A New Run Algorithm for Solving the Continuous Linear-Quadratic Optimal Control Problem with Unseparated Boundary Conditions
A sweep method is proposed for solving the optimal control problem with unseparated boundary conditions. This model reduces the boundary conditions to the initial condition. Using the properties of the J -symmetry of the corresponding Hamiltonian matrix and Euler–Lagrange equations, it is shown that...
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| Published in | Journal of computer & systems sciences international Vol. 60; no. 1; pp. 48 - 55 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Moscow
Pleiades Publishing
01.01.2021
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1064-2307 1555-6530 |
| DOI | 10.1134/S1064230721010020 |
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| Abstract | A sweep method is proposed for solving the optimal control problem with unseparated boundary conditions. This model reduces the boundary conditions to the initial condition. Using the properties of the
J
-symmetry of the corresponding Hamiltonian matrix and Euler–Lagrange equations, it is shown that linear algebraic equations for determining the missing initial data of the system being solved have a symmetric matrix of coefficients. The proposed algorithm allows us to reduce the dimension of the problem of finding the fundamental matrix of the Hamiltonian system. The results are illustrated by the example of a linear-quadratic optimal control problem (stationary case) with the minimal control actions. |
|---|---|
| AbstractList | A sweep method is proposed for solving the optimal control problem with unseparated boundary conditions. This model reduces the boundary conditions to the initial condition. Using the properties of the
J
-symmetry of the corresponding Hamiltonian matrix and Euler–Lagrange equations, it is shown that linear algebraic equations for determining the missing initial data of the system being solved have a symmetric matrix of coefficients. The proposed algorithm allows us to reduce the dimension of the problem of finding the fundamental matrix of the Hamiltonian system. The results are illustrated by the example of a linear-quadratic optimal control problem (stationary case) with the minimal control actions. |
| Author | Mutallimov, M. M. Maharramov, I. A. Aliev, F. A. Guseinova, N. Sh |
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| Cites_doi | 10.1186/s13662-016-0816-4 10.1186/s13662-015-0569-5 |
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| Copyright | Pleiades Publishing, Ltd. 2021. ISSN 1064-2307, Journal of Computer and Systems Sciences International, 2021, Vol. 60, No. 1, pp. 48–55. © Pleiades Publishing, Ltd., 2021. Russian Text © The Author(s), 2021, published in Izvestiya Akademii Nauk, Teoriya i Sistemy Upravleniya, 2021, No. 1, pp. 52–59. |
| Copyright_xml | – notice: Pleiades Publishing, Ltd. 2021. ISSN 1064-2307, Journal of Computer and Systems Sciences International, 2021, Vol. 60, No. 1, pp. 48–55. © Pleiades Publishing, Ltd., 2021. Russian Text © The Author(s), 2021, published in Izvestiya Akademii Nauk, Teoriya i Sistemy Upravleniya, 2021, No. 1, pp. 52–59. |
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| References | MutallimovM. M.Zul’fugarovaR. T.AmirovaL. I.Sweep algorithm for solving optimal control problem with multi-point boundary conditionsAdv. Differ. Equat.20152015233337370710.1186/s13662-015-0569-5 AlievF.LarinV.VelievaN.GasimovaK.FaradjovaSh.Algorithm for solving the systems of the generalized sylvester-transpose matrix equations using LMITWMS J. Pure Appl. Math.20191023924539711091431.65056 V. E. Shamanskii, Methods for the Numerical Solution of Boundary Value Problems on a Digital Computer, Part 1 (Kiev, Nauk. Dumka, 1966, 1963) [in Russian]. AshyralyevA.ErdoganA. S.TekalanS. N.An investigation on finite difference method for the first order partial differential