Wireless Ad Hoc and Sensor Networks

70 Wireless Ad Hoc and Sensor Networksnx ∈R . Assume again that the origin is an equilibrium point. For nonautonomoussystems, the basic concepts just introduced still hold, but theexplicit time dependence of the system must be taken into account. Thebasic issue is that the Lyapunov function may now depend on time. Inthis situation, the definitions of definiteness must be modified, and thenotion of “decrescence” is needed.nLet Lxk ( ( ), k):R ×R→R be a scalar function such that L(0,k) = 0 and Sbe a compact subset of R n . Then Lxk ( ( ), k)is said to be:Locally positive definite if Lxk ( ( ), k) ≥ L 0 ( xk ( )) for some time-invariantpositive definite L0( x( k)), for all k ≥ 0 and x∈ S.(Denoted byLxk ( ( ), k)> 0.)Locally positive semidefinite if Lxk ( ( ), k) ≥ L 0 ( xk ( )) for some time-invariantpositive semidefinite L0( x( k)), for all k ≥ 0 and x∈ S.(Denoted byLxk ( ( ), k)≥ 0.)Locally negative definite if Lxk ( ( ), k) ≤ L 0 ( xk ( )) for some time-invariantnegative definite L0( x( k)), for all k ≥ 0 and x∈ S.(Denoted byLxk ( ( ), k)< 0.)Locally negative semidefinite if Lxk ( ( ), k) ≤ L 0 ( xk ( )) for some time-invariantnegative semidefinite L0( x( k)), for all k ≥ 0 and x∈ S.(Denotedby Lxk ( ( ), k)≥ 0.)Thus, for definiteness of time-varying functions, a time-invariant definitefunction must be dominated. All these definitions are said to hold globallynif S ∈R .nA time-varying function Lxk ( ( ), k):R ×R→R is said to be decrescentif L(0,k) = 0 and there exists a time-invariant positive definite functionL1( x( k))such thatLxk ( ( ), k) ≤L( xk ( )), ∀k≥1 0(2.54)The notions of decrescence and positive definiteness for time-varyingfunctions are depicted in Figure 2.4.Example 2.4.4: Decrescent FunctionConsider the time-varying functionx kLxk ( ( ), k) = ( )x1 2 ( k)+ 223 + sin kT