Show that every solution $x(t)$ of $x'(t)= A(t)x(t)$ converges to some limit

Question

(Long-time asymptotics). Suppose

$$\int_0^∞\|A(t)\|\,dt < ∞.$$

Show that every solution $x(t)$ of $x'(t)= A(t)x(t)$ converges to some limit: $\lim_{t→∞} x(t) = x_∞.$

(Hint: First show that all solutions are bounded and then use the corresponding integral equation.)

I was able to solve a first part:

I used that the solutions are limited by $\|\phi(t,t_0)\| \leq e^ {\int_0^∞\|A(t)\|\,dt}$. As $\int_0^∞\|A(t)\|\,dt < ∞$, then $\|\phi(t,t_0)\| \leq K $. The problem now is to show that the above limit always exists. Someone can help

Answer 1

Hint

Once you know that $x$ is bounded by some constant $K$ you can use the integral equation to show that $x(t)$ is Cauchy, for $t_1 \leq t_2$ you have: $$\|x(t_1)-x(t_2)\| =\left\|\int_{t_1}^{t_2} A(s) x(s) ds \right\| \leq K \int_{t_1} ^{t_2} \|A(s)\| ds$$ And $\int_{t_1} ^{t_2} \|A(s)\|ds$ goes to $0$ as $t_1$ goes to $+\infty$.

So using the sequential characterization of convergence and Cauchy sequences you can show that, as for Cauchy sequences, as long as your space is complete the function converges.

Answer 2

To expand on my comment regarding the right form of the solution estimate:

Let $\pi$ be an anti-derivative of the scalar function $\|A(t)\|$. Then for the solution $x(t)$ with initial value $x(t_0)=x_0$ one gets the estimate from the integral form of the ODE $$ \|x(t)-x_0\|\le (\pi(t)-\pi(t_0))\|x_0\|+\int_{t_0}^t\pi'(s)\|x(s)-x_0\|\,ds $$ In the fashion of the Grönwall lemma, we get that $\|x(t)-x_0\|$ is smaller than the exact solution of $$ u'(t)=\pi'(t)(\|x_0\|+u(t)),~~u(0)=0 $$ which is $$ \|x_0\|+u(t)\le \|x_0\|e^{\pi(t)-\pi(t_0)} $$ or in terms of the flow mapping the initial point $(t_0,x_0)$ to the solution $x(t)=ϕ(t;t_0,x_0)$ $$ \|ϕ(t;t_0,x_0)-x_0\|≤\|x_0\|\left(\exp\left(\int_{t_0}^{t}\|A(t)\|\,dt\right)-1\right) $$ This formula now contains both the boundedness of any solution for $t\to\infty$ as well as the Cauchy property that forces the solution to stay inside ever decreasing balls that contract toward the limit point.