Determine stability of non autonomous system at the origin

Question

I am trying to determine the stability of the zero solution of the system $x'= \begin{bmatrix} -t & 1 \\ 1 & -t \end{bmatrix}x $

Even though, a Liapunov method can only be applied to autonomous systems I was thinking we can use a similar approach as follows:

To show that the system is asymptotically stable at the point $x_0$ it is enough to show that it is stable and that

$$\displaystyle \lim_{t \to +\infty} |\phi(t,t_0)-x_0|=0 $$ for all $x \in U(x_0)$ for some neighborhood $U$ of $x_0$.

Then define $L(x(t),y(t))=x^2+y^2$ which would be the $|| \, .||^2$.

And if we show that $L'(x,y)<0$ along trajectories then $L(x,y)$ will be strictly decreasing continuous function and lower bounded by $0$ and therefore the limit condition mentioned above will be satisfied.

The problem is that this is not the case at least with $L$ the way I am defining it. So another option is that may be I need to find another norm where this does happen.

Does that sound like something we could do?

I tried $L(x,y)=ax^n+by^m$ but still I can't get that $L'<0$.

Is there any other approach someone can suggest?

Any help would be greatly appreciate it. Thanks!

Answer 1

You are almost there.

Stability: For $L(x,y)=x^2+y^2$ we have $$ \dot L(x,y)=4xy-2t(x^2+y^2), $$ which is negative definite for $t$ sufficiently large, independently of $x,y$, and so we are done with stability.

Asymptotic stability: The determinant of any fundamental matrix is $\exp\int_0^t(-2s)\,ds=e^{-t^2}$ and so necessarily all solutions tend to the origin.

Answer 2

You get that all the matrices $A(t)=J-tI$ commute, where $I=\pmatrix{1&0\\0&1}$, $J=\pmatrix{0&1\\1&0}$, thus $$\exp(\int A(s)ds)=\exp(Jt-\frac12t^2I)=e^{-t^2/2}(\cosh(t)I+\sinh(t)J)$$ is a fundamental matrix of the LDE system and the solution can be written as $$ x(t)=e^{-t^2/2}(\cosh(t)I+\sinh(t)J)x_0 $$ and the first factor falls, eventually, faster than the second factor grows.