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Can anybody give me a good reference which under suitable assumptions discusses exponential stability of $0$ for the equation

$\dot{u}_t = A(t)u(t) + b(t)$

Here $u_t\in\mathbb R^n$ is the unknown, $b_t\in\mathbb R^n$, $A(t)\in \mathbb R^{n\times n}$ and all functions are smooth as you want. In fact I'm interested in the case in which also $u$ and $b$ are matrix valued, but this shouldn't make much difference I think.

I didn't find anything on this problem in Arnold's classic ODE book.

I know that $0$ is exponentially stable when $A$ is constant and $b=0$ under the assumption that the eigenvalues of $A$ all have strictly negative real part.

What happens in this more general setting?

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1 Answer 1

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The short answer is "Anything can happen in this more general case". Here is an example, which I take from Wiggings (2003).

Consider $$ \dot x=A(t)x,\quad x\in\mathbb R^2, $$ where $$ A=\begin{pmatrix} -1+\frac 32 \cos^2 t&& 1-\frac 32 \cos t\sin t\\ -1-\frac 32\cos t\sin t&& -1+\frac 32 \sin^2 t \end{pmatrix} $$

It is easy to check that the eigenvalues of this matrix are independent of $t$ have negative real parts. However, the direct calculation shows that $$ x_1(t)=\begin{pmatrix} -\cos t\\ \sin t \end{pmatrix}e^{t/2}\quad x_1(t)=\begin{pmatrix} \sin t\\ \cos t \end{pmatrix}e^{-t} $$ are two linearly independent solutions, hence the origin is actually of a saddle type.

Having said all this, I would like to add that there is rich theory for different classes of differential equations of the form $$ \dot x=A(t)x,\quad x\in \mathbb R^n. $$ In particular, this is true for the cases $A(t+T)=A(t),\forall t$ (periodic coefficients) and $B=\lim_{t\to\infty}A(t)$ (asymptotically autonomous differential equations). But you won't find this theory in Arnold's book.

In addition to the book by Wiggins (which is rather thick), I would suggest to look through the book by Verhulst "Nonlinear Differential Equations and Dynamical Systems", which is easier to read and still contains a lot of pertinent material.

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