensional systems in Fornasini-Marchesini state-space model. Multidimensional Systems and Signal Processing 22(4): 319–333

MathSciNet MATH CrossRef Google Scholar Fornasini, E.& Marchesini, G. (1976). State space realization of two dimensional filters. IEEE Transactions on Automatic Control , AC-21(4), 484–491.

Fornasini, E. & Marchesini, G. (1978/1979). Doubly-indexed dynamical systems: State-space models and structural properties. Mathematical Systems Theory , 12, 59–72.

Idczak D., Kibalczyc K., Walczak S. (1994) On an optimization problem with cost of rapid variation of control. Journal of the Australian Mathematical Society, Series B 36: 117–131

MathSciNet MATH CrossRef Google Scholar Idczak D., Walczak S. (2000) On the existence of a solution for some distributed optimal control hyperbolic system. International Journal of Mathematics and Mathematical Sciences 23(5): 297–311

MathSciNet MATH CrossRef Google Scholar Idczak D., Walczak S. (1994) On Helly’s theorem for functions of several variables and its applications to variational problems. Optimization 30: 331–343

MathSciNet MATH CrossRef Google Scholar Idczak D. (2008) Maximum principle for optimal control of two-directionally continuous linear repetitive processes. Multidimensional Systems and Signal Processing 19(3–4): 411–423

MathSciNet MATH CrossRef Google Scholar Kaczorek T. (1985) Two-dimensional linear systems. Springer, Berlin, Germany

MATH Google Scholar Kisielewicz, M. (1991). Differential inclusions and optimal control , volume 44. Kluwer, Dordrecht, Boston, London, Higher School of Engineering, Zielona Góra, Poland.

Klamka J. (1991) Controllability of dynamical systems. Kluwer, Dordrecht, Holland

MATH Google Scholar Łojasiewicz S. (1988) An Introduction to the theory of real functions. Wiley, Chichester

MATH Google Scholar Majewski M. (2006) On the existence of optimal solutions to an optimal control problem. Journal of Optimization Theory and Applications 128(3): 635–651

MathSciNet MATH CrossRef Google Scholar Rehbock V., Wang S., Teo K. L. (1998) Computing optimal control with hyperbolic partial differential equation. Journal of the Australian Mathematical Society, Series B 40(2): 266–287

MathSciNet MATH CrossRef Google Scholar Tikhonov A. N., Samarski A. A. (1990) Equations of mathematical physics. Dover Publications, Inc., New York

Google Scholar Walczak S. (1987) Absolutely continuous functions of several variables and their application to differential equations. Bulletin of the Polish Academy of Sciences 35(11–12): 733–744

MathSciNet MATH Google Scholar Yang R., Zhang C., Xie L. (2007) Linear quadratic Gaussian control of 2-dimensional systems. Multidimensional Systems and Signal Processing 18(4): 273–295

MathSciNet MATH CrossRef Google Scholar © The Author(s) 2012

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations 1. Faculty of Mathematics and Computer Science University of Lodz Lodz Poland