Geometric control theory, developed by Basile and Marro, and independently, by Wonham and Morse in the 1970s revolves around characterizing the properties of finite dimensional, linear and time-invariant systems using geometry. Some examples of these properties are invariance, controllability and observability. The task addressed in this paper is to develop the geometric tools for fractional systems using the diffusive representation (also known as the distributed frequency) model. The mathematics involved in this approach is different from the classical case as fractional systems are inherently infinite dimensional. Unlike the integer order derivative, the fractional derivative is a non-local operator, and so the evolution of the so-called pseudo-state depends on not just its current value but also its past history. Thus, the notion of an initial condition for a fractional system can come in different forms. This leads to different kinds of fractional derivative operators. With some of these operators, the semigroup property is lost. With the distributed frequency model, the initial condition comes in the form of an initialization function. This takes care of the infinite dimensional nature of fractional systems. Furthermore, the distributed frequency model retains the semigroup property. This is useful in developing invariance and controlled invariance for fractional systems. Moreover, these properties of fractional systems are verified numerically using a higher order finite-dimensional approximation, which retains all the geometric properties of the distributed frequency model.

中文翻译：

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分布式频率模型的几何框架


几何控制理论由 Basile 和 Marro 以及 Wonham 和 Morse 在 20 世纪 70 年代独立开发，主要围绕使用几何来表征有限维、线性和时不变系统的特性。这些属性的一些例子是不变性、可控性和可观察性。本文解决的任务是使用扩散表示（也称为分布式频率）模型开发分数系统的几何工具。这种方法涉及的数学与经典情况不同，因为分数系统本质上是无限维的。与整数阶导数不同，分数阶导数是非局部算子，因此所谓伪状态的演化不仅取决于其当前值，还取决于其过去的历史。因此，分数系统的初始条件的概念可以有不同的形式。这导致了不同种类的分数阶导数算子。对于其中一些运算符，半群性质会丢失。对于分布式频率模型，初始条件以初始化函数的形式出现。这考虑到了分数系统的无限维性质。此外，分布式频率模型保留了半群性质。这对于开发分数系统的不变性和受控不变性很有用。此外，分数系统的这些属性使用高阶有限维近似进行数值验证，保留了分布式频率模型的所有几何属性。