Problem 3.8

Determining of various asymptotics of

solutions of nonlinear time-optimal

problems via right ideals in the moment

algebra

G. M. Sklyar

Szczecin University

Wielkopolska str. 15, 70-451 Szczecin, Poland;

Kharkov National University

Svoboda sqr. 4, 61077 Kharkov

Ukraine

[email protected], [email protected]

S. Yu. Ignatovich

Kharkov National University

Svoboda sqr. 4, 61077 Kharkov, Ukraine

[email protected]

1 MOTIVATION AND HISTORY OF THE PROBLEM

The time-optimal control problem is one of the most natural and at the same time difficult problems in the optimal control theory.

For linear systems, the maximum principle allows to indicate a class of optimal controls. However, the explicit form of the solution can be given only in a number of particular cases [1-3]. At the same time [4], an arbitrary linear time-optimal problem with analytic coefficients can be approximated (in a neighborhood of the origin) by a certain linear problem of the form

In the nonlinear case, the careful analysis is required for any particular system [5, 6]. However, in a number of cases the time-optimal problem for a nonlinear system can be approximated by a linear problem of the form (1) [7]. We recall this result briefly. Consider the time-optimal problem in the

2 FORMULATION OF THE PROBLEM.

BIBLIOGRAPHY

[1] V. I. Korobov and G.M. Sklyar, “Time optimality and the power moment problem, ” Math. USSR Sb., 62, pp. 185-205, 1989.

[2] V. I. Korobov and G. M. Sklyar, “Time optimality and the trigonometric moment problem, ” Math. USSR Izv., 35, pp. 203-220, 1990.

[3] V. I. Korobov and G. M. Sklyar, “Markov power min-problem with periodic gaps, ” J. Math. Sci., 80 , pp. 1559-1581, 1996.

[4] G. M. Sklyar and S. Yu. Ignatovich, “A classification of linear time-optimal control problems in a neighborhood of the origin, ” J. Math.

Anal. Applic., 203, pp. 791-811, 1996.

[5] A. Bressan, “The generic local time-optimal stabilizing controls in dimension 3, ” SIAM J. Control Optimiz., 24, pp. 177-190, 1986.

[6] B. Bonnard, “On singular estremals in the time minimal control problem in , ” SIAM J. Control Optimiz., 23, pp. 794-80, 1985.

[7] G. M. Sklyar and S. Yu. Ignatovich, “Moment approach to nonlinear time optimality, ” SIAM J. Control Optimiz., 38, pp. 1707-1728, 2000.

[8] G. M. Sklyar and S. Yu. Ignatovich, “Approximation in the sense of time optimality via nonlinear moment problems, ” SIAM J. Control Optimiz.

[9] V. I. Korobov, “On continuous dependence of the solution of the optimal control problem with free time on initial data, ” Differen. Uravn., 7, pp. 1120-1123, 1971, (in Russian).