Skip to main content
SpringerLink
Log in

Existence of optimal solutions of two-directionally continuous linear repetitive processes

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

In the paper, an existence theorem for a Lagrange optimal control problem connected with a two-directionally continuous variant of a linear autonomous repetitive process is derived. As a corollary, existence of an optimal solution in the case of cost functional depending on a fixed “end-function” is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Berkovitz L. D. (1974) Optimal control theory. Springer, New York

    MATH  Google Scholar 

  • Bisiacco M., Fornasini E. (1990) Optimal control of two-dimensional systems. SIAM Journal of Control and Optmization 28(3): 582–601

    Article  MATH  MathSciNet  Google Scholar 

  • Idczak, D. (1990). Optimization of nonlinear systems described by partial differential equations, Doctoral Thesis, University of Łódź, Łódź.

  • Idczak D. (1995) Optimal control of a continuous Roesser model. Bulletin of the Polish Academy of Sciences. Technical Series 43(2): 227–234

    MATH  Google Scholar 

  • Idczak, D. (2005). On a continuous variant of a linear repetitive process. In Proceedings of the fourth international workshop on multidimensional systems—NDS 2005 (pp. 247–252). Wuppertal (Germany).

  • Idczak, D. (2006). The bang-bang principle for a continuous repetitive process with distributed and boundary controls. In Proceedings of the 17th international symposium on mathematical theory of networks and systems, Kyoto, Japan.

  • Idczak D. (2008) Maximum principle for optimal control of two-directionally continuous linear repetitive processes. Multidimensional Systems and Signal Processing 19: 411–423

    Article  MATH  MathSciNet  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 Science 23(5): 297–311

    Article  MATH  MathSciNet  Google Scholar 

  • Kaczorek T., Klamka J. (1986) Minimum energy control of 2-D linear systems with variable coefficients. International Journal of Control 44(3): 645–650

    Article  MATH  MathSciNet  Google Scholar 

  • Li Ch., Fadali M. S. (1991) Optimal control of 2-D systems. IEEE Transaction on Automatic Control 36(2): 223–228

    Article  MATH  MathSciNet  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

    Article  MATH  MathSciNet  Google Scholar 

  • Ntogramatzidis L., Cantoni M. (2009) LQ optimal control for 2D Roesser models of finite extent. Systems and Control Letters 58: 482–490

    Article  MATH  MathSciNet  Google Scholar 

  • Olech Cz. (1977) A charakterization of L 1-weak lower semicontinuity of integral functionals. Bulletin of the Polish Academy of Science Series Mathematics Astronomy and Physics 25(2): 135–142

    MATH  MathSciNet  Google Scholar 

  • Owens D. H., Amann N., Rogers E., French M. (2000) Analysis of linear iterative learning control schemes—a 2D systems/repetitive processes approach. Multidimensional Systems and Signal Processing 11(1/2): 125–177

    Article  MATH  MathSciNet  Google Scholar 

  • Roberts P. D. (2000) Numerical investigations of a stability theorem arising from 2-dimensional analysis of an iterative optimal control algorithm. Multidimensional Systems and Signal Processing 11(1/2): 109–124

    Article  MATH  MathSciNet  Google Scholar 

  • Rogers, E., Gałkowski, K., Owens, D. H. (2007). Control systems theory and applications for linear repetitive processes, Lecture Notes in Control and Inform. Sci., vol. 349. Berlin, Heidelberg, New York: Springer.

  • Rogers, E., Owens, D. H. (1982). Stability analysis for linear repetitive processes, Lecture Notes in Control and Inform. Sci., vol. 175. Berlin, Heidelberg, New York: Springer.

  • Walczak S. (1987) Absolutely continuous functions of several variables and their applications to differential equations. Bulletin of the Polish Academy of Science Series Mathematics 35(11–12): 733–744

    MATH  MathSciNet  Google Scholar 

  • Walczak S. (1988) On the differentiability of absolutely continuous functions of several variables. Remarks on the Rademacher theorem. Bulletin of the Polish Academy of Science Series Mathematics 36(9–10): 513–520

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Lodz, Lodz, Poland

    Dariusz Idczak & Marek Majewski

Authors
  1. Dariusz Idczak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marek Majewski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marek Majewski.

Additional information

This work is a part of the research project N514 027 32/3630 supported by the Ministry of Science and Higher Education (Poland).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Idczak, D., Majewski, M. Existence of optimal solutions of two-directionally continuous linear repetitive processes. Multidim Syst Sign Process 23, 155–162 (2012). https://doi.org/10.1007/s11045-010-0105-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-010-0105-4

Keywords

Search

Navigation