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

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

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

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Abstract

The analytic solution of the time-optimal control problem for the system ˙ x1 =u, ˙ xk =xk¡1, k =2; : : : ;n, juj·1, 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 Pn(x;q) such that for any initial point x0 the optimal time q0 coincides with the maximal real root of the equation Pn(x0;q) = 0. When q0 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.