Document Type : Research Paper
Authors
Institute of Applied Mathematics, BSU, Baku, Azerbaijan.
Abstract
In the paper a linear-quadratic optimization problem (LCTOR) with unseparated two-point boundary conditions is considered. To solve this problem is proposed a new sweep algorithm which increases doubles the dimension of the original system. In contrast to the well-known methods, here it refuses to solve linear matrix and nonlinear Riccati equations, since the solution of such multi-point optimization problems encounters serious difficulties in passing through nodal points. The results are illustrated with a specific numerical example.
Keywords
[1] A.A. Abramov, On the transfer of boundary conditions for systems of linear ordinary differential equations (a variant of the sweep method), Journal of Computational Mathematics and Mathematical Physics, 1 (1961), pp. 542-545. (in Russian)
[2] F.A. Aliev, Comments on `Sweep algorithm for solving optimal control problem with multi-point boundary conditions' by M. Mutallimov, R. Zulfuqarova and L. Amirova, Adv. Differ. Equ., 131 (2016).
[3] F.A. Aliev, Methods for Solving Applied Problems of Optimizing Dynamic Systems, Elm, Baku, 1989. (in Russian)
[4] F.A. Aliev, The problem of optimal control of a linear system with unseparated two- point boundary conditions, Differential equations, 2 (1986), pp. 345-347. (in Russian)
[5] F.A. Aliev, Optimization problem with unseparated two-point boundary conditions, News AN USSR, cybernetics, 6(1985), pp. 138-146. (in Russian)
[6] F.A. Aliev, N.A. Aliev, N.A. Safarova, et al., Sweep method for solving the Roesser type equation describing the motion in the pipeline, Appl. Math. Comput., 295 (2017), pp. 16-23.
[7] F.A. Aliev and N.A. Ismailov, Methods for solving optimization problems with point-to-point boundary conditions, Preprint AS Azerb. SSR, Institute of Physics, Baku, 151 (1985). (in Russian)
[8] F.A. Aliev, V.B. Larin, N.I. Velieva, et al., On periodic solution of generalized Sylvester matrix equations, Appl. Comput. Math., 16 (2017), pp. 78-84.
[9] F.A. Aliev, M.M. Mutallimov, N.A. Ismailov, and M.F. Radjabov, Algorithms for the construction of optimal controllers for gas-lift operation, Autom. Remote Control, 73 (2012), pp. 1279-1289.
[10] R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems, Elsevier, New York, 1965.
[11] V.L. Bordyug, V.B. Larin, and A.G. Timoshenko, The Control Problem of Walking Apparatus, Nauk.Dumka, Kiev, 1985. (in Russian)
[12] C. Bota, B. Caruntu, and C. Lazureanu, The least square homotopy perturbation method for boundary value problems, Appl. Comput. Math., 16 (2017), pp. 39-47.
[13] A.E. Bryson and Y.C. Jr. & Ho, Applied Optimal Control: Optimization, Estimation, and Control, Waltham, MA Blaisdell, 1969.
[14] F.R. Gantmacher, The Theory of Matrices, Nauka, Moscow, 1967. (in Russian)
[15] A. Khan and G. Zaman, Asymptotic behavior of an age-structured SIRS endemic model, Appl. Comput. Math., 17:2 (2018), pp. 185-204.
[16] V.B. Larin, The Control of Walking Apparatus, Nauka Dumka, Kiev, 1980. (in Russian)
[17] N.I. Mahmudov, Finite-approximate controllability of evolution equations, Appl. Comput. Math., 16 (2017), pp. 159-167.
[18] F.G. Maksudov and F.A. Aliyev, Optimization of pulse systems with unseparated two-point boundary conditions, Dokl.AS SSSR, 280(1985), pp. 796-798. (in Russian)
[19] K. Moszynski, A method of solving the boundary value problem for a system of linear ordinary differensial equations, Algorytmy, 11 (1964), pp.25-43.
[20] M.M. Mutallimov and F.A. Aliyev, Methods for Solving Optimization Problems in the Operation of Oil Wells, LAP LAMBERT, r 2012. (in Russian)
[21] M.M. Mutallimov, L.I. Amirova, F.A. Aliev, Sh.A. Faradjova, and I.A. Maharramov, Remarks to the paper: sweep algorithm for solving optimal control problem with multi-point boundary conditions, TWMS J. Pure Appl. Math., 9 (2018), pp. 243-246.
[22] M.M. Mutallimov, R.T. Zulfugarova, and L.I. Amirova, Sweep algorithm for solving optimal control problem with multi-point boundary conditions, Adv. Differ. Equ., 233 (2015).
[23] E. Polak, Computational Methods in Optimization. Unified Approach, Academic Press, 1971.