equation with the nonlocal boundary conditionAppl. Comput. Math.20191824726039716161443.65115 AlievF. A.AlievN. A.GulievA. P.TagievR. M.DzhamalbekovM. A.A method for solving a boundary value problem for a system of hyperbolic equations describing motion in a gas-lift processPrikl. Mat. Mekh.201882512519 AbramovA. A.On the transfer of boundary conditions for systems of linear ordinary differential equations (a variant of the sweep method)Zh. Vychisl. Mat. Mat. Fiz.19611542545 AndreevYu. N.Controlling Finite-Dimensional Line Features1976MoscowNauka M. M. Mutallimov and F. A. Aliev, Methods for Solving Optimization Problems during the Operation of Oil Wells (LAP Lambert, Saarbrücken, Deutscland, 2012). BordyugB. A.LarinV. B.TimeshenkoA. G.Control Tasks for Walking Vehicles1985KievNauk. Dumka SamkoS. G.KilbasA. A.MarichevO. I.Fractional Integrals and Derivatives: Theory and Applications1993AmsterdamGordon and Breach Science0818.26003 G. Ahmad, N. Omid, and Mahboubeh Molavi-Arabshahi, “Numerical approximation of time fractional advection-dispersion model arising from solute transport in rivers,” TWMS J. Pure Appl. Math. 10, 117–131 (2019). LarinV. B.Walking Apparatus Control1980KievNauk. Dumka0702.93041 F. A. Aliev, “Optimal control problem for a linear system with nonseparated two-point boundary conditions,” Differ. Uravn., No. 2, 345–347 (1986). HamidovR. H.MutallimovM. M.Dimension reduction of one multivariable decision making problemAppl. Comput. Math.201918626838881971427.90273 MutallimovM. M.Sweep method for the solution of optimization problem with three point boundary conditions with unseparabilities in the internal and end pointsDokl. NAN Azerb.20076324292416749 A. Bryson and Yu-Chi Ho, Applied Optimal Control: Optimization, Estimation and Control (CRC, Boca Raton, FL, 1975). M. M. Mutallimov, R. T. Zul’fugarova, and A. P. Guliev, “A sweeping algorithm for solving optimal control problems with nonseparated boundary conditions,” Vestn. BGU, Ser. Fiz. Mat. Nauk., No. 2, 153–160 (2009). Zakhar-ItkinM. Kh.Riccati matrix differential equation and the semigroup of linear fractional transformationsUsp. Mat. Nauk.197328831204073580293.34056 MutallimovM. M.AmirovaL. I.AlievF. A.FaradjovaSh. A.MaharramovI. A.Remarks to the paper: Sweep algorithm for solving optimal control problem with multi-point boundary conditionsTWMS J. Pure Appl. Math.2018924324638230121423.49003 AlievF. A.AlievN. A.SafarovaN. A.GasimovaK. G.VelievaN. I.Solution of linear fractional-derivative ordinary differential equations with constant matrix coefficientsAppl. Comput. Math.201817317322383947907122374 ZhangJ.ShenYu.HeJ.Some analytical methods for singular boundary value problem in a fractal space: A reviewAppl. Comput. Math.20191822523539716141440.34025 F. A. Aliev and N. A. Ismailov, “Methods for solving optimization problems with two-point boundary conditions,” Preprint No. 151 (Inst. Phys. Acad. Sci. AzSSR, Baku, 1985). AlievF. A.Comments on 'Sweep algorithm for solving optimal control problem with multi-point boundary conditions' by M. Mutallimov, R. Zulfuqarova and L. AmirovaAdv. Differ. Equat.20162016131350048910.1186/s13662-016-0816-4 F. A. Aliev, “Optimization problem with two-point boundary conditions,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 6, 138–146 (1985). F. A. Aliev, “Optimal control problem for a linear system with nonseparated two-point boundary conditions,” Izv. Akad. Nauk Az SSR, Ser. Fiz. Tekh. Mat. Nauk., No. 2, 345–347 (1986). F. A. Aliev, Methods for Solving Applied Problems of Optimization of Dynamic Systems (Elm, Baku, 1989) [in Russian]. F. A. Aliev, N. S. Hajiyeva, N. A. Ismailov, and S. M. Mirsaabov, “Calculation algorithm defining the coefficient of hydraulic resistance on different areas of pump-compressor pipes in gas lift process by lines method,” SOCAR Proc., No. 4, 13–17 (2019). MoszynskiK.A method of solving the boundary value problem for a system of linear