Wireless Ad Hoc and Sensor Networks

62 Wireless Ad Hoc and Sensor Networkswhere ∆L( x)is evaluated along the trajectories of (2.4.1) (as shown insubsequent example). That is,∆L( xk ( )) = Lxk ( ( + 1)) −Lxk( ( ))(2.34)THEOREM 2.4.1 (LYAPUNOV STABILITY)If there exists a Lyapunov function for a system as shown in Equation 2.31, thenthe equilibrium point is stable in the sense of Lyapunov (SISL).This powerful result allows one to analyze stability using a generalized notionof energy. The Lyapunov function performs the role of an energy function.If Lx ( ) is positive definite and its derivative is negative semidefinite, then Lx ( ) isnonincreasing, which implies that the state xt () is bounded. The next result showswhat happens if the Lyapunov derivative is negative definite — then Lx ( ) continuesto decrease until |( xk)|vanishes.THEOREM 2.4.2 (ASYMPTOTIC STABILITY)If there exists a Lyapunov function Lx ( ) for system as shown in Equation 2.31with the strengthened condition on its derivative,∆L( x)is negative definite,∆L( x)< 0then the equilibrium point is AS.To obtain global stability results, one needs to expand the set S to all ofalso required is an additional radial unboundedness property.(2.35)R n , butTHEOREM 2.4.3 (GLOBAL STABILITY)Globally SISL: If there exists a Lyapunov function L(x) for the systemas shown In Equation 2.31 such that Equation 2.32 and Equation2.33 hold globally, andLx ( )→∞ as | x|→∞(2.36)then the equilibrium point is globally SISL.Globally AS: If there exists a Lyapunov function Lx ( ) for a system asshown in Equation 2.31 such that Equation 2.32 and Equation 2.35hold globally, and also the unboundedness condition as in Equation2.36 holds, then the equilibrium point is GAS.The global nature of this result of course implies that the equilibriumpoint mentioned is the only equilibrium point.