Perturbed system of ODEs

Question

I have a system of ordinary differential equations that comes up in control theory for $\mathbf{x}: [0, \infty) \to \mathbb{R}^n$ that looks like

$$\frac{d\mathbf{x}}{dt} = [A + B(t)]\mathbf{x}\\\mathbf{x}(0) = \mathbf {c} \neq 0$$

The matrix $A$ is real with distinct eigenvalues $\lambda_1, ...,\lambda_n$ where $\Re(\lambda_j) < 0$ for all $j$.

Without $B(t)$ I know the solution must behave as $\lim_{t \to \infty} \mathbf{x}(t) = 0.$ Now I am unable to solve the system in closed form with $B(t)$ present, but I would like to know if the solution has the same asymptotic behavior ($\to 0$) as $t \to \infty$.

Conditions on $B(t)$:

$\lim_{t \to \infty}b_{ij}(t) = 0 \\ b_{ij} \in L^1([0,\infty))$

I think this may be enough to ensure the asymptotic behavior $\mathbf{x}(t) \to 0$ but maybe more is needed.

Thank you for any help.

Answer 1

The claim is correct. In fact, convergence of the elements of $B(t)$ to zero suffices and the $L_1$ property of $b_{ij}$ is not needed.

Since the eigenvalues of $A$ have all negative real parts for every $Q=Q^T>0$ there exists some $P=P^T>0$ such that $$PA+A^TP=-Q$$ Consider now the Lyapunov function candidate $V:=(1/2)x^TPx$. Its derivative has the form $$\dot{V}=\frac{1}{2}x^T(PA+A^TP)x+x^TPB(t)x\\ \leq -\frac{1}{2}x^TQx+\|P\|\|B(t)\|\|x\|^2$$ Due to the limit property of $b_{ij}$ there exists time $T>0$ such that $$\|P\|\|B(t)\|\|x\|^2\leq \frac{1}{4}x^TQx \qquad \forall t\geq T$$ and therefore $$\dot{V}\leq -\frac{1}{4}x^TQx\leq -\frac{\lambda_{\min}(Q)}{2\lambda_{\max}(P)}V \qquad t\geq T$$ The above inequality proves the exponential convergence of $V$ (and therefore $x$) to zero.

Answer 2

In general, Idon't see why it wouldn't. If anything your $B$ matrix is like an initial perturbation that dies out. Now if it is stochastic and involves something like a Wiener process, then things could change.

Another potential issue could be with induced instabilities like those in the Lorenz equations that propagate with time and are affected by numerical roundoff as Ed Lorenz found out by examining the data on his punch cards which showed very slight changes in the initial conditions! A lot depends on the relative magnitudes of $a_{ij}$ and $b_{ij}$.