Let $V: \mathbb R^n \rightarrow \mathbb R$ be a Lyapunov function for system $\dot{x}=f(x)$. Let $\dot V \leq 0$ hold. Do there exist conditions on th system dynamics such that we can ensure that the system converges to a steady state instead of showing an undamped oscillation?
For example, consider an LC circuit (or equivalently, a mass-spring system) with system dynamics $$ \dot x = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} x. $$ We clearly see that there is no damping in any state and thus periodic oscillations occur. If we include damping, i.e. a resistor or friction in the mechanical equivalent), we have damping and the system converges to a steady state instead of undamped oscillations. Can we generalize that and state something like: Lyapunov stability + damping in somestates $\rightarrow$ convergence to a steady state?
Thanks in advance!
In general, no, you cannot conclude that your equilibrium point $x_e$ is asymptotically stable provided $\dot{V}(x) \le 0$ unless by LaSalle's invariance principle, the only trajectory satisfying $\dot{V}(x) = 0$ is $x(t) = x_e$.
In order to use Lyapunov theory to prove $x_e$ is asymptotically stable, you will need to search for another Lyapunov function where $\dot V(x) < 0$ when $x\ne x_e$. There are a few techniques (e.g. replacing $x^Tx$ terms with $x^TPx$ where $P$ is positive definite symmetric matrix in your Lyapunov function and then solving for elements of $P$ to ensure $\dot{V} < 0$, or the variable gradient method) that can help you search for another Lyapunov function. Please refer to Nonlinear Systems by Khalil, Section 4.1 for examples.
There are other, more advanced techniques to prove stability, such as Control Contraction Mapping (CCM), which may be of interest.