More general stability conditions for the Lie-bracket-based unicycle ESC

Derive more general sufficient conditions on the parameters c1=4! c a^3 C1, c2=4! c a^3 C2, and the angular velocity Ω that guarantee exponential stability of the equilibrium (x_d, y_d) for the third-order Lie bracket system associated with the proposed unicycle extremum seeking control when the objective function is J(x, y) = C1(x − x_d)^4 + C2(y − y_d)^4, for example by employing a Lyapunov function with time-periodic coefficients or by applying Barbalat's lemma, thereby relaxing the current sufficient conditions stated in the paper.

Background

The paper proves exponential stability for the equilibrium of the third-order Lie bracket system corresponding to the proposed unicycle ESC when the objective function behaves locally as a fourth-degree polynomial. Theorem 1 provides sufficient (but not necessary) parameter conditions on C1, C2 (through c1 and c2) and the angular velocity Ω that ensure exponential stability.

In Remark 1, it is acknowledged that these conditions stem from a particular Lyapunov function choice and may be conservative. The authors expect that more general (less restrictive) conditions could be derived using alternative analysis tools, such as Lyapunov functions with time-periodic coefficients or Barbalat's lemma, and explicitly leave this task for future work.