Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

About History Editorial Page Addresses Scope Editorial Staff Submission Review For Authors Publication Ethics Issues Русская версия

Observation Stability and Convergence for Neural-type Evolutionary Variational Inequalities

Author(s):

Stefan Reitmann

Freiberg University of Mining and Technology,
Institute of Computer Science, Germany
Chair of Virtual Reality and Multimedia
Doctor of Engineering
Technological University Bergakademie Freiberg
Akademiestrasse 6
09599 Freiberg,Germany

stefan.reitmann@informatik.tu-freiberg.de

Bernhard Jung

Freiberg University of Mining and Technology,
Institute of Computer Science,Geremany
Chair of Virtual Reality and Multimedia
Doctor of Science
Professor
Technological University Bergakademie Freiberg
Akademiestrasse 6
09599 Freiberg,
Germany

jung@informatik.tu-freiberg.de

Elena V. Kudryashova

Department of Applied Cybernetics,
St. Petersburg State University
Doctor of Physical and Mathematical Sciences,
Leading Researcher
198504 St.Peterburg Starii Petergof
Universitetski pr.28

e.kudryashova@spbu.ru

Volker Reitmann

Department of Applied Cybernetics,
St. Petersburg State University
Doctor of Physical and Mathematical Sciences,
Professor
198504 St.Peterburg Starii Petergof
Universitetski pr.28

vreitmann@aol.com

Abstract:

We derive absolute observation stability and instability results for controlled evolutionary inequalities which are based on frequency-domain characteristics of the linear part of the inequalities. The uncertainty parts of the inequalities (nonlinearities which represent external forces and constitutive laws) are described by certain local and integral quadratic constraints. Other terms in the considered evolutionary inequalities represent contact-type properties of a mechanical system with dry friction. The absolute stability criteria with respect to a class of observation operators (or measurement operators) gives the opportunity to prove the weak convergence of arbitrary solutions of inequalities to their stationary sets. In particular the obtained results can be used for the investigation of continuum type memories in neural networks. In such a way we can introduce a new type of neurons with hysteretic nonlinearities which are described by evolutionary variational inequalities.

Keywords

References:

