Theoretical and computational issues of optimal control for distributed Hopfield neural network equations with diffusion term

SIAM Journal on Scientific Computing

Volumn 29, Issue 2, 2007, Pages 890-911

  Wang, Quan Fang ^a  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Boundary control; Distributed control; Finite element methods; Hopfield neural network equations; Initial control; Numerical approximation

Indexed keywords

BOUNDARY CONDITIONS; DISTRIBUTED PARAMETER CONTROL SYSTEMS; FINITE ELEMENT METHOD; NUMERICAL METHODS;

BOUNDARY CONTROL; HOPFIELD NEURAL NETWORK EQUATIONS; INITIAL CONTROL; NUMERICAL APPROXIMATION;

HOPFIELD NEURAL NETWORKS;

EID: 39549095262     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/050647943     Document Type: Article

References (24)

1. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. N. ARADA AND J. P. RAYMOND, Dirichlet Boundary Control of Semilinear Parabolic Equations, Part 1: Problem With No State Constraints, Report 98-05, UMR-CNRS 5640, Toulouse, France.
3. V. BARBU, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, New York, 1993.
4. E. CASAS, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), pp. 1297-1327.
5. T. CARLEMAN, Sur un probléme d'unicité pour les systémes déquations aux dérivées partielles á deux variables indépendantes, Ark. Mat. Astr. Fys., 26, 17 (1939), pp. 1-9.
6. R. DAUTRAY AND J, L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Evolution Problems I, Springer-Verlag, Berlin, Heidelberg, New York, 1992.
7. P. VAN DEN DRIESSCHE AND X. ZOU, Global attractivity in delayed Hopfield neural network models, SIAM J. Appl. Math., 58 (1998), pp. 1878-1890.
8. R. FITZ-HUGH, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys, 17 (1955), p. 257.
9. M. FORTI AND A. TESI, New conditions for global stability of neural networks with applications to linear and quadratic programming problems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), pp. 354-366.
10. A. L. HODGKIN AND A. F. HUXLEY, A qualitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), pp. 500-544.
11. J. J. HOPFIELD, Neural network and physics system with collective computational abilities, Proc. Natl. Acad. Sci. USA, 79 (1982), pp. 2554-2558.
12. J. J. HOPFIELD, Neurons with graded response have collective properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA, 81 (1984), pp. 3088-3092.
13. J. J. HOPFIELD AND D. W. TANK, Computing with neural circuits: A Model, Science, 233 (1986), pp. 625-632.
14. L. S. LASDON, S. K. MITTER, AND A. D. WARREN, The conjugate gradient method for optimal control problems, IEEE Trans. Automat. Control, 12 (1967), pp. 132-138.
15. J. L. LIONS, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1971.
16. J. NAGUMO, S. ARIMOTO, AND S. YOSHIZAWA, An active pulse transmission line simulating nerve axon, Proc. Inst. Radio Engineers, 50 (1962), pp. 2061-2070.
17. S. NAKAGIRI AND Q. F. WANG, Optimal control problems for distributed Hopfield-type neural networks, Nonlinear Funct. Anal. Appl., 7 (2002) pp. 167-186.
18. R. TEMAM, Navier Stokes Equation, North-Holland, Amsterdam, 1984.
19. Q. F. WANG, Numerical analysis for optimal control problems of Hopfield-type neural networks equations, Poster, International Symposium on Intervention and Adaptation in Complex Systems, Beijing, China, 2002.
20. Q. F. WANG, D. FENG, AND D. CHENG, Parameter identification problems for Hopfield-type neural network equations, Appl. Math. Comput., 152 (2004), pp. 535-550.
21. Q. F. WANG, Numerical approximation of optimal control for distributed diffusion Hopfield neural network, in Proceedings of the 2004 Annual Conference of the Japan Society for Industrial and Applied Mathematics, 2004, pp. 178-179.
22. Q. F. WANG AND S. NAKAGIRI, Numerical solution of distributed diffusion Hopfield neural network equations, Appl. Math. Comput., 174 (2006), pp. 1343-1362.
23. P. DE. WILDE, Neural Network Models, 2nd ed., Springer-Verlag, London, 1997.
24. R. WINTHER, Initial value methods for parabolic control problems, Math. Comp., 34 (1980), pp. 115-125.