Feedback design of differential equations of reconstruction for second–order distributed parameter systems

Maksimov, Vyacheslav I.; Mordukhovich, Boris S.

doi:10.1515/amcs-2017-0032

Abstract

The paper aims at studying a class of second-order partial differential equations subject to uncertainty involving unknown inputs for which no probabilistic information is available. Developing an approach of feedback control with a model, we derive an efficient reconstruction procedure and thereby design differential equations of reconstruction. A characteristic feature of the obtained equations is that their inputs formed by the feedback control principle constructively approximate unknown inputs of the given second-order distributed parameter system.

References

Banks, H.T. and Kunisch, K. (1980). Estimation Techniques for Distributed Parameter Systems, Birkhäuser, Boston, MA.10.21236/ADA088244
Search in Google Scholar Back to article
Barbashin, E.A. (1970). Introduction to the Theory of Stability, Wolters/Noordhoff, Groningen.
Search in Google Scholar Back to article
Gajewski, H., Groger, K. and Zacharias, K. (1974). Nichtlineare operatorgleichungen und operator differentialgleichungen, Academic Verlag, Berlin.10.1002/mana.19750672207
Search in Google Scholar Back to article
Krasovskii, N.N. and Subbotin, A.I. (1988). Game-Theoretial 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. and Osipov, Yu.S. (1995). Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, Gordon and Breach, London.
Search in Google Scholar Back to article
Kryazhimskii, A.V. and Maksimov, V.I. (2010). Resource-saving infinite-horizon tracking under uncertain input, Applied Mathematics and Computations217(3): 1135–114010.1016/j.amc.2010.01.014
Search in Google Scholar Back to article
Kapustyan, V. and Maksimov, V. (2014). On attaining the prescribed quality of a controlled fourth order system, International Journal of Applied Mathematics and Computer Science24(1): 75–85, DOI: 10.2478/amcs-2014-0006.10.2478/amcs-2014-0006
Open DOI 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
Maksimov, V.I. (1995). Approximation of an inverse problem for variational inequalities, Differential and Integral Equations8(8): 2189–2196.10.57262/die/1369056147
Back to article
Maksimov, V.I. (1996). Some dynamical inverse problems for hyperbolic systems, Control and Cybernetics25(3): 465–481.
Search in Google Scholar Back to article
Maksimov, V.I. and Pandolfi, L. (2001). The reconstruction of unbounded controls in non-linear dynamical systems, Journal of Applied Mathematics and Mechanics65(3): 371–380.10.1016/S0021-8928(01)00042-9
Search in Google Scholar Back to article
Maksimov, V.I. (2002). Dynamics Inverse Problems of Distributed Systems, VSP, Utrecht/Boston, MA.10.1515/9783110944839
Search in Google Scholar Back to article
Maksimov, V.I. and Tröltzsch, F. (2006). Dynamical state and control reconstruction for a phase field model, Dynamics of Continuous, Discrete and Impulsive Systems A: Mathematical Analysis13(3–4): 419–436.
Search in Google Scholar Back to article
Mordukhovich, B.S. and Zhang, K. (1997). Minimax control of parabolic equations with Dirichlet boundary conditions and state constraints, Applied Mathematics and Optimization36(3): 323–360.10.1007/s002459900066
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
Mordukhovich, B.S. (2011). Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions, Applicable Analysis90(6): 1075–110910.1080/00036811003735840
Open DOI Search in Google Scholar Back to article
Schwaller, B., Ensminger, D., Dresp-Langley, B. and Ragot, J. (2013). State estimation for a class of nonlinear systems, International Journal of Applied Mathematics and Computer Science23(2): 383–394, DOI: 10.2478/amcs-2013-0029.10.2478/amcs-2013-0029
Open DOI Search in Google Scholar Back to article
Warga, J. (1972). Optimal Control of Differential and Functional Equations, Academic Press, New York, NY.10.1016/B978-0-12-735150-6.50011-7
Search in Google Scholar Back to article

Feedback design of differential equations of reconstruction for second–order distributed parameter systems

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