The existence of a solution of the controllability problem for linear integral and differential equations with restrictions
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Keywords:
controllability, integral and differential equations, optimal control, minimizing sequencesAbstract
A controlled process described by linear integral and differential equation with boundary conditions in the presence of phase and integral constraints with the limited control resources is considered. Research of controllability of integral and differential equations is a new direction in the theory of integral and differential equations. The controllability problem that arose from the need to solve urgent problems in the field of natural sciences, medicine, economics, technical sciences sets more sophisticated problems for integral and differential equations. In the previous works, the author studied the controllability problems of the processes described by ordinary differential equations with boundary conditions in the presence of phase and integral constraints with the limited control resources. In this paper an attempt is made to extend these results to the integral and differential equations. The necessary and sufficient conditions for the controllability processes for linear integral and differential equations by non-singular transformation and construction of the general solution of a class of integral equations are obtained.
References
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[28] Semenov Yu. M., «O polnoy upravlyaemosti lineynyih neavtonomnyih sistem [On the complete controllability of linear nonautonomous systems]», Differentsialnyie uravneniya, vol 48, Issue 9 (2012) : 1265-1277.
[2] Aisagaliev S. A., Kabidoldanova A. A. «Optimalnoe upravlenie lineynyimi sistemami s lineynyim kriteriem kachestva i ogranicheniyami [On the optimal control of linear systems with linear performance criterion and constraints]», Differentsialnyie uravneniya, vol. 48, Issue 6 (2012) : 826–836.
[3] Aisagaliev S. A. Konstruktivnaya teoriya kraevyih zadach obyiknovennyih differentsialnyih uravneniy [Constructive theory of boundary value problems for ordinary differential equations](Almaty: Kazakh universiteti, 2015), 207.
[4] Aisagaliev S. A., «Obschee reshenie odnogo klassa integralnyih uravneniy [The general solution of a class of integral equations]», Matematicheskiy zhurnal, vol. 5, No 4 (2005) : 17–34.
[5] Aisagaliev S. A. Teoriya upravlyaemosti dinamicheskih sistem [Controllability theory of dynamical systems] (Kazakh universiteti, 2014), 158.
[6] Aisagaliev S. A., «Upravlyaemost nekotoroy sistemyi differentsialnyih uravneniy» [Controllability of a differential-equation system], Differentsialnyie uravneniya, vol. 27, Issue 9 (1991) : 1037–1045.
[7] Aisagaliev S. A., Belogurov A. P., «Upravlyaemost i byistrodeystvie protsessa, opisyivaemogo parabolicheskim uravneniem s ogranichennyim upravleniem [Controllability and speed of the process described by a parabolic equation with bounded control]», Sibirskiy matematicheskiy zhurnal, vol. 53, Issue 1 (2012) : 13–28.
[8] Aisagaliev S. A., Kalimoldaev M. N., «Konstruktivnyiy metod resheniya kraevoy zadachi dlya obyiknovennyih differentsialnyih uravneniy [Constructive method for solving a boundary value problem for ordinary differential equations]», Differentsialnyie uravneniya, vol. 51, Issue 2 (2015) : 149–162.
[9] Ananevskiy I. M., Anahin N. V., Ovseevich A. I., «Sintez ogranichennogo upravleniya lineynyimi dinamicheskimi sistemami s pomoschyu obschey funktsii Lyapunova [Synthesis of limited control for linear dynamical systems with the using of the general Lyapunov function]», Dokladyi RAN, vol. 434, No 3 (2010) : 319-323.
[10] Bellman R. Matematicheskie metodyi v meditsine [Mathematical Methods in Medicine] (M.: Mir, 1987), 282.
[11] Byikov Ya.V., O nekotoryih zadachah teorii integro-differentsialnyih uravneniy [On some problems of the theory of integro- differential equations] (Frunze: Ilim, 1957), 312.
[12] Varga Dzh. Optimalnoe upravlenie differentsialnyimi i funktsionalnyimi uravneniyami [Optimal control of differential and functional equations] (M.: Nauka, 1977), 585.
[13] Volterra V. Matematicheskaya teoriya borbyi za suschestvovaniya [The mathematical theory of the struggle for existence] (M.: Nauka, 1976), 320.
[14] Gabasov R., Kirillov F. M. Kachestvennaya teoriya optimalnyih protsessov [Qualitative theory of optimal processes] (M.: Nauka, 1971), 421.
[15] Glushkov V.M., Ivanov V.V., Yanenko V.M. Modelirovanie razvivayuschihsya sistem [Modeling of developing systems] (M.: Nauka, 1983), 285.
[16] Emelyanov S. V., Krischenko A. P., «Stabilizatsiya neregulyarnyih sistem [Stabilization of irregular systems]», Differentsialnyie uravneniya, vol. 48, Issue 11 (2012) : 1515-1524.
[17] Zubov V. I. Lektsii po teorii upravleniya [Lectures on the control theory] (M.: Nauka, 1975), 502.
[18] Imanaliev M.I. Metodyi resheniya nelineynyih obratnyih zadach i ih prilozheniya [Methods for solving nonlinear inverse problems and their applications] (Frunze: Ilim, 1977), 302.
[19] Imanaliev M.I. Obobschennyie resheniya integralnyih uravneniy pervogo roda [Generalized solutions of integral equations of the first kind] (Frunze: Ilim, 1981), 265.
[20] Kalman R. E., «Ob obschey teorii sistem upravleniya [On the General Theory of Control Systems]». Trudyi Kongressa IFAK. Vol. 2. – M.: AN SSSR, 1961.
[21] Korovin S. K., Kapalin I. V., Fomichev V. V., «Minimalnyie stabilizatoryi dlya lineynyih dinamicheskih sistem [Minimum stabilizers for linear dynamic systems]». Dokladyi RAN, vol. 441, No 5. 2011.
[22] Krasnov M.L. Integralnyie uravneniya [Integral equations] (M.: Nauka, 1975), 304.
[23] Krasovskiy N. N. Teoriya upravleniya dvizheniem [Theory of motion control] (M.: Nauka, 1968), 475.
[24] Li E. V., Markus L. Osnovyi teorii optimalnogo upravleniya [Fundamentals of Optimal Control Theory] (M.: Nauka, 1972), 575.
[25] Nikolis G., Prigozhin I. Samoorganizatsiya v neravnovesnyih sistemah [Self-organization in nonequilibrium systems] (M.: Mir, 1979), 245.
[26] Romanovskiy Yu.M., Stepanova N.V., Chernavskiy D.S. Matematicheskaya biofizika [Mathematical Biophysics] (M.: Nauka, 1984), 290.
[27] Rubin A.B. Termodinamika biologicheskih protsessov [Thermodynamics of biological processes] (Izd-vo MGU, 1984), 352.
[28] Semenov Yu. M., «O polnoy upravlyaemosti lineynyih neavtonomnyih sistem [On the complete controllability of linear nonautonomous systems]», Differentsialnyie uravneniya, vol 48, Issue 9 (2012) : 1265-1277.
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Aisagaliev S. А., & Aisagalieva, S. S. (2018). The existence of a solution of the controllability problem for linear integral and differential equations with restrictions. Journal of Mathematics, Mechanics and Computer Science, 93(1), 3–20. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/430
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