The paper continues the author's previous studies. We consider 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
$$ \left\{\begin{array}{llll} \phantom{\\varepsilon^3}\dot{x}=y,\,& x,\,y\in \mathbb{R}^{2},\quad u\in \mathbb{R}^{2},\\[1ex]
\varepsilon^3\dot{y}=Jy+u,&\,\|u\|\leqslant 1,\quad 0<\varepsilon,\mu\ll 1,\\[1ex] x(0)=x_0(\varepsilon,\mu)=(x_{0,1}, \varepsilon^3\mu\xi)^*,\quad y(0)=y_0,\\[1ex]
x(T_{\varepsilon,\mu})=0,\quad y(T_{\varepsilon,\mu})=0,\quad T_{\varepsilon,\mu} \longrightarrow \min,& \end{array} \right. $$
where
$$ J=\left(\begin{array}{rr} 0&1 \\ 0&0\end{array}\right).\qquad $$
The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix $J$ at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter $\mu$. We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence $\varepsilon^\gamma(\varepsilon^k+\mu^k)$, $0<\gamma<1$.
Keywords: optimal control, time-optimal control problem, asymptotic expansion, singularly perturbed problem, small parameter
Received January 10, 2019
Revised February 4, 2019
Accepted February 11, 2019
Funding Agency: The second author was 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).
Aleksei Rufimovich Danilin, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: dar@imm.uran.ru
Ol’ga Olegovna Kovrizhnykh, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: koo@imm.uran.ru
REFERENCES
1. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The mathematical theory of optimal processes, ed. L.W. Neustadt, N Y; London, Interscience Publ. John Wiley & Sons, Inc., 1962, 360 p. ISBN: 0470693819 . Original Russian text published in Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal’nykh protsessov, Moscow: Fizmatgiz Publ., 1961, 391 p.
2. Dmitriev M.G., Kurina G.A. Singular perturbations in control problems. Automation and Remote Control, 2006, vol. 67, no. 1, pp. 1–43. doi: 10.1134/S0005117906010012
3. Kokotovic P.V., Haddad A.H. Controllability and time-optimal control of systems with slow and fast modes. IEEE Trans. Automat. Control., 1975, vol. 20, no. 1, pp. 111–113. doi: 10.1109/TAC.1975.1100852
4. Dontchev A.L. Perturbations, approximations and sensitivity analisis of optimal control systems, Berlin; Heidelberg; N Y; Tokio: Springer-Verlag, 1983, 161 p. doi: 10.1007/BFb0043612. Translated to Russian under the title Sistemy optimal’nogo upravleniya: Vozmushcheniya, priblizheniya i analiz chuvstvitel’nosti, Moscow: Mir Publ., 1987, 156 p.
5. Gichev T.R., Donchev A.L. Convergence of the solution of the linear singularly perturbed problem of time-optimal response. J. Appl. Math. Mech., 1979, vol. 43, no. 3, pp. 502–511. doi: 10.1016/0021-8928(79)90098-4
6. Danilin A.R., Il’in A.M. On the structure of the solution of a perturbed optimal-time control problem. Fundament. Prikl. Matematika, 1998, vol. 4, no. 3, pp. 905–926 (in Russian).
7. Danilin A.R., Parysheva Y.V. Asymptotics of the optimal cost functional in a linear optimal control problem. Dokl. Math., 2009, vol. 80, no. 1, pp. 478–481. doi: 10.1134/S1064562409040073
8. Shaburov A.A. Asymptotic expansion of a solution for the singularly perturbed optimal control problem with a convex integral quality index and smooth control constraints. Proc. of the Institute of Math. and Inf. at Udmurt State University, 2017, vol. 50, no 2, pp. 110–120.
9. Danilin A.R., Kovrizhnykh O.O. On the dependence of the time-optimal control problem for a linear system of two small parameters. Vest. Chelyabinsk. Univer., Ser. Matematika, Mekhanika, Informatika, 2011, vol. 14, no. 27, pp. 46–60 (in Russian).
10. Danilin A.R., Kovrizhnykh O.O. Time-optimal control of a small mass point without environmental resistance. Dokl. Math., 2013, vol. 88, no. 1, pp. 465–467. doi: 10.1134/S1064562413040364
11. Lee E.B., Markus L. Foundations of optimal control theory. N Y; London; Sydney: John Wiley & Sons, Inc., 1967, 576 p. Translated to Russian under the title Osnovy teorii optimal’nogo upravleniya, Moscow: Nauka Publ., 1972, 576 p. ISBN: 0471522635 .
12. Danilin A.R. Asymptotic behavior of the optimal cost functional for a rapidly stabilizing indirect control in the singular case. Comput. Math. Math. Physics, 2006, vol. 46, no. 12, pp. 2068–2079. doi: 10.1134/S0965542506120062
13. Il’in A.M., Danilin A.R. Asimptoticheskie metody v analize [Asymptotic methods in analysis]. Moscow: Fizmatlit Publ., 2009, 248 p. ISBN: 978-5-9221-1056-3 .
14. Kantorovich L.V., Akilov G.P. Functional analysis. 2nd ed. N Y: Pergamon Press Ltd, 1982, 589 p. ISBN: 0-08-023036-9 . Original Russian text (3rd ed.) published in Kantorovich L.V., Akilov G.P. Funktsional’nyi analiz. Moscow: Nauka Publ., 1984, 752 p.
Cite this article as: A.R.Danilin, O.O.Kovrizhnykh. Asymptotics of the solution to a singularly perturbed time-optimal control problem with two small parameters, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 88–101.
