Abstract
For the control systems whose dynamics obeys a nonlinear regular integral Volterra equation with additional constraints in the form of equalities, the necessary optimality conditions were established on the basis of the abstract Yakubovich-Matveev theory of optimal control and, in particular, the abstract principle of maximum. Consideration was given to two kinds of the nonlinear controllable singular integral equations with unrestricted multipliers under the integral—with the power kernel of the Cauchy kernel type and with the logarithmic kernel. Attention was paid mostly to the nonlinear controlled dynamic systems obeying an integro-differential Volterra equation of the first order. As before, the study relied on the abstract theory of optimal control. The necessary optimality conditions were established by deriving the corresponding conjugate equation, transversality conditions, and principle of maximum.
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References
Heisenberg, W., Physics and philosophy, New York: Harper, 1958. Translated under the title Fizika i filosofiya, Moscow: Inostrannaya Literatura, 1963.
Volterra, V., Theory of Functionals and of Integral and Integro-differential Equations, London: Blackie and Son, 1930. Translated under the title Teoriya funktsionalov, integral’nykh i integro-differentsial’nykh uravnenii, Moscow: Nauka, 1982.
Mikhlin, S.G., Integral’nye uravneniya i ikh prilozheniya k nekotorym problemam mekhaniki, matematicheskoi fiziki i tekhniki (Integral Equations and Their Application to Some Problems of Mechanics, Mathematical Physics, and Engineering), Moscow: Gostekhizdat, 1949.
Muskhelishvili, N.I., Singulyarnye integral’nye uravneniya (Singular Integral Equations), Moscow: Fizmatgiz, 1962.
Krasnov, M.L., Integral’nye uravneniya (Integral Equations), Moscow: Nauka, 1975.
Mikhlin, S.G., Morozov, N.F., and Paukshto, M.V., Integral’nye uravneniya v teorii uprugosti (Integral Equations in the Elasticity Theory), St. Petersburg: S.-Peterburg. Univ., 1994.
Vinokurov, V.R., Optimal Control of Processes Obeying Integral Equations. I–III, Izv. Vuzov, Mat., 1967, no. 7, pp. 21–33; no. 8, pp. 16–23, no. 9, pp. 16–25.
Abuladze, A.A., On Necessary Optimality Conditions for Systems Obeying Integral Equations, Soobshch. Akad. Nauk GSSR, 1985, vol. 119, no. 1, pp. 49–52.
Warga, J., Optimal Control of Differential and Functional Equations, New York: Academic, 1972. Translated under the title Optimal’noe upravlenie differentsial’nymi i funktsional’nymi uravneniyami, Moscow: Nauka, 1977.
Gabasov, R., Necessary Conditions for a System of Integral Equations, Diff. Uravn., 1969, vol. 5, no. 5, pp. 952–953.
Matveev, A.S. and Yakubovich, V.A., Optimal’nye sistemy upravleniya: obyknovennye differentsial’nye uravneniya. Spetsial’nye zadachi (Optimal Control Systems. Ordinary Differential Equations. Special Problems), St. Petersburg: S.-Peterburg. Univ., 2003.
Matveev, A.S., Problems of Optimal Control with General Delays and Phase Constraints, Izv. Akad. Nauk SSSR, Mat., 1988, vol. 52, no. 6, pp. 1200–1229.
Bittner, L., On Optimal Control of Processes, Governed by Abstract Functional, Integral and Hyperbolic Differential Equation, Math. Oper. Statist., 1975, no. 6, pp. 107–134.
Butkovskii, A.G., Teoriya optimal’nogo upravleniya sistemami s raspredelennymi parametrami (Theory of Optimal Control of the Distributed-parameter Systems), Moscow: Nauka, 1965.
Tertychnyi-Dauri, V.Yu., Adaptivnaya mekhanika (Adaptive Mechanics), Moscow: Nauka, 1998.
Tertychnyi-Dauri, V.Yu., Optimal Stabilization in Problems of Adaptive Nuclear Kinetics, Diff. Uravn., 2006, vol. 42, no. 3, pp. 374–384.
Bukina, A.V., Identification of the Speciation Model by the Methods of the Optimal Control Theory, J. Siber. Federal Univ., Math. Phys., 2008, vol. 1, no. 3, pp. 231–235.
Dzhdeed, M., Methods and Algorithms of Optimal Control of the Dynamic Systems Obeying Integrodifferential Equations, Cand. Sc. (Phys. Math.) Dissertation, Tver: Tver. Gos. Univ., 2004.
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F., Matematicheskaya teoriya optimal’nykh protsessov (Mathematical Theory of Optimal Processes), Moscow: Nauka, 1983.
Boltyanskii, V.G., Matematicheskie metody optimal’nogo upravleniya (Mathematical Methods of Optimal Control), Moscow: Nauka, 1969.
Dwhight, H.B., Tables of Integrals and Other Mathematical Data, New York: Macmillan, 1961. Translated under the title Tablitsy integralov i drugie matematicheskie formuly, Moscow: Nauka, 1977.
Volterra, V., Thèorie mathèmatique de la lutte pour la vie, Gauthier-Villars, 1931. Translated under the title Matematicheskaya teoriya bor’by za sushchestvovanie, Moscow: Nauka, 1976.
Rozovskii, M.P., Application of the Integro-differential Equations to Some Dynamic Problems of Elasticity Theory in the Presence of Memory, Prikl. Mat. Mekh., 1947, vol. 11, no. 3, pp. 329–338.
Tertychnyi-Dauri, V.Yu., Solution of Dynamic Variational Problems under Parametric Uncertainty, Probl. Peredachi Inf., 2005, vol. 41, no. 1, pp. 53–67.
Vladimirov, V.S., On the Integro-differential Equation of Particle Transfer, Proc. Akad. Nauk SSSR, Mat., 1957, vol. 21, no. 5, pp. 681–710.
Vladimirov, V.S., Mathematical Problems of the One-speed Theory of Particle Transfer, Proc. Steklov Mat. Inst., 1961, vol. 61, pp. 1–159.
Burton, T.A., An Integro-differential Equation, Proc. Am. Math. Soc., 1980, vol. 79, no. 3, pp. 393–399.
Matematicheskaya entsiklopediya (Mathematical Encyclopedia), vol. 1, Moscow: Sovetskaya Entsiklopediya, 1977.
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Original Russian Text © V.Yu. Tertychnyi-Dauri, 2009, published in Avtomatika i Telemekhanika, 2009, No. 10, pp. 45–74.
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Tertychnyi-Dauri, V.Y. Integral and integro-differential control plants: Optimality conditions. Autom Remote Control 70, 1635–1661 (2009). https://doi.org/10.1134/S0005117909100051
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DOI: https://doi.org/10.1134/S0005117909100051