Problem 3.8 Determining of various asymptotics of solutions of nonlinear time-optimal problems via right ideals in the moment algebra

Abstract

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][2][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 forṁ(1)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

Time-Optimal Control Problem for a Special Class of Control Systems: Optimal Controls and Approximation in the Sense of Time Optimality

“…It is, obviously, satisfied if the system (20) is linear, because in this case K includes only piecewise constant functions having no more than n − 1 switchings. However, in the general case this is an open question about a class of systems satisfying this condition [6]. The results of Sect.…”

“…Now, we recall some concepts and results concerning the application of the free algebras technique proposed and developed in [4][5][6][7][8][9][10]. Different approaches based on series representations close to (12) can be found in [13][14][15][16][17][18][19].…”

“…Further progress was achieved by developing the algebraic approach [4][5][6][7][8][9][10]. As a result, it was shown that the approximating system can be constructed with the use of some special structures in the algebra of nonlinear power moments.…”