The study of reachable sets of controlled objects is an important research area in optimal control theory. Such sets describe in a rough form the dynamical possibilities of the objects, which is important for theory and applications. Many optimization problems for controlled objects use the reachable set \(D(T)\) in their statements. In the study of properties of controlled objects, it is useful to have some constructive estimates of \(D(T)\) from above with respect to inclusion. In particular, such estimates are helpful for the approximate calculation of \(D(T)\) by the pixel method. In this paper, we consider two nonlinear models of direct regulation known in the theory of absolute stability with a control term added to the right-hand side of the corresponding system of differential equations. To obtain the required upper estimates with respect to inclusion, we use Lyapunov functions from the theory of absolute stability. Note that the upper estimates for \(D(T)\) are obtained in the form of balls in the phase space centered at the origin.

中文翻译：

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关于某些非线性控制系统的包容性，从上方估计可到达集合

可控对象可达集的研究是最优控制理论的重要研究领域。这样的集合以粗糙的形式描述了物体的动力学可能性，这对于理论和应用很重要。受控对象的许多优化问题在其语句中使用可达集\（D（T）\）。在研究受控对象的属性时，从包含的角度对\（D（T）\）进行一些建设性的估计非常有用。特别地，这样的估计有助于\（D（T）\）的近似计算。通过像素方法。在本文中，我们考虑了绝对稳定性理论中已知的两个直接调节的非线性模型，在相应的微分方程组的右侧添加了一个控制项。为了获得所需的关于包含的上限估计，我们使用来自绝对稳定性理论的李雅普诺夫函数。注意，\（D（T）\）的较高估计是在以原点为中心的相空间中以球的形式获得的。