Hausdorff moment problem and nonlinear time optimality

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2022-01-28 DOI:10.1051/cocv/2022007

G. Sklyar, S. Ignatovich

引用次数: 0

Abstract

A complete analytic solution for the time-optimal control problem for nonlinear control systems of the form $\dot x_1=u$, $\dot x_j=x_1^{j-1}$, $j=2,\ldots,n$, 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 a certain class of affine systems 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.

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Hausdorff矩问题与非线性时间最优性

对于任意~$n$，得到了$\dot x_1=u$， $\dot x_j=x_1^{j-1}$， $j=2，\ldots,n$的非线性控制系统的时间最优控制问题的完整解析解。在本文中，我们给出了以下令人惊讶的观察结果:这个非线性最优性问题导致一个截断的豪斯多夫矩问题，它提供了寻找最优时间和最优控制的分析工具。由于系统是齐次的，所以在时间最优性意义上近似于某类仿射系统。因此，所得结果可用于求解该类系统的时间最优控制问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

自引率

7.10%

发文量

期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.