The geometric structure of small-time reachable sets plays an important role in control theory, in particular, in solving problems of local synthesis. In this paper, we consider the problem of approximate description of reachable sets on small time intervals for control-affine systems with integral quadratic constraints on the control. Using a time substitution, we replace such a set by the reachable set on a unit interval of a control system with a small parameter, which is the length of the time interval for the original system. The constraints on the control are given by a ball of small radius in the Hilbert space \(\mathbb{L}_{2}\). Under certain conditions imposed on the controllability Gramian of the linearized system, this reachable set turns out to be convex for sufficiently small values of the parameter. We show that in this case the shape of the reachable set in the state space is asymptotically close to an ellipsoid. The proof of this fact is based on the representation of the reachable set as the image of a Hilbert ball of small radius in \(\mathbb{L}_{2}\) under a nonlinear mapping to \(\mathbb{R}^{n}\). In particular, this asymptotic representation holds for a fairly wide class of second-order nonlinear control systems with integral constraints. We give three examples of systems whose reachable sets demonstrate both the presence of the indicated asymptotic behavior and the absence of the latter if the necessary conditions are not satisfied.

中文翻译：

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可到达Setson小时间间隔的渐近行为

小时间可及集的几何结构在控制理论中，特别是在解决局部合成问题中，起着重要的作用。在本文中，我们考虑了具有仿射二次约束的仿射系统的小时间间隔上的可达集的近似描述问题。使用时间替换，我们用一个小的参数（即原始系统的时间间隔的长度），用一个控制系统的单位间隔上的可达集合替换此集合。控制的约束由希尔伯特空间\（\ mathbb {L} _ {2} \）中的小半径球给出 。在对线性化系统的可控制性Gramian施加一定条件的情况下，对于参数的足够小，此可达到的结果证明是凸的。我们表明，在这种情况下，状态空间中的可达集的形状渐近地接近椭圆形。该事实的证明基于可到达集合的表示形式，该表示形式是在\（\ mathbb {L} _ {2} \）中的小半径希尔伯特球的图像，该非线性映射到 \（\ mathbb {R} ^ {n} \）。特别地，这种渐近表示适用于具有积分约束的相当广泛的一阶非线性控制系统。我们给出系统的三个示例，这些系统的可达集证明了所指示的渐近行为的存在与否（如果不满足必要条件的话，则没有后者）。