ordinary differensial equationsAlgorytmy1964112543183120 Yu. N. Andreev (7182_CR15) 1976 M. M. Mutallimov (7182_CR10) 2007; 63 7182_CR16 S. G. Samko (7182_CR26) 1993 F. Aliev (7182_CR24) 2019; 10 7182_CR8 7182_CR9 A. Ashyralyev (7182_CR23) 2019; 18 7182_CR6 A. A. Abramov (7182_CR12) 1961; 1 7182_CR7 K. Moszynski (7182_CR11) 1964; 11 F. A. Aliev (7182_CR13) 2016; 2016 7182_CR4 7182_CR5 F. A. Aliev (7182_CR27) 2018; 17 7182_CR3 M. M. Mutallimov (7182_CR19) 2015; 2015 7182_CR28 7182_CR1 M. Kh. Zakhar-Itkin (7182_CR17) 1973; 28 R. H. Hamidov (7182_CR20) 2019; 18 J. Zhang (7182_CR25) 2019; 18 V. B. Larin (7182_CR18) 1980 7182_CR21 B. A. Bordyug (7182_CR2) 1985 M. M. Mutallimov (7182_CR14) 2018; 9 F. A. Aliev (7182_CR22) 2018; 82 |
| References_xml | – reference: F. A. Aliev, “Optimization problem with two-point boundary conditions,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 6, 138–146 (1985). – reference: AlievF. A.AlievN. A.SafarovaN. A.GasimovaK. G.VelievaN. I.Solution of linear fractional-derivative ordinary differential equations with constant matrix coefficientsAppl. Comput. Math.201817317322383947907122374 – reference: V. E. Shamanskii, Methods for the Numerical Solution of Boundary Value Problems on a Digital Computer, Part 1 (Kiev, Nauk. Dumka, 1966, 1963) [in Russian]. – reference: MutallimovM. M.Sweep method for the solution of optimization problem with three point boundary conditions with unseparabilities in the internal and end pointsDokl. NAN Azerb.20076324292416749 – reference: Zakhar-ItkinM. Kh.Riccati matrix differential equation and the semigroup of linear fractional transformationsUsp. Mat. Nauk.197328831204073580293.34056 – reference: A. Bryson and Yu-Chi Ho, Applied Optimal Control: Optimization, Estimation and Control (CRC, Boca Raton, FL, 1975). – reference: AshyralyevA.ErdoganA. S.TekalanS. N.An investigation on finite difference method for the first order partial differential equation with the nonlocal boundary conditionAppl. Comput. Math.20191824726039716161443.65115 – reference: F. A. Aliev, N. S. Hajiyeva, N. A. Ismailov, and S. M. Mirsaabov, “Calculation algorithm defining the coefficient of hydraulic resistance on different areas of pump-compressor pipes in gas lift process by lines method,” SOCAR Proc., No. 4, 13–17 (2019). – reference: BordyugB. A.LarinV. B.TimeshenkoA. G.Control Tasks for Walking Vehicles1985KievNauk. Dumka – reference: MutallimovM. M.Zul’fugarovaR. T.AmirovaL. I.Sweep algorithm for solving optimal control problem with multi-point boundary conditionsAdv. Differ. Equat.20152015233337370710.1186/s13662-015-0569-5 – reference: AlievF. A.AlievN. A.GulievA. P.TagievR. M.DzhamalbekovM. A.A method for solving a boundary value problem for a system of hyperbolic equations describing motion in a gas-lift processPrikl. Mat. Mekh.201882512519 – reference: SamkoS. G.KilbasA. A.MarichevO. I.Fractional Integrals and Derivatives: Theory and Applications1993AmsterdamGordon and Breach Science0818.26003 – reference: M. M. Mutallimov and F. A. Aliev, Methods for Solving Optimization Problems during the Operation of Oil Wells (LAP Lambert, Saarbrücken, Deutscland, 2012). – reference: HamidovR. H.MutallimovM. M.Dimension reduction of one multivariable decision making problemAppl. Comput. Math.201918626838881971427.90273 – reference: G. Ahmad, N. Omid, and Mahboubeh Molavi-Arabshahi, “Numerical approximation of time fractional advection-dispersion model arising from solute transport in rivers,” TWMS J. Pure Appl. Math. 10, 117–131 (2019). – reference: F. A. Aliev, “Optimal control problem for a linear system with nonseparated two-point boundary conditions,” Differ. Uravn., No. 2, 345–347 (1986). – reference: LarinV. B.Walking Apparatus Control1980KievNauk. Dumka0702.93041 – reference: F. A. Aliev and N. A. Ismailov, “Methods for solving optimization problems with two-point boundary conditions,” Preprint No. 151 (Inst. Phys. Acad. Sci. AzSSR, Baku, 