Wireless Ad Hoc and Sensor Networks

Background 57nGiven a function ft ():[ 0, ∞→R ) , its L p (function) norm is given interms of the vector norm |()| ft p at each value of t by⎛∞⎞p|()| f ⋅ p = ⎜ f( t)dt⎟p⎝⎜⎠⎟∫01p(2.23)and if p =∞,|()| f ⋅ = sup|()| f t∞t∞(2.24)If the L p norm is finite, we say ft () ∈L p . Note that a function is in L ∞ ifand only if it is bounded. For detailed treatment, refer to Lewis, Abdallah,and Dawson (1993) and Lewis, Jagannathan, and Yesilderek (1999).In the discrete-time case, let Z + = { 012… , , , } be the set of natural numbersnand f( k): Z+ →R . The L p (function) norm is given in terms of the vector|()| at each value of k byff k p|()| ( ) ,⎛⋅ p =⎝⎜∞∑k=0f kpp⎞⎠⎟1p(2.25)and, if p =∞,|()| f ⋅ = sup|()| f k∞k∞(2.26)If the L p norm is finite, we say f( k) ∈L p . Note that a function is in L ∞ ifand only if it is bounded.2.3 Properties of Dynamical SystemsIn this section are discussed some properties of dynamical systems,including stability. For observability and controllability, please refer toJagannathