V.R. Barseghyan. A control problem for string vibrations with nonseparated conditions on the velocities of deflection points at intermediate times ... P. 24-33

We consider the problem of control of string vibrations with given nonseparated values of the derivative of the deflection function at intermediate times. By the method of separation of variables, the problem is reduced to a control problem with countably many ordinary differential equations with given initial, terminal, and nonseparated multipoint intermediate conditions. We solve this problem using the methods of the theory of control of finite-dimensional systems with multipoint intermediate conditions. As an application of the proposed approach, we construct a control action for the problem of control of string vibrations with given nonseparated conditions on the values of the velocities of points of the string at two intermediate times.

Keywords: control of vibrations, string vibrations, intermediate times, nonseparated multipoint conditions

Received July 5, 2019

Revised July 18, 2019

Accepted July 21, 2019

Vanya Rafaelovich Barseghyan, Dr. Phys.-Math. Sci., Prof., Leading Scientific Researcher of Institute of Mechanics of NAS of RA; Prof. of Mathematics and Mechanics Department of Yerevan State University, Yerevan, 0025 Armenia; Yerevan, 0019 Armenia, e-mail: barseghyan@sci.am

REFERENCES

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

2.   Sirazetdinov T.K. Optimizatsiya sistem s raspredelennymi parametrami [Optimization of systems with distributed parameters]. Moscow: Nauka Publ., 1977, 480 p.

3.   Znamenskaya L.N. Upravlenie uprugimi kolebaniyami [Control of elastic vibrations]. Moscow: Fizmatlit Publ., 2004, 176 p.

4.   Aschepkov L.T. Optimal control of the system with intermediate conditions. Prikl. Mat. Mekh., 1981, vol. 45, no. 2, pp. 215–222 (in Russian).

5.   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 .

6.   Barseghyan V.R., Barseghyan T.V. On an approach to the problems of control of dynamic system with nonseparated multipoint intermediate conditions. Automation and Remote Control, 2015, vol. 76, no. 4, pp. 549–559. doi: 10.1134/S0005117915040013 

7.   Barseghyan V.R., Saakyan, M.A. The optimal control of wire vibration in the states of the given intermediate periods of time. Proc. of NAS RA: Mechanics, 2008, vol. 61, no. 2, pp. 52–60 (in Russian).

8.   Barseghyan V.R. Optimal control of a membrane vibration with fixed intermediate states. Proc. of Yerevan State University, 1998, vol. 188, no. 1, pp. 24–29 (in Russian).

9.   Barseghyan V.R. On the problem of boundary control of string oscillations with given states at intermediate moments of time. Proc. XIth All-Russian Congress on Basic Problems of Theoretical and Applied Mechanics (Kazan, August 20–24), 2015, vol. 1, pp. 354–356 (in Russian).

10.   Barseghyan V.R. On one problem of optimal boundaery control of string vibrations with restrictions in the intermediate moment of time. Proc. 11th Internat. Chetaev Conf. “Analytical mechanics, stability and control”( Kazan, June 14–18), 2017, vol. 3, part 1, pp. 119–125 (in Russian).

11.   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. NAS of Belarus, 2010, vol. 18, no. 2, pp. 22–35 (in Russian).

12.   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. NAS of Belarus, 2011, vol. 19, no. 1, pp. 62–70 (in Russian).

13.   Makarov A.A., Levkin D.A. Multipoint boundary value problem for pseudo-differential equations in multilayer. Vistnyk of V.N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 2014, vol. 69, no. 1120, pp. 64–74 (in Ukrainian).

14.   Assanova A.T., Imanchiev A.E On the solvability of a nonlocal boundary value problem for a loaded hyperbolic equations with multi-point conditions. Bulletin of the Karaganda University. Ser.: Mathematics, 2016, no. 1 (81), pp. 15–25 (in Russian).

15.   Bakirova E.A., Kadirbayeva Zh.M. On a solvability of linear multipoint boundary value problem for the loaded differential equations. Izvestiya NAS RK. Ser. Fiz.-Mat., 2016, vol. 5, no. 309, pp. 168–175 (in Russian).

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

17.   Zubov V.I. Lektsii po teorii upravleniya [Lectures on Control Theory]. Moscow: Nauka Publ., 1975, 496 p.

Cite this article as: V.R. Barseghyan. A control problem for string vibrations with nonseparated conditions on the velocities of deflection points at intermediate times, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 3, pp. 24–33.