1985). – reference: F. A. Aliev, “Optimal control problem for a linear system with nonseparated two-point boundary conditions,” Izv. Akad. Nauk Az SSR, Ser. Fiz. Tekh. Mat. Nauk., No. 2, 345–347 (1986). – reference: AndreevYu. N.Controlling Finite-Dimensional Line Features1976MoscowNauka – reference: MoszynskiK.A method of solving the boundary value problem for a system of linear ordinary differensial equationsAlgorytmy1964112543183120 – reference: AlievF. A.Comments on 'Sweep algorithm for solving optimal control problem with multi-point boundary conditions' by M. Mutallimov, R. Zulfuqarova and L. AmirovaAdv. Differ. Equat.20162016131350048910.1186/s13662-016-0816-4 – reference: F. A. Aliev, Methods for Solving Applied Problems of Optimization of Dynamic Systems (Elm, Baku, 1989) [in Russian]. – reference: AbramovA. A.On the transfer of boundary conditions for systems of linear ordinary differential equations (a variant of the sweep method)Zh. Vychisl. Mat. Mat. Fiz.19611542545 – reference: M. M. Mutallimov, R. T. Zul’fugarova, and A. P. Guliev, “A sweeping algorithm for solving optimal control problems with nonseparated boundary conditions,” Vestn. BGU, Ser. Fiz. Mat. Nauk., No. 2, 153–160 (2009). – reference: ZhangJ.ShenYu.HeJ.Some analytical methods for singular boundary value problem in a fractal space: A reviewAppl. Comput. Math.20191822523539716141440.34025 – reference: MutallimovM. M.AmirovaL. I.AlievF. A.FaradjovaSh. A.MaharramovI. A.Remarks to the paper: Sweep algorithm for solving optimal control problem with multi-point boundary conditionsTWMS J. Pure Appl. Math.2018924324638230121423.49003 – reference: AlievF.LarinV.VelievaN.GasimovaK.FaradjovaSh.Algorithm for solving the systems of the generalized sylvester-transpose matrix equations using LMITWMS J. Pure Appl. Math.20191023924539711091431.65056 – ident: 7182_CR7 – ident: 7182_CR28 – volume-title: Fractional Integrals and Derivatives: Theory and Applications year: 1993 ident: 7182_CR26 – volume: 11 start-page: 25 year: 1964 ident: 7182_CR11 publication-title: Algorytmy – volume: 1 start-page: 542 year: 1961 ident: 7182_CR12 publication-title: Zh. Vychisl. Mat. Mat. Fiz. – volume-title: Control Tasks for Walking Vehicles year: 1985 ident: 7182_CR2 – volume: 28 start-page: 83 year: 1973 ident: 7182_CR17 publication-title: Usp. Mat. Nauk. – ident: 7182_CR16 – ident: 7182_CR3 – volume: 82 start-page: 512 year: 2018 ident: 7182_CR22 publication-title: Prikl. Mat. Mekh. – ident: 7182_CR1 – volume: 9 start-page: 243 year: 2018 ident: 7182_CR14 publication-title: TWMS J. Pure Appl. Math. – ident: 7182_CR5 – volume: 17 start-page: 317 year: 2018 ident: 7182_CR27 publication-title: Appl. Comput. Math. – ident: 7182_CR9 – volume: 2016 start-page: 131 year: 2016 ident: 7182_CR13 publication-title: Adv. Differ. Equat. doi: 10.1186/s13662-016-0816-4 – volume: 18 start-page: 247 year: 2019 ident: 7182_CR23 publication-title: Appl. Comput. Math. – ident: 7182_CR21 – volume: 10 start-page: 239 year: 2019 ident: 7182_CR24 publication-title: TWMS J. Pure Appl. Math. – volume: 2015 start-page: 233 year: 2015 ident: 7182_CR19 publication-title: Adv. Differ. Equat. doi: 10.1186/s13662-015-0569-5 – volume: 18 start-page: 225 year: 2019 ident: 7182_CR25 publication-title: Appl. Comput. Math. – volume: 18 start-page: 62 year: 2019 ident: 7182_CR20 publication-title: Appl. Comput. Math. – volume: 63 start-page: 24 year: 2007 ident: 7182_CR10 publication-title: Dokl. NAN Azerb. – volume-title: Walking Apparatus Control year: 1980 ident: 7182_CR18 – volume-title: Controlling Finite-Dimensional Line Features year: 1976 ident: 7182_CR15 – ident: 7182_CR4 – ident: 7182_CR6 – ident: 7182_CR8 |
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| Title | A New Run Algorithm for Solving the Continuous Linear-Quadratic Optimal Control Problem with Unseparated Boundary Conditions |
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