On the construction of the Lyapunov function with sign-definite derivative with the help of auxiliary functions with sign-constant derivatives

Trajectory Convergence From Coordinate-Wise Decrease of Quadratic Energy Functions, and Applications to Platoons

“…Moreover, It is easy to confirm that y is absolutely continuous as it is the integral of a measurable locally bounded function. Hence Theorem 4 shows that y converges to some y * and y i (t) remains at all time in [min y i (t), max y i (t)], which implies the convergence of x and the inclusion (13).…”

“…where d(i) is the degree of agent i in G and w the bound on the disturbance. (b) For every agent i and all time t there holds p i + min(x j (0) − p j ) ≤ x i (t) ≤ p i + max j (x j (0) − p j ), (13) Proof: Let us perform a change of variable, defining y…”

“…Condition (1) appears naturally in certain multi-agent dynamics, including the platooning problem we will analyze in Section III, which involves control laws with deadzones to remove potential instabilities resulting from incoherent measurements of the same distance by different agents. And Classical approaches for establishing convergence based on energy functions rely on variation of the Lyapunov -Kraskowski -LaSalle theorems [13], [15]. These approaches apply to unforced dynamical systems of the form ẋ = f (x, t), where the vector field f often satisfies some (uniform) continuity condition [4], [8].…”