V.R. Barseghyan. Problems of boundary control and optimal control of string vibrations with multipoint intermediate conditions on the state functions ... P. 38-52

For the vibrating string equation with given initial and final conditions, the problems of boundary control and optimal control with given various multipoint intermediate conditions on the values of the deflection function and on the velocities of points of the string are considered. The control is performed both by displacement of one end with the other end fixed and by displacement at the two ends. The quality criterion is given for the whole time interval. Using the method of separation of variables, the problem is reduced to the problem of control and optimal control of ordinary differential equations with given initial, final, and unseparated multipoint intermediate conditions. For all problems according to a single scheme using methods of control theory of finite-dimensional systems with multipoint intermediate conditions, a constructive approach is proposed for finding functions of boundary control and optimal control of string vibrations that ensure the fulfillment of multipoint intermediate conditions.

Keywords: string vibrations, boundary control, vibration control, optimal control of vibrations, multipoint intermediate conditions

Received July 7, 2022

Revised July 7, 2022

Accepted July 11, 2022

Vanya R. Barseghyan, Dr. Phys.-Math. Sci., Prof., Institute of Mechanics of National Academy of Sciences of RA; Yerevan State University, Yerevan, Armenia, e-mail: barseghyan@sci.am


1.   Butkovskii A.G. Metody upravleniya sistemami s raspredelennymi parametrami [Control methods for systems with distributed parameters]. Moscow: Nauka Publ., 1975, 568 p.

2.   Znamenskaya L.N. Upravlenie uprugimi kolebaniyami [Control of elastic vibrations]. Moscow: Fizmatlit Publ., 2004, 176 p. ISBN: 5-9221-0473-X .

3.   Il’in V.A., Moiseev E.I. Optimization of boundary controls of string vibrations. Russ. Math. Surv., 2005, vol. 60, no. 6, pp. 1093–1119. doi: 10.1070/RM2005v060n06ABEH004283 

4.   Moiseev E.I., Kholomeyeva A.A., Frolov A.A. Boundary displacement control for the oscillation process with boundary conditions of damping type for a time less than critical. Proccedings of the Intern. Conference on Mathematical Modelling in Applied Sciences ICMMAS–17. St. Petersburg Polytechnical University, July, 24–28, 2017. Results of science and technology. Series Modern Mathematics and Its Applications, Thematic Overview, no. 160. Moscow: VINITI Publ., 2019, pp. 74–84 (in Russian).

5.   Abdukarimov M.F. On optimal boundary control of displacements in the process of forced vibrations on both ends of a string. Dokl. Akad. Nauk Republic of Tadzhikistan, 2013, vol. 56, no. 8, pp. 612–618 (in Russian).

6.   Kopets M.M. The problem of optimal control of the string vibration process. In: The theory of optimal solutions. Kiev: V. M. Glushkov Institute of Cybernetics NAS of Ukraine Publ., 2014, pp. 32–38 (in Russian).

7.   Zuazua E. Controllability of partial differential equations. Madrid: Universidad Autonoma, 2002, 311 p.

8.   Andreev A.A., Leksina S.V. The boundary control problem for the system of wave equations. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 2008, vol. 1(16), pp. 5–10. doi: 10.14498/vsgtu565 . (in Russian)

9.   Barseghyan V.R. Optimal control of string vibrations with nonseparate state function conditions at given intermediate instants. Autom. Remote Control, 2020, vol. 81, no. 2, pp. 226–235. doi: 10.31857/S0005231020020038 

10.   Barsegyan V.R. The problem of optimal control of string vibrations. Int. Appl. Mech., 2020, vol. 56, no. 4, pp. 471–480. doi: 10.1007/s10778-020-01030-w 

11.   Barseghyan V.R., Solodusha S.V. On one boundary control problem of string vibrations with given velocity of points at an intermediate moment of time. J. Phys.: Conf. Ser., 2021, vol. 1847, art. no. 012016. doi: 10.1088/1742-6596/1847/1/012016 

12.   Barseghyan V., Solodusha S. Control of string vibrations by displacement of one end with the other end fixed, given the deflection form at an intermediate moment of time. Axioms, 2022, vol. 11, no. 4, art. no. 157. doi: 10.3390/axioms11040157 

13.   Barseghyan V.R. Control problem of string vibrations with inseparable multipoint conditions at intermediate points in time. Mech. Solids, 2019, vol. 54, no. 8, pp. 1216–1226. doi: 10.3103/S0025654419080120 

14.   Barseghyan V., Solodusha S. Optimal boundary control of string vibrations with given shape of deflection at a certain moment of time. In: Mathematical Optimization Theory and Operations Research. Lecture Notes in Computer Science, vol. 12755. Cham: Springer, 2021, pp. 299–313. doi: 10.1007/978-3-030-77876-7_20 

15.   Korzyuk V.I., Kozlovskaya I.S. Two-point boundary problem for string oscillation equation with given velocity in arbitrary point of time. I. Tr. Inst. Mat., 2010, vol. 18, no. 2, pp. 22–35 (in Russian).

16.   Korzyuk V.I., Kozlovskaya I.S. Two-point boundary problem for string oscillation equation with given velocity in arbitrary point of time. II. Tr. Inst. Mat., 2011, vol. 19, no. 1, pp. 62–70 (in Russian).

17.   Barseghyan V.R. Upravlenie sostavnykh dinamicheskikh sistem i sistem s mnogotochechnymi promezhutochnymi usloviyami [Control of composite dynamic systems and systems with multipoint intermediate conditions]. Moscow: Nauka Publ., 2016, 230 p. ISBN: 978-5-02-039961-7/hbk .

18.   Zubov V.I. Lektsii po teorii upravleniya [Lectures on control theory]. Moscow: Nauka Publ., 1975, 496 p.

19.   Krasovskii N.N. Teoriya upravleniya dvizheniem [Theory of motion control]. Moscow: Nauka Publ., 1968, 476 p.

Cite this article as: V.R. Barseghyan. Problems of boundary control and optimal control of string vibrations with multipoint intermediate conditions on the state functions. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 38–52; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S66–S79.