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Section
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Mathematics
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Title
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Computational solution of time-optimal control problem for linear systems with delay
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Author(-s)
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Shevchenko G.V.a
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Affiliations
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Institute of Mathematics, Siberian Branch of the Russian Academy of Sciencesa
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Abstract
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A computational method of solving time-optimal control problem for linear systems with delay is proposed. It is proved that the method converges in a finite number of iterations to an $\varepsilon$-optimal solution, which is understood as a pair $\{T,u\},$ where $u=u(t)$, $t\in[0,T]$ is an admissible control that moves the system into an $\varepsilon$-neighborhood of the origin in time $T\leqslant T_{\min}$, and the optimal time is $T_{\min}$. An enough general time-optimal control problem with delay is studied in [Vasil'ev F.P, Ivanov R.P. On an approximated solving of time-optimal control problem with delay, Zh. Vychisl. Mat. Mat. Fiz., 1970, vol. 10, no. 5, pp. 1124-1140 (in Russian)], an approximate solution is proposed for it, and computational aspects are discussed. However, to solve some auxiliary optimal control problems arising there, it is suggested to use methods of gradient and Newton type, which possess only a local convergence. The method proposed in the present paper has a global convergence.
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Keywords
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admissible control, optimal control, time-optimal control
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UDC
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517.97
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MSC
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49J15, 49M05
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DOI
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10.20537/vm120209
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Received
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20 February 2012
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Language
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Russian
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Citation
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Shevchenko G.V. Computational solution of time-optimal control problem for linear systems with delay, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 2, pp. 100-105.
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References
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- Vasil'ev F.P., Ivanov R.P. On approximated solution of a time-optimal control problem with delay, Zh. Vychisl. Mat. Mat. Fiz., 1970, vol. 10, no. 5, pp. 1124-1140.
- Boltyanskii V.G. Matematicheskie metody optimal'nogo upravleniya (Mathematical methods of optimal control), Moscow: Nauka, 1969, 408 p.
- Kwakernaak Kh., Sivan R. Lineinye optimal'nye sistemy upravleniya (Linear optimal systems of control), Мoscow: Mir, 1977 650 p.
- Shevchenko G.V. A numerical algorithm for solving a linear time-optimality problem, Comput. Math. Math. Phys., 2002, vol. 42, no. 8, pp. 1123-1134.
- Shevchenko G.V. Numerical method for solving a nonlinear time-optimal control problem with additive control, Comput. Math. Math. Phys., 2007, vol. 47, no. 11, pp. 1768-1778.
- Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal'nykh protsessov (Mathematical theory of optimal processes), Moscow: Phizmatgiz, 1983, 393 p.
- von Hohenbalken B. A finite algorithm to maximize certain pseudo concave functions on polytopes, Math. Program, 1975, vol. 9, pp. 189-206.
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