Summary: This paper is motivated by giving the detailed proofs and some interesting remarks on the results the author obtained in a series of papers, where an adaptive-feedback algorithm was proposed to effectively stabilize and synchronize chaotic systems. This note proves in detail the strictness of this algorithm from the viewpoint of mathematics, and gives some interesting remarks for its potential applications to chaos control and synchronization. In addition, a significant comment on synchronization-based parameter estimation is given, which shows some techniques proposed in literature less strict and ineffective in some cases.

### MSC:

 93B52 Feedback control 93C40 Adaptive control/observation systems 34H10 Chaos control for problems involving ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior