Abstract
We suggest to apply the Bubnov–Galerkin method to solving control problems for bilinear systems. We reduce the solution of a control problem to a finite-dimensional system of linear problem of moments. We show a specific example of applying this procedure and give its numerical solution.
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Banichuk, N.V., Optimizatsiya form uprugikh tel (Optimization of the Form of Elastic Bodies), Moscow: Nauka, 1980.
Bendsøe, M.P. and Sigmund, O., Topology Optimization, Berlin: Springer, 2003.
Christensen, P. and Klarbring, A., An Introduction to Structural Optimization, Berlin: Springer, 2009.
Haslinger, J. and Neittaanmäki, P., Finite Element Approximation for Optimal Shape, Material and Topology Design, New York: Wiley, 1996, 2nd ed.
Pardalos, P.M. and Yatsenko, V., Optimization and Control of Bilinear Systems, Berlin: Springer, 2008.
Krotov, V.F., Bulatov, A.V., and Baturina, O.V., Optimization of Linear Systems with Controllable Coefficients, Autom. Remote Control, 2011, vol. 72, no. 6, pp. 1199–1212.
Rasina, I.V. and Baturina, O.V., Control Optimization in Bilinear Systems, Autom. Remote Control, 2013, vol. 74, no. 5, pp. 802–810.
Bradley, M.E. and Lenhart, S., Bilinear Optimal Control of a Kirchhoff Plate, Syst. Control Lett., 1994, vol. 22, no. 1, pp. 27–38.
Liang, M., Bilinear Optimal Control for a Wave Equation, Math. Mod. Meth. Appl. Sci., 1999, vol. 9, no. 1, pp. 45–68.
Jilavyan, Sh., Khurshudyan, As.Zh., and Sarkisyan, As., On Adhesive Binding Optimization of Elastic Homogeneous Rod to a Fixed Rigid Base as a Control Problem by Coefficient, Archiv. Control Sci., 2013, vol. 23 (LIX), no. 4, pp. 413–425.
Baudouin, L., Cerpa, E., Crepeau, E., and Mercado, A., On the Determination of the Principal Coefficient from Boundary Measurements in a KdV Equation, J. Inverse Ill–Posed Probl., doi:10.1515/jip- 2013-0015.
Ouzahra, M., Controllability of theWave Equation with Bilinear Controls, Eur. J. Control, 2014, vol. 20, no. 2, pp. 57–63.
Beauchard, K. and Rouchon, P., Bilinear Control of Schrödinger PDEs, in Encyclopedia of Systems and Control, Berlin: Springer, 2014.
Mikhlin, S.G., Variatsionnye metody v matematicheskoi fizike (Variational Methods in Mathematical Physics), Moscow: Nauka, 1970, 2nd ed.
Butkovskii, A.G., Metody upravleniya sistemami s raspredelennymi parametrami (Methods of Control for Systems with Distributed Parameters), Moscow: Nauka, 1975.
Krasovskii, N.N., Teoriya upravleniya dvizheniem (Theory of Motion Control), Moscow: Nauka, 1968.
Krein, M.G. and Nudel’man, A.A., Problema momentov Markova i ekstremal’nye zadachi (The Markov Moments Problem and Extremal Problems), Moscow: Nauka, 1973.
Khurshudyan, As.Zh., On Optimal Boundary Control of Non-Homogeneous String Vibrations under Impulsive Concentrated Perturbations with Delay in Controls, Math. Bull. T. Shevchenko Scientific Soc., 2013, vol. 10, pp. 203–209.
Khurshudyan, As.Zh., On Optimal Boundary and Distributed Control of Partial Integro–Differential Equations, Arch. Control Sci., 2014, vol. 24 (LX), no. 1, pp. 5–25.
Vibration Damping, Control and Design, de Silva, C.W., Ed., Boca Raton: CRC Press, 2007.
Soares, R.M., del Prado, Z., and Gonsalves, P.B., On the Vibration Control of Beams Using a Moving Absorber and Subjected to Moving Loads, Mecanica Comput., 2010, vol. 29, pp. 1829–1840.
Museros, P., Moliner, E., and Martinez–Rodrigo, M.D., Free Vibrations of Simply Supported Beam Bridges under Moving Loads. Maximum Resonance, Cancellation and Resonant Vertical Acceleration, J. Sound Vibrat., 2013, vol. 332, no. 2, pp. 326–345.
Museros, P. and Martinez-Rodrigo, M.D., Vibration Control of Simply Supported Beams under Moving Loads Using Fluid Viscous Dampers, J. Sound Vibrat., 2007, vol. 300, nos. 1–2, pp. 292–315.
Qu, Ji-Ting, and Li, Hong-Nan, Study on Optimal Placement and Reasonable Number of Viscoelastic Dampers by Improved Weight Coefficient Method, Math. Probl. Eng., 2013, vol. 2013, ID 358709.
Fujita, K., Moustafa, A., and Takewaki, I., Optimal Placement of Viscoelastic Dampers and Supporting Members under Variable Excitations, Earthquak. Struct., 2010, vol. 1, no. 1, pp. 43–67.
Shilov, G.E., Matematicheskii analiz. Vtoroi spets. kurs (Calculus. Second Special Course), Moscow: Mosk. Gos. Univ., 1984, 2nd ed.
Betts, J.T., Methods of Optimal Control and Estimation Using Nonlinear Programming, Philadelphia: SIAM, 2010, 2nd ed.
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Original Russian Text © As.Zh. Khurshudyan, 2015, published in Avtomatika i Telemekhanika, 2015, No. 8, pp. 46–55.
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Khurshudyan, A.Z. The Bubnov–Galerkin method in control problems for bilinear systems. Autom Remote Control 76, 1361–1368 (2015). https://doi.org/10.1134/S0005117915080032
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DOI: https://doi.org/10.1134/S0005117915080032