# A Mechanical Feedback Classification of Linear Mechanical Control Systems
## 1 Introduction  
In this paper we consider the problem of classification of linear mechanical control systems under mechanical feedback transformations. Therefore, by providing a solution to the classification problem, we answer several questions:  
1. Given two linear mechanical systems, how to determine whether they are equivalent? 2. Is there a set of complete invariants that are computable in terms of objects on the configuration space only? 3. Is there a distinguished normal (or canonical) form? We consider the above-defined classification problem in three important cases, namely for controllable and uncontrollable mechanical systems, and for the subclass of Lagrangian systems. A classification of controllable linear (first-order) systems \(\hat{x}=Ax+Bu\) under general linear transformations and general linear feedback has been solved in the celebrated Brunovsky classification [1], see also [2]. A general classification (including not necessarily controllable systems) leads to the canonical decomposition [3, 4] and a short yet concise note [5]. Here, we consider the novel problem of classification of linear mechanical systems under linear transformations that respect splitting into configurations and velocities, and linear mechanical feedback. It turns out that the special form of considered (second-order) systems and the special form of (mechanical) transformations yield the counterpart of the above-mentioned classical results. In other words, we deal with a smaller class of control systems (than general linear systems) and we use more subtle mechanical feedback transformations (than general feedback transformations) and yet the invariants are perfectly analogous to those of the general case. Moreover, they can be computed on a half of the state space, namely using objects defined on the configurations space only. What is more, we show that mechanical feedback transformations are perfectly adaptedto the class of mechanical systems, namely the classification of mechanical systems is the same if, instead of mechanical feedback transformations, we use all linear feedback transformations. Our analysis of the classification problem implies a series of results for stability and stabilization of linear mechanical systems that we present in Section 6. The problem of stability of motion of linear mechanical systems has been studied extensively, see e.g., [6; 7], Section 5.2 in [8] and the reference therein. In those papers, certain structural assumptions (symmetry, positive definiteness, etc.) on matrices of the system are assumed, and what is crucial, dissipative forces are allowed an thus asymptotic stability is concerned. We present new results in this field. We do not assume, a priori, any structure on the mechanical system (see Section 2). Moreover, we consider mechanical systems without dissipative terms. This leads to important conclusions that, first, such systems are never asymptotically stable nor can be asymptotically stabilized by feedback. Second, only the Lagrangian subclass of mechanical control systems can be stable, therefore the problem of stabilization of (general) mechanical system reduces, actually, to the problem of finding mechanical transformations that make the system Lagrangian, which fits well to our classification problem. The paper is organized as follows. In Section 2, we define the class of linear mechanical control systems and its special subclass, namely Lagrangian control systems, and we introduce linear mechanical feedback transformations. In Sections 3 and 4, the complete classification under mechanical feedback transformations of, respectively, controllable (Theorem 1) and uncontrollable (Theorem 2) linear mechanical control systems is presented. Then, the classification of linear Lagrangian control systems is given in Section 5. In Section 6, we formulate results about stability and stabilization and show in Theorem 3 that, within the class of linear mechanical non-dissipative control systems, only Lagrangian ones can be stabilized (not asymptotically).