Input Reconstruction by Feedback Control for the Schlögl and FitzHugh–Nagumo Equations

Maksimov, Vyacheslav; Tröltzsch, Fredi

doi:10.34768/amcs-2020-0001

Abstract

Dynamical reconstruction of unknown time-varying controls from inexact measurements of the state function is investigated for a semilinear parabolic equation with memory. This system includes as particular cases the Schlögl model and the FitzHugh–Nagumo equations. A numerical method is suggested that is based on techniques of feedback control. An error analysis is performed. Numerical examples confirm the theoretical predictions.

References

Avdonin, S. and Bell, J. (2015). Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph, Journal of Inverse Problems and Imaging9(3): 645–659.10.3934/ipi.2015.9.645
Search in Google Scholar Back to article
Banks, H.T. and Kunisch, K. (1989). Estimation Techniques for Distributed Parameter Systems, Birkhäuser, Boston, MA.10.1007/978-1-4612-3700-6
Search in Google Scholar Back to article
Barbu, V. (1990). The inverse one phase Stefan problem, Differential Integral Equations3(2): 209–218.
Search in Google Scholar Back to article
Breiten, T. and Kunisch, K. (2014). Riccati-based feedback control of the monodomain equations with the FitzHugh–Nagumo model, SIAM Journal of Control and Optimization52(6): 4057–4081.10.1137/140964552
Search in Google Scholar Back to article
Buchholz, R., Engel, H., Kammann, E. and Tröltzsch, F. (2013). On the optimal control of the Schlögl-model, Computational Optimization and Application56(1): 153–185.10.1007/s10589-013-9550-y
Search in Google Scholar Back to article
Casas, E., Ryll, C. and Tröltzsch, F. (2013). Sparse optimal control of the Schlögl and FitzHugh–Nagumo systems, Computational Methods in Applied Mathematics13(4): 415–442.10.1515/cmam-2013-0016
Search in Google Scholar Back to article
Evans, L.C. (1998). Partial Differential Equations, American Mathematical Society, Providence, RI.
Search in Google Scholar Back to article
Gugat, M. and Tröltzsch, F. (2015). Boundary feedback stabilization of the Schlögl system, Automatica51(C): 192–199.10.1016/j.automatica.2014.10.106
Search in Google Scholar Back to article
Jackson, D.E. (1990). Existence and regularity for the FitzHugh–Nagumo equations with inhomogeneous boundary conditions, Nonlinear Analysis: Theory, Methods & Applications A: Theory and Methods14(3): 201–216.10.1016/0362-546X(90)90029-G
Search in Google Scholar Back to article
Kabanikhin, S.I. (2011). Inverse and Ill-posed Problems, De Gruyter, Berlin.10.1515/9783110224016
Search in Google Scholar Back to article
Krasovskii, N.N. and Subbotin, A.I. (1988). Game-Theoretical Control Problems, Springer Verlag, New York, NY/Berlin.10.1007/978-1-4612-3716-7
Search in Google Scholar Back to article
Kryazhimskii, A.V., Osipov, Yu.S., (1995). Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, Gordon and Breach, London.
Search in Google Scholar Back to article
Lasiecka, I., Triggiani, R., and Yao, P.F. (1999). Inverse/observability estimates for second order hyperbolic equations with variable coefficients, Journal of Mathematical Analysis and Applications235(1): 13–57.10.1006/jmaa.1999.6348
Search in Google Scholar Back to article
Lavrentiev, M.M., Romanov, V.G., and Shishatskii, S.P. (1980). Ill-posed Problems of Mathematical Physics and Analysis, Nauka, Novosibirsk, (in Russian).
Search in Google Scholar Back to article
Le, M.H.L. (2019). Eine Methode der Steuerungs-Rekonstruktion bei semilinearen parabolischen Differentialgleichungen, Master thesis, Technische Universität Berlin, Berlin.
Search in Google Scholar Back to article
Maksimov, V. (2002). Dynamical Inverse Problems of Distributed Systems, VSP, Utrecht/Boston, MA.10.1515/9783110944839
Search in Google Scholar Back to article
Maksimov, V. (2009). On reconstruction of boundary controls in a parabolic equations, Advances in Differential Equations14(11–12): 1193–1211.
Search in Google Scholar Back to article
Maksimov, V. (2016). Game control problem for a phase field equation, Journal Optimization Theory and Applications170(1): 294–307.10.1007/s10957-015-0721-0
Search in Google Scholar Back to article
Maksimov, V. (2017). Some problems of guaranteed control of the Schlögl and FitzHugh–Nagumo systems, Evolution Equations and Control Theory6(4): 559–586.10.3934/eect.2017028
Search in Google Scholar Back to article
Maksimov, V. and Mordukhovich, B.S. (2017). Feedback design of differential equations of reconstruction for second-order distributed systems, International Journal of Applied Mathematics and Computer Science27(3): 467–475, DOI: 10.1515/amcs-2017-0032.10.1515/amcs-2017-0032
Search in Google Scholar Back to article
Maksimov, V. and Pandolfi, L. (2002). The problem of dynamical reconstruction of Dirichlet boundary control in semilinear hyperbolic equations, Journal of Inverse and Ill-Posed Problems8(4): 399–411.10.1515/jiip.2000.8.4.399
Search in Google Scholar Back to article
Mordukhovich, B.S. (2008). Optimization and feedback design of state-constrained parabolic systems, Pacific Journal of Optimization4(3): 549–570
Search in Google Scholar Back to article
Ryll, C., Löber, J., Martens, S., Engel, H., and Tröltzsch, F. (2016). Analytical, optimal, and sparse optimal control of traveling wave solutions to reaction-diffusion systems, in F. Schöll et al. (Eds.), Control and Self-Organizing Nonlinear Systems, Springer, Berlin, pp. 189–210.10.1007/978-3-319-28028-8_10
Search in Google Scholar Back to article
Samarskii, A.A. (1971). Introduction to the Theory of Difference Schemes, Nauka, Moscow, (in Russian).
Search in Google Scholar Back to article
Shitao, L. and Triggiani, R. (2013). Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace, Discrete and Continuous Dynamical Systems33(11&12): 5217–5252.10.3934/dcds.2013.33.5217
Search in Google Scholar Back to article
Tikhonov, A.N. and Arsenin, V.Y. (1977). Solutions of Ill-posed Problems, V.H. Winston & Sons, Washington, DC.
Search in Google Scholar Back to article

Input Reconstruction by Feedback Control for the Schlögl and FitzHugh–Nagumo Equations

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