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?
Best Answer
This is known as partial stability and was pioneered by Rumyantsev. You may want to read:
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.