ON THE NUMERICAL SOLUTION TO OPTIMAL CONTROL PROBLEMS WITH NON-LOCAL CONDITIONS

Abstract

Optimal control problems involving non-separated multipoint and integral conditions are investigated. For numerical solution to the problem, we propose to use rst order optimization methods with application of the formulas for the gradient of the functional obtained in the work. To solve the adjoint boundary problems, we propose an approach. This approach makes it possible to reduce solving initial boundary problems to solving supplementary Cauchy problems and a linear algebraic system of equations. Results of numerical experiments are given.

Keywords

• Optimal control,integral conditions,multipoint conditions,transfer method for intermediate conditions.

References

1. Nicoletti, O., (1897), Sulle condizioni iniziali che determinano gli integrali delle equazioni differenziali ordinarie, Atti R. Sci. Torino, 33, pp. 746–748.
2. Tamarkin, Ya. D., (1917), On some general problems of ordinary differential equations theory and on series expansion of arbitrary functions,Petrograd.
3. Vallee-Poussin, Ch. J., (1929), Sur l’quation diffrentielle linaire du second ordre. Dtermination d’une intgrale par deux valeurs assignes. Extension aux quations d’ordre n., J. Math. Pures Appl., 8, pp.125– 144.
4. Aida-zade, K. R., Abdullaev, V. M., (2012) On an approach to designing control of the distributed- parameter processes, Autom. Remote Control, 73 (9), pp. 1443–1455.
5. Abdullayev, V. M., Aida-zade, K. R., (2018) Numerical solution of the problem of determining the number and locations of state observation points in feedback control of a heating process, Comput. Math. Math. Phys., 58 (1), pp.78–89.
6. Bouziani, A., (2002), On the solvability of parabolic and hyperbolic problems with a boundary integral condition, Intern.J. Math. Sci., 31 (4), pp. 202–213.
7. Pulkina, L. S., (2004), Non-local problem with integral conditions for a hypergolic equation, Differen- tial Equations, 40 (7), pp. 887–891.
8. Abdullayev, V. M., (2017), Identification of the functions of response to loading for stationary systems, Cybern. Syst. Analysis, 53 (3), pp. 417–425.

9. Abdullaev, V. M., Aida-zade, K. R., (2006), Numerical solution of optimal control problems for loaded lumped parameter systems, Comput. Math. Math. Phys., 46 (9), pp.1487–1502.
10. Abdullayev, V. M., Aida-zade, K. R., (2017), Optimization of loading places and load response func- tions for stationary systems, Comput. Math. Math. Phys., 57 (4), pp. 634–644.
11. Aschepkov, L. T., (1981), Optimal control of system with intermediate conditions, Journal of Applied Mathematics and Mechanics, 45 (2), pp. 215–222.
12. Vasileva, O. O., Mizukami, K., (2000), Dynamical processes described by boundary problem: necessary optimality conditions and methods of solution, Journal of Computer and System Sciences International (A Journal of Optimization and Control), 1, pp.95–100.
13. Vasilev, O. V., Terleckij, V. A., (1995), Optimal control of a boundary problem, Proceedings of the Steklov Institute of Mathematics, (211), pp. 221–130.
14. Abdullaev, V. M., Aida-zade, K. R., (2014), Numerical method of solution to loaded nonlocal boundary-value problems for ordinary differential equations, Comput. Math. Math. Phys., 54 (7), pp. 1096–1109.
15. Abramov, A. A., (1961), On the transfer of boundary conditions for systems of ordinary linear differ- ential equations (a variant of the dispersive method), Zh. Vychisl. Mat. Mat. Fiz., 3 (1), pp.542–545. [16] Aida-zade, K. R., Abdullaev, V. M., (2013), On the solution of boundary-value problems with non- separated multipoint and integral conditions, Differential Equations, 49 (9), pp. 1114–1125.
16. Moszynski, K., (1964), A method of solving the boundary value problem for a system of linear ordinary differential equation, Algorytmy. Varshava, 11 (3), pp. 25–43.
17. Bondarev, A. N., Laptinskii, V. N., (2011), A multipoint boundary value problem for the Lyapunov equation in the case of strong degeneration of the boundary conditions, Differential Equations, 47 (6), pp. 776–784.
18. Dzhumabaev, D. S., Imanchiev, A. E., (2005), Well-posed solvability of linear multipoint boundary problem, Matematicheskij zhurnal. Almaaty, 5 (1(15)), pp. 30–38.
19. Kiguradze, I. T., (1987), Boundary value problems for systems of ordinary differential equations, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., 30, pp. 3–103.
20. Samoilenko, A. M., Laptinskii, V. N., Kenzhebaev, K. K., (1999), Constructive Methods of Investi- gating Periodic and Multipoint Boundary - Value Problems, Kiev.
21. Vasilyev, F. P., (2002), Optimization Methods, M: Faktorial.
22. Aida-zade, K. R., Abdullayev, V. M., (2016), Solution to a class of inverse problems for system of loaded ordinary differential equations with integral conditions, J. of Inverse and Ill-posed Problems, 24 (5), pp. 543–558.
23. Karchevsky, A.L., (2000), Numerical solution of the one-dimensional inverse problem for the elasticity system, Doklady RAS. 375 (2), pp. 235-238.
24. Karchevsky, A.L., (2010), Reconstruction of pressure velocities and boundaries of thin layers in thinly- stratified layers, J. Inv. Ill-Posed Problems. 18 (4), pp. 371-388.