Sufficient conditions for global stability from linear stability.

Consider $$\dot x = -Ax+f(x),$$ where $A>0$ and $f(x)$ smooth. Question: What are sufficient conditions for $f(x)$ such that the origin of the system is globally asymptotically stable?

I know that the local version holds. But I am looking for global version. A trivial example would be $f(x)=Bx$ with $-A+B<0$.

My approach: Let $x_l(t)$ denote the solution of the linear system and $x(t)$ of the nonlinear. Moreover, suppose that $f(x)$ is uniformly bounded, say $||f(x)||<\beta$ with $\beta>0$, and that $f(x)=0\iff x=0$. It follows that $$\begin{split} || x(t)-x_l(t)|| &= ||\int_0^te^{-A(t-\tau)}f(x(\tau))d\tau||\\ &\leq \int_0^t ||e^{-A(t-\tau)}||\cdot||f(x(\tau))||d\tau\\ &\leq \beta\int_0^t ||e^{-A(t-\tau)}||d\tau,\\ \end{split}$$ where the second inequality is due to our assumptions. The above error does not converge to $0$ (but to $\beta||A^{-1}||$) as $t\to\infty$. So, the assumptions only lead to stability.

Naturally I can pick some specific cases where the integral can be explicitly solved to ensure that the error converges to zero, but I fail to see a general assumption on $f(x)$.

• Chapter 13 of Coddington and Levinson has some theorems about asymptotic stability. – Adrian Keister Feb 19 '18 at 18:35

Assume $f(x)$ is globally Lipschitz, i.e., $\|f(x)\| \leq L\|x\|$, $L>0$, for all $x \in \mathbb R^n$.

Let

$V(x) = x^T P x$ with $P = P^T >0$

a Lyapunov function for the system $\dot x = \tilde A x$, with $\tilde A = -A$.

If $\tilde A$ is Hurwitz, the derivative of $V$ along the trajectories is

$\dot V = -x^T Q x$, where $Q=Q^T>0$

is the solution of the Lyapunov equation

$-Q = \tilde A^TP + P \tilde A$.

Now for the full system $\dot x = \tilde A x+f(x)$, it yields

\begin{align} \dot V &= -x^T Q x + 2P x \,f(x)\\& \leq -\lambda_{\min}(Q)\|x\|^2 + 2\lambda_{\max}(P)L\|x\|^2 \\& = -(\lambda_{\min}(Q) - 2\lambda_{\max}(P)L) \|x\| ^2\\& <0 \end{align} for

\begin{align} L<\frac{\lambda_{\min}(Q)}{2\lambda_{\max}(P)}. \end{align}

Provided that this condition holds, the origin is globally exponentially stable. Note that it is only a sufficient condition since the structure of $f$ is not considered, and the inequality with minimal and maximal eigenvalues leads generally to conservative results.

The minimal and the maximal eigenvalues of $Q$ and $P$ are associated to the eigenvalues of $A$. Note that the ratio $\frac{\lambda_{\min}(Q)}{\lambda_{\max}(P)}$ is maximal for $Q$ being chosen as the identity matrix.

For further details, see e.g. Khalil, H.K. Nonlinear systems