## Communications in Mathematical Analysis

### Solving of the polynomial systems arising in the linear time-optimal control problem

#### Abstract

The analytic solution of the time-optimal control problem for the system $\dot x_1=u$, $\dot x_k=x_{k-1}$, $k=2,\ldots,n$, $|u|\le1$, for arbitrary $n$ is given. The paper relies on the approach originated in V.I.Korobov, G.M.Sklyar, Mat. Sb. (N.S.) 134(176) (1987), pp 186-206 which proved to be closely connected with ideas from Markov moment problem theory. We give the explicit form of a polynomial $P_n(x,\theta)$ such that for any initial point $x^0$ the optimal time $\theta_0$ coincides with the maximal real root of the equation $P_n(x^0,\theta)=0$. When $\theta_0$ is known, the switching times of the optimal control are found as the roots of a single polynomial. The approach leads to the transparent and easy algorithm for solving of the time-optimal control problem mentioned above. We present two programs using Maple and discuss several examples.

#### Article information

Source
Commun. Math. Anal., Conference 3 (2011), 153-171.

Dates
First available in Project Euclid: 25 February 2011

https://projecteuclid.org/euclid.cma/1298670010

Mathematical Reviews number (MathSciNet)
MR2772059

Zentralblatt MATH identifier
1209.49043

Subjects
Secondary: 49M30

#### Citation

Korobov, V. I.; Sklyar, G. M.; Ignatovich, S. Yu. Solving of the polynomial systems arising in the linear time-optimal control problem. Commun. Math. Anal. (2011), no. 3, 153--171. https://projecteuclid.org/euclid.cma/1298670010

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