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Abstract
A complete analytic solution for the time-optimal control problem for nonlinear control systems of the form ẋ1 = u, ẋj = ẋ1j−1, j = 2, …, n, is obtained for arbitrary n. In the paper we present the following surprising observation: this nonlinear optimality problem leads to a truncated Hausdorff moment problem, which gives analytic tools for finding the optimal time and optimal controls. Being homogeneous, the mentioned system approximates affine systems from a certain class in the sense of time optimality. Therefore, the obtained results can be used for solving the time-optimal control problem for systems from this class.
Highlights
The time-optimal control problem is one of the most well-studied topics in the optimal control theory
In [19] we described a class of nonlinear systems (1.8) that can be approximated by linear ones in the sense close to (1.7): after some change of variables in the system (1.8), optimal times and optimal controls in the problems (1.1)–(1.2) and (1.8)–(1.9) become equivalent as x0 → 0
We have shown that this problem reduces to a truncated Hausdorff moment problem
Summary
The time-optimal control problem is one of the most well-studied topics in the optimal control theory. M=0 m=0 θ τ mu(τ )dτ, which coincides with (1.5) Using this representation, in [19] we described a class of nonlinear systems (1.8) that can be approximated by linear ones in the sense close to (1.7): after some change of variables in the system (1.8), optimal times and optimal controls in the problems (1.1)–(1.2) and (1.8)–(1.9) become equivalent as x0 → 0. The crucial point of our study is the following surprising observation: this optimality problem, like (1.1), (1.2), leads to a moment problem, in this case we deal with the classical truncated Hausdorff moment problem [6, 13] This allows us to use profound ideas and methods of the classical moment theory and, as a result, leads to finding an analytic solution of the time-optimal control problem for the system (1.14). In Appendix A we recall solvability conditions for a classical truncated Hausdorff moment problem and explain their connection with the results of Lemma 3.1
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