A Classification of Linear Time-Optimal Control Problems in a Neighborhood of the Origin

Abstract

The main goal of the paper is to classify time-optimal control problems for linear controllable systems with analytic coefficients. The reduction to the Markov moment min-problem is the main tool of this investigation. The constructive solution of the time-optimal control problem using mentioned reduction is obtained under certain conditions. ᮊ

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

“…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 x i = −t qi u, i = 1, . .…”

“…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].…”

Hausdorff moment problem and nonlinear time optimality

“…As was proved in [9], [18], the integrator system (1.1) plays a special role since it approximates an arbitrary controllable linear system (1.3) in the following sense. Denote by ( θ x 0 , u x 0 (t)) and (θ x 0 , u x 0 (t)) the optimal times and the optimal controls for the problems (1.1)-(1.2) and (1.3)-(1.4) respectively.…”

“…It would be interesting to find conditions under which the successive approximation method is applicable for solving the time-optimal problem for systems (1. 15), where the time-optimal problem for the system (1.14) is solved on each step, like the case of linear systems studied in [18]. One can think of the solution of the time-optimal problem for the system (1.14) as the first step in solving the time-optimal problems for general nonlinear affine systems.…”

Invariants of linear control systems with analytic matrices and the linearizability problem

“…We also require that a target linear system has analytic matrices, i.e., A(t) and b(t) are real analytic on the interval [α, β]. Such systems were considered, e.g., in [13,14]. It was shown that the Markov moment problem [15,16] can be efficiently applied for solving the time-optimal control problem for such systems.…”