Are there Lyapunov stability conditions that focus on a subset of the available state equations? Are there Lyapunov stability conditions that focus on a subset of the available state equations?  Essentially, I have a system of nonlinear ODEs and I only care that some of the variables converge to zero and not others.  For example, consider the contrived system:
\begin{align}
\dot{y_1} =& -y_1 + e^{-t}y_2 \\\\
\dot{y_2} =& \frac{5}{2}\pi\cos(10\pi t) + e^{-t}y_1\\\\
y_1(0) =& 1\\\\
y_2(0) =& 1
\end{align}
This has a solution seen here:

Clearly, $y_1$ converges to $0$ and $y_2$ does not.  Now, normally Lyapunov stability conditions come in the form of something like:
If $V (y, t)$ is locally positive definite and decrescent, and $\dot{V} (y, t) \leq 0$
locally in y and for all t, then the origin of the system is uniformly
locally stable (in the sense of Lyapunov).
where
$$
\dot{V}\bigg\rvert_{\dot{y}=f(y,t)} = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial y} f.
$$
However, this doesn't work in the above case because $y_2$ continues to oscillate around some nonzero value.  In truth, I don't care about $y_2$, I only care about $y_1$, so is there a transformation or alternative Lyapunov stability theorem that only focuses on this quantity?
 A: This is known as partial stability and was pioneered by Rumyantsev. You may want to read:

*

*Rumyantsev, V. V. "On the stability of motion relative to some of the variables." Vestnik Moskov. Gos. Univ., Ser. Mat., Mekh., Fiz., Astron., Khim. 5 (1957).

*Rumyantsev, V. V. "On the stability with respect to a part of the variables." Nonlinear mechanics and stability (1970): 243-265.

*Vorotnikov, V. I. "On the theory of partial stability." Journal of Applied Mathematics and Mechanics 59.4 (1995): 525-531.

*Vorotnikov, V. I.. Partial stability and control. Springer Science & Business Media (2012).

In short, if you have
$$
\begin{align}
\dot{x}_1 &= f_1(x_1, x_2) \\
\dot{x}_2 &= f_2(x_1, x_2)
\end{align}
$$
with $f_1(0, x_2) = 0$ for all $x_2$ and you only care whether $x_1\rightarrow 0$ then you can show that with $V(x_1)$ with $V(0) = 0$ and $V(x_1) > 0$ for all $x_1 \neq 0$ if you also can show that
$$
\dot{V}(x_1, x_2) < 0
$$
for all $x_1$ and $x_2$. The problem is (as always) that you first need to find $V$.
Edit: I just noticed that for your system the equilibrium condition is not met so unfortunatly this won't work for your system. But there are many other ways to check partial stability which can be found in the book by Vorotnikov. Maybe one of these can be used here.
A: For the given contrived system one can do a coordinate transformation. When checking the stability of $y_1$ a natural set of coordinates would be $x_1 = y_1$ and $x_2 = \dot{y}_1$, yielding
\begin{align}
\dot{x}_1 &= x_2, \\
\dot{x}_2 &= -2\,x_2 - \left(1 - e^{-2\,t}\right) x_1 + \frac{5}{2}\pi\cos(10\,\pi\,t)\,e^{-t},
\end{align}
which is a linear time varying inhomogeneous system. This isn't super trivial to analyse, but I think it is possible to show that this is exponentially stable.
The transformation back to the original coordinates yields $y_1 = x_1$ and $y_2 = e^t \left(x_1 + x_2\right)$. This transformation for $y_2$ at $t\to\infty$ is not well defined, which explains why even though $x_1,x_2\to0$ as $t\to\infty$ does not imply that $y_2\to0$.
