Section

Mathematics

Title

Computational solution of timeoptimal control problem for linear systems with delay

Author(s)

Shevchenko G.V.^{a}

Affiliations

Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences^{a}

Abstract

A computational method of solving timeoptimal 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 timeoptimal control problem with delay is studied in [Vasil'ev F.P, Ivanov R.P. On an approximated solving of timeoptimal control problem with delay, Zh. Vychisl. Mat. Mat. Fiz., 1970, vol. 10, no. 5, pp. 11241140 (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.

Keywords

admissible control, optimal control, timeoptimal control

UDC

517.97

MSC

49J15, 49M05

DOI

10.20537/vm120209

Received

20 February 2012

Language

Russian

Citation

Shevchenko G.V. Computational solution of timeoptimal control problem for linear systems with delay, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 2, pp. 100105.

References

 Vasil'ev F.P., Ivanov R.P. On approximated solution of a timeoptimal control problem with delay, Zh. Vychisl. Mat. Mat. Fiz., 1970, vol. 10, no. 5, pp. 11241140.
 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 timeoptimality problem, Comput. Math. Math. Phys., 2002, vol. 42, no. 8, pp. 11231134.
 Shevchenko G.V. Numerical method for solving a nonlinear timeoptimal control problem with additive control, Comput. Math. Math. Phys., 2007, vol. 47, no. 11, pp. 17681778.
 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. 189206.

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