[24] V.E. Shamansky, Methods for numerical solution of boundary value problems at the ECM, Nauk. Dumka, Kiev, Part I, 1963, Part II, 1966. (in Russian)
[25] Y.A. Sharifov, Optimal controlof impulsive nonlocale boundary conditions, Russian Mathematics, 57 (2013), pp. 65-72.
[26] V. Yucesoy and H. Ozbay, Optimal nevanlinna pick interpolant and its application to robust repetitive control of time delay systems, Appl. Comput. Math., 17 (2018), pp. 96-108.
[2] F.A. Aliev, Comments on `Sweep algorithm for solving optimal control problem with multi-point boundary conditions' by M. Mutallimov, R. Zulfuqarova and L. Amirova, Adv. Differ. Equ., 131 (2016).
[3] F.A. Aliev, Methods for Solving Applied Problems of Optimizing Dynamic Systems, Elm, Baku, 1989. (in Russian)
[4] F.A. Aliev, The problem of optimal control of a linear system with unseparated two- point boundary conditions, Differential equations, 2 (1986), pp. 345-347. (in Russian)
[5] F.A. Aliev, Optimization problem with unseparated two-point boundary conditions, News AN USSR, cybernetics, 6(1985), pp. 138-146. (in Russian)
[6] F.A. Aliev, N.A. Aliev, N.A. Safarova, et al., Sweep method for solving the Roesser type equation describing the motion in the pipeline, Appl. Math. Comput., 295 (2017), pp. 16-23.
[7] F.A. Aliev and N.A. Ismailov, Methods for solving optimization problems with point-to-point boundary conditions, Preprint AS Azerb. SSR, Institute of Physics, Baku, 151 (1985). (in Russian)
[8] F.A. Aliev, V.B. Larin, N.I. Velieva, et al., On periodic solution of generalized Sylvester matrix equations, Appl. Comput. Math., 16 (2017), pp. 78-84.
[9] F.A. Aliev, M.M. Mutallimov, N.A. Ismailov, and M.F. Radjabov, Algorithms for the construction of optimal controllers for gas-lift operation, Autom. Remote Control, 73 (2012), pp. 1279-1289.
[10] R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems, Elsevier, New York, 1965.
[11] V.L. Bordyug, V.B. Larin, and A.G. Timoshenko, The Control Problem of Walking Apparatus, Nauk.Dumka, Kiev, 1985. (in Russian)
[12] C. Bota, B. Caruntu, and C. Lazureanu, The least square homotopy perturbation method for boundary value problems, Appl. Comput. Math., 16 (2017), pp. 39-47.
[13] A.E. Bryson and Y.C. Jr. & Ho, Applied Optimal Control: Optimization, Estimation, and Control, Waltham, MA Blaisdell, 1969.
[14] F.R. Gantmacher, The Theory of Matrices, Nauka, Moscow, 1967. (in Russian)
[15] A. Khan and G. Zaman, Asymptotic behavior of an age-structured SIRS endemic model, Appl. Comput. Math., 17:2 (2018), pp. 185-204.
[16] V.B. Larin, The Control of Walking Apparatus, Nauka Dumka, Kiev, 1980. (in Russian)
[17] N.I. Mahmudov, Finite-approximate controllability of evolution equations, Appl. Comput. Math., 16 (2017), pp. 159-167.
[18] F.G. Maksudov and F.A. Aliyev, Optimization of pulse systems with unseparated two-point boundary conditions, Dokl.AS SSSR, 280(1985), pp. 796-798. (in Russian)
[19] K. Moszynski, A method of solving the boundary value problem for a system of linear ordinary differensial equations, Algorytmy, 11 (1964), pp.25-43.
[20] M.M. Mutallimov and F.A. Aliyev, Methods for Solving Optimization Problems in the Operation of Oil Wells, LAP LAMBERT, r 2012. (in Russian)
[21] M.M. Mutallimov, L.I. Amirova, F.A. Aliev, Sh.A. Faradjova, and I.A. Maharramov, Remarks to the paper: sweep algorithm for solving optimal control problem with multi-point boundary conditions, TWMS J. Pure Appl. Math., 9 (2018), pp. 243-246.
[22] M.M. Mutallimov, R.T. Zulfugarova, and L.I. Amirova, Sweep algorithm for solving optimal control problem with multi-point boundary conditions, Adv. Differ. Equ., 233 (2015).
[23] E. Polak, Computational Methods in Optimization. Unified Approach, Academic Press, 1971.
[24] V.E. Shamansky, Methods for numerical solution of boundary value problems at the ECM, Nauk. Dumka, Kiev, Part I, 1963, Part II, 1966. (in Russian)
[25] Y.A. Sharifov, Optimal controlof impulsive nonlocale boundary conditions, Russian Mathematics, 57 (2013), pp. 65-72.
[26] V. Yucesoy and H. Ozbay, Optimal nevanlinna pick interpolant and its application to robust repetitive control of time delay systems, Appl. Comput. Math., 17 (2018), pp. 96-108.
)
