The differential equation below describes a phase lock loop (PLL) in communication networks: y¨ +(a + b cos y)y˙ + c sin y = 0 with the constant c > 0
a)Using a Lyapunov analysis, show that y = 0, y˙ = 0 is a locally stable equilibrium if a ≥ b ≥ 0. Define the domain D for the Lyapunov analysis. Hint: You may consider the Lyapunov candidate V = c (1 − cos y) + 0.5 ˙y 2 (don’t forget to show that this is a valid Lyapunov function.)
b) Using a Lyapunov (and possibly LaSalle) analysis, show that y = 0, y˙ = 0 is locally asymptotically stable if a > b ≥ 0. Define the domain D for the Lyapunov analysis.