Wireless Ad Hoc and Sensor Networks

Background 71L 1 (x)L(x, t)L 0 (x)0 xFIGURE 2.4Time-varying function Lxk ( ( ), k) that is positive definite (( L0 ( x( k))) < L( x( k), k))and decrescent( Lxk ( ( ), k) ≤ L 1 ( xk ( ))).2 3 4Note that≤ + sin kT ≤ ,so thatandLxk ( ( ), k)so that it is decrescent.x ( k) Lxk ( ( ), k) ≥ L( xk ( )) ≡ x( k)+ ,0 1 2 22 4is globally positive definite. Also,Lxk ( ( ), k) ≤L( xk ( )) ≡ x( k) + x( k)1 1 2 2 2THEOREM 2.4.5 (LYAPUNOV RESULTS FOR NONAUTONOMOUS SYSTEMS)1. Lyapunov stability: If, for system described in Equation 2.53, thereexists a function L(x(k), k) with continuous partial derivatives,nsuch that for x in a compact set S ⊂RLxk ( ( ), k) is positive definite, Lxk ( ( ), k)> 0 (2.55)∆L( xk ( ), k) is negative semidefinite, ∆L( xk ( ), k)≤ 0 (2.56)then the equilibrium point in SISL.2. Asymptotic stability: If, furthermore, condition (2.56) is strengthenedto∆L( xk ( ), k) is negative definite, ∆L( xk ( ), k)< 0 (2.57)then the equilibrium point is AS.