# A Mechanical Feedback Classification of Linear Mechanical Control Systems
## 6 Stability and Stabilization
It turns out that \(\ddot{x}=Ex\) is never asymptotically stable either (although it may have, contrary to \(\ddot{x}=E^{L}x\), complex eigenvalues with non-zero real part) and, moreover, it is stable if and only if it is Lagrangian. So there are no stable second order differential equations (in other words, dynamical systems) \(\ddot{x}=Ex\) others than Lagrangian ones. A mechanical control system (2) is called asymptotically stabilizable if there exists a mechanical feedback of the form \(u=Fx\) such that the closed loop  
\[ \begin{split}\dot{x}&=y\\ \dot{y}&=(E+BF)x\end{split} \]  
is asymptotically stable and is called stabilizable if all equilibria of the above closed loop system are stable. Recall that the map \(E:\mathbb{R}^{n}\to\mathbb{R}^{n}\) induces the map \(\hat{E}:\mathbb{R}^{n}/\mathcal{X}_{c}\to\mathbb{R}^{n}/\mathcal{X}_{c}\), where \(\mathcal{X}_{c}\cong\mathbb{R}^{k}\), that is represented by the matrix \(E_{d}\in\mathbb{R}^{l\times l}\), where \(l=n-k\). Proposition 7, applied to the uncontrolled system \(\dot{x}_{d}=y_{d}\), \(\dot{y}_{d}=E_{d}x_{d}\), leads to the following result describing stabilizability of mechanical control systems (2). Theorem 3: _Consider a mechanical control system (2)._  
1. _The system (_2_) is never asymptotically stabilizable._
2. _The following conditions are equivalent:_ 1. _the following conditions are equivalent:_ 2. _the matrix_ \(E_{d}\) _has all eigenvalues_ \(\lambda_{j}\in\mathbb{R}_{\leq 0}\) _and, moreover, is diagonalizable, i.e., its Jordan form_ \(E_{d}^{l}\) _consists of diagonal blocks_ \(D_{j}^{\mathbb{R}}\) _only, corresponding to_ \(\lambda_{j}\in\mathbb{R}_{\leq 0}\)_,_ 3. _all eigenvalues_ \(\lambda_{j}\) _of_ \(E_{d}\) _satisfy_ \(\lambda_{j}\in\mathbb{R}_{\leq 0}\) _and, moreover, their algebraic and geometric multiplicities coincide, i.e.,_ \(\mu_{j}=\gamma_{j}\)_, for_ \(1\leq j\leq q\)_, where_ \(q\) _is the number of distinct eigenvalues of_ \(E_{d}\)_;_ _the uncontrolled subsystem_ \(\dot{x}_{d}=y_{d}\)_,_ \(\dot{y}_{d}=E_{d}x_{d}\) _is T-equivalent to a Lagrangian system,_ _(2) is LMF-equivalent to (6) and, moreover, the matrix_ \(E_{d}\) _satisfies the conditions of item_ \((ii)\) _above (or of the equivalent items_ \((iii)\) _or_ \((iv)\)_)._  
Proof. : Item \((AS)\) follows directly from Proposition 7. For \((S)\), the crucial observation is that the controllable system is stabilizable, thus we deal with the uncontrollable system only, which must already be stable for the whole system to be stabilizable. Therefore, the equivalence \((i)\iff(iii)\) follows directly from Proposition 7. Equivalence of \((ii)\iff(iii)\) is given by Proposition 1 and the comments below it. For \((ii)\iff(iv)\), note that the matrix \(E_{d}^{J}\) of \((ii)\) coincides with \(E_{d}^{L}\) of (19), so the transformation \(T\) that diagonalizes \(E_{d}\) renders the uncontrolled subsystem Lagrangian. Finally, assume that (2) is LMF-equivalent to (6) and \(E_{d}\) satisfies \((ii)\), or equivalent conditions \((iii)\) or \((iv)\), i.e., it is stable. Then, by Proposition 5\((ii)\) it is also LMF-equivalent to a Lagrangian system that is stable. Hence, LMF-transformation stabilizes the system (2). The inverse follows from the previous arguments. Corollary 2: _Any stabilizing feedback \(u=Fx\), for a mechanical control system (2), renders the system \(\dot{x}=y\), \(\dot{y}=(E+BF)x\) Lagrangian._  
In other words, for (2) to be stabilizable, all unstable modes \(\dot{x}_{j}=\lambda_{j}x_{j}\), where \(\text{Re }\lambda_{j}>0\) must be controllable, i.e., must be contained in the controllable \(x_{c}\)-subsystem, or, equivalently, all uncontrollable modes must be stable. Checking that requires, however, a decomposition of (2) into controllable and uncontrollable subsystems. Therefore we provide below invariant conditions that can be verified for any (2). This is analogous to [7], where Lagrangian systems with dissipative forces were considered.