Abstract
In this paper, we investigate a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball. The main difference between this case and the case of systems with fast and slow variables studied earlier is that the matrix at the fast variables is a multidimensional analog of the second-order Jordan cell with zero eigenvalue and, thus, does not satisfy the standard condition of asymptotic stability. Continuing our previous studies, we consider initial conditions depending on the second small parameter. In the degenerate case, this results in an asymptotic expansion of the solution of a fundamentally different kind. The solvability of the problem is proved. We also construct and justify a complete power asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control in a small parameter at the derivatives in the equations of the systems.
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References
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Wiley, New York, 1962).
N. N. Krasovskii, Theory of Motion Control: Linear Systems (Nauka, Moscow, 1968) [in Russian].
A. B. Vasil’eva and M. G. Dmitriev, “Singular perturbations in optimal control problems,” J. Math. Sci. 34 (3), 1579–1629 (1986).
P. V. Kokotović and A. H. Haddad, “Controllability and time-optimal control of systems with slow and fast models,” IEEE Trans. Automat. Control 20 (1), 111–113 (1975). doi https://doi.org/10.1109/TAC.1975.1100852
A. L. Dontchev, Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems (Springer, Berlin, 1983; Mir, Moscow, 1987).
T. R. Gichev and A. L. Donchev, “Convergence of the solution of the linear singularly perturbed problem of time-optimal response,” J. Appl. Math. Mech. 43 (3), 502–511 (1979).
A. R. Danilin and A. M. Il’in, “On the structure of the solution of a perturbed time-optimal problem,” Fundam. Prikl. Mat. 4 (3), 905–926 (1998).
A. R. Danilin and O. O. Kovrizhnykh, “On the dependence of a time-optimal problem for a linear system on two parameters,” Vestn. Chelyab. Gos. Univ., No. 27, 46–60 (2011).
A. R. Danilin and O. O. Kovrizhnykh, “Time-optimal control of a small mass point without environmental resistance,” Dokl. Math. 88 (1), 465–467 (2013).
A. Erdelyi and M. Wyman, “The asymptotic evaluation of certain integrals,” Arch. Rational Mech. Anal. 14, 217–260 (1963).
E. B. Lee and L. Markus, Foundations of Optimal Control Theory (Wiley, New York, 1967).
A. R. Danilin, “Asymptotic behavior of the optimal cost functional for a rapidly stabilizing indirect control in the singular case,” Comp. Math. Math. Phys. 46 (12), 2068–2079 (2006).
A. M. Il’in and A. R. Danilin, Asymptotic Methods in Analysis (Fizmatlit, Moscow, 2009) [in Russian].
L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1984) [in Russian].
Funding
This work was partially supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).
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Danilin, A.R., Kovrizhnykh, O.O. On a Singularly Perturbed Time-Optimal Control Problem with Two Small Parameters. Proc. Steklov Inst. Math. 307 (Suppl 1), 34–50 (2019). https://doi.org/10.1134/S0081543819070046
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DOI: https://doi.org/10.1134/S0081543819070046