标题： 总变分减少用于对动态机械系统施加的精确控制 显示英文标题

标题： Total variations reduction for an exact control applied to a dynamical mechanical system

摘要： 在最优控制策略中，一个重要的是定义控制的成本。通常，它被加到控制准则中，并乘以一个用$\varepsilon$表示的小系数，这被称为控制的边际成本。本文的关键思想是，在控制成本中引入一个平滑项，旨在减少消耗的能量和减少控制的振荡。然后，利用所谓的渐近控制，基于$\varepsilon$的小性，我们构建了一个可以闭环实现的精确控制。控制中涉及的能量主要取决于控制的变化。因此，将这个量（总变差）包含在最优控制中的准则中似乎是自然的。可以通过引入控制的一阶导数的$L^1$范数来近似地做到这一点。我们在本文中开发的控制策略可以应用于这种线性模型。一个重要的新点是，由于控制的成本，我们关注非可微准则的精确控制策略。根据Tykhonov正则化方法的思想，使用所谓的基于控制边际成本小性的渐近方法，证明了所建议的精确控制是所有精确控制中代表边际成本最小的那个。此外，这也是主要的技术要点，可以通过对控制回路的各种参数进行适当调整来减少控制的变化。我们在三个例子中测试了该方法。 显示英文摘要

摘要： In an optimal control strategy, an important point is to define the cost of the control. Usually it is added to the control criterion and multiplied by a small coefficient denoted by $\varepsilon$ which is known as the marginal cost of the control. The key idea of this paper, is to introduce a smoothing term in the control cost which aims at reducing the quantity of energy spent and reducing the oscillations of the control. Then using a so-called asymptotic control based on the smallness of $\varepsilon$, we construct an exact control which can be implemented in a close loop. The energy involved in the control depends mainly on the variation of the control. Therefore it seems natural to include this quantity (the total variations) in the criterion involved in the optimal control. This can be done approximately by introducing the $L^1$ norm of the first order derivative of the control. The control strategy that we develop in this paper can be applied to such linear models. One important and new point is that we focus on exact control strategies for a non-differentiable criterion because of the cost of the control. Following the ideas of Tykhonov regularization method, it is proved using the so-called asymptotic method based on the smallness of the marginal cost of the control, that the exact control suggested is the one which represents the minimum of the marginal cost among exact controls. Furthermore, and it is the main technical point, it can reduce the variations of the control with an adequate tuning of the various parameters of the control loop. We test the method on three examples.