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We consider the control system $\dot x = A x + \alpha(t) b u$, where the pair $(A, b)$ is controllable, $x \in \mathbb R^2$, $u$ is a scalar control, and the unknown signal $\alpha$ satisfies a persistent excitation condition. We study the stabilization of this system, and we prove that it is globally asymptotically stable with arbitrarily large exponential rate uniformly with respect to all signals satisfying a common persistent excitation condition and a common Lipschitz continuity bound.

Stabilization of Two-Dimensional Persistently Excited Linear Control Systems with Arbitrary Rate of Convergence