  1. Banks, H. T., Gilliam, D. S. and V. Shubov, Global solvability for damped abstract nonlinear hyperbolic systems. Differential and Integral Equations. 10, 309 - 332 (1997)
  2. Banks, H. T. and K. Ito, A unified framework for approximation in inverse problems for distributed parameter systems. Control-Theory and Advanced Technology. 4, 73 - 90 (1988)
  3. Br´ ezis, H., Problemes unilateraux. J. Math. Pures Appl. 51, 1 - 168 (1972)
  4. Brusin, V. A., The Lur´ e equations in Hilbert space and its solvability. Prikl. Math. Mekh. 40 5, 947 - 955 (1976) (in Russian)
  5. Chueshov, I. D., Introduction to the Theory of Infinite-Dimensional Dissipative Systems. ACTA. Kharkov, (1999)
  6. Curtain, R. F., Logemann, H. and O. J. Staffans, Stability results of Popov type for infinitedimensional systems with applications to integral control. Mathematics Preprint 01/09, University of Bath, (2001), Proc. London Math. Soc. 86, 3, 779 - 816 (2003)
  7. Datko, R., Extending a theorem of A. M. Liapunov to Hilbert spaces. J. Math. Anal. Appl. 32, 610 - 616 (1970)
  8. Del Rosario, R. C. H. and R. C. Smith, LQR control of thin shell dynamics: Formulation and numerical implementation. J. of. Intelligent Material Systems and Structures. 9, 301 - 320 (1998)
  9. Duvant, G. and J. -L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976)
  10. Flandoli, F., Lasiecka J. and R. Triggiani, Algebraic Riccati equations with nonsmoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems. Annali di Matematica Pura Applicata. 153, 307 - 382 (1988)
  11. Gelig, A. Kh., Leonov, G. A. and V. A. Yakubovich, Stability of Nonlinear Systems with Nonunique Equilibrium States. Nauka, Moscow (1978) (in Russian)
  12. Han, W. and M. Sofonea, Evolutionary variational inequalities arising in viscoelastic contact problems. SIAM J. Numer. Anal. 38, 2, 556 - 579 (2000)
  13. Jaruˇ sek, J. and Ch. Eck, Dynamic contact problem with friction in linear viscoelasticity. C. R. Acad. Sci. Paris 322, 497 - 502 (1996)
  14. Kalinichenko, D. Yu., Skopinov, S. N. and V. Reitmann, Stability and bifurcations on a finite time interval in variational inequalities. Electr. J. Diff. Equ. and Contr. Processes, 4 (2012) (Russian); English transl. J. Diff. Equ., 48 (13), 1721 - 1732 Pleiades Publishing, Ltd. (2012)
  15. Kuttler, K. L. and M. Shillor, Set-valued pseudomonotone maps and degenerate evolution inclusions. Comm. Contemp. Math. 1, 1, 87 - 123 (1999)
  16. Leonov, G. A. and V. Reitmann, Absolute observation stability for evolutionary variational inequalities. Book Series: World Scientific Series on Nonlinear Science Ser. B, 14, 29 - 42, World Scientific Publishing Co Pte Ltd (2010)
  17. Leonov, G. A., Reitmann, V. and V. B. Smirnova, Non-Local Methods for Pendulum- Like Feedback Systems. Teubner-Texte zur Mathematik, B. G. Teubner Verlagsgesellschaft, Stuttgart-Leipzig, (1992)
  18. Likhtarnikov, A. L., Absolute stability criterion for nonlinear operator equations. Izv. Akad. Nauk SSSR, Ser. Mat. 41 5, 1064 - 1083 (1977)
  19. Likhtarnikov, A. L. and V. A. Yakubovich, The frequency theorem for equations of evolutionary type. Siberian Math. J. 17 5, 790 - 803 (1976)
  20. Likhtarnikov, A. L. and V. A. Yakubovich, Abstract criteria for absolute stability of nonlinear systems relative to a linear output and their applications. I, II. Siberian Math. J. 23 4, 103 - 121; 24 5, 129 - 148 (1983)
  21. Likhtarnikov, A. L. and V. A. Yakubovich, Dichotomy and stability of uncertain nonlinear systems in Hilbert spaces. Algebra and Analysis. 9 6, 132 - 155 (1997)
  22. Lions, J. -L., Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, New York, (1971)
  23. Louis, J. and D. Wexler, The Hilbert space regulator problem and operator Riccati equation under stabilizability. Annales de la Societe Scientifique de Bruxelles. 105 4, 137 - 165 (1991)
  24. Neˇ cas, J. and I. Hlavaˇ cek, Mathematical Theory of Elastic and Elastoplastic Bodies, An Introduction, Elsevier, Amsterdam (1981)
  25. Orlov, Y. V. and V. I. Utkin, Sliding mode control in infinite-dimensional systems. Automatica, 23, 753 - 757 (1987)
  26. Pankov, A. A., Bounded and Almost Periodic Solutions of Nonlinear Differential Operator Equations. Naukova dumka, Kiev (1986) (in Russian)
  27. Reitmann, V., Anikushin, M. and S. Romanov, Dimension-like properties and almost periodicity for cocycles generated by variational inequalities with delay. Proc. of the Int. Conf. Equadiff 2019, Leiden, Netherlands (2019). https://www.universiteitleiden.nl/equadiff2019
  28. Reitmann, S., Jung, B., Kudryashova, E. V. and Reitmann, V., Classification of point clouds with neural networks and continuum-type memories. 627, IFIP Advances in Information and Communication Technology series, Springer (2021)
  29. Reitmann, S., Neumann, L. and B. Jung, BLAINDER—A Blender AI Add-On for generation of semantically labeled Depth-Sensing Data. Sensors 2021, 21, 2144. https://doi.org/10.3390/s21062144 (2021)
  30. Rochdi, M., Shillor, M. and M. Sofonea, Quasistatic nonlinear viscoelastic contact with normal compliance and friction. J. Elasticity 51, 105 - 126 (1998)
  31. Shestakov, A. A., The Generalized Direct Lyapunov Method for Systems with distributed parameters. Nauka, Moscow (1990) (in Russian)
  32. Staffans, O. J., J-energy preserving well-posed linear systems. Int. J. Appl. Math. Comp. Sci., 11 1361 - 1378 (2001)
  33. Triebel, H., Interpolation Theorie, Function Spaces, Differential Operators. Amsterdam, North-Holland (1978)
  34. Triggiani, R., On the lack of exact controllability for mild solutions in Banach spaces. J. Math. Anal. and Appl. 50 438 - 446 (1975)
  35. Visintin, A., Differential Models for Hysteresis. Springer-Verlag, Berlin (1994)
  36. Wexler, D., On frequency domain stability for evolution equations in Hilbert spaces via the algebraic Riccati equation. SIAM J. Math. Analysis. 11, 969 - 983 (1980)
  37. Willems, J. L., Stability Theory of Dynamical Systems. Nelson, London (1970)
  38. Yakubovich, V. A., On the abstract theory of absolute stability in nonlinear systems. Vestn. Leningr. Univers,. Ser. Mat., Mekh., Astr., 13, 99 - 118 (1977) (in Russian)

Full text (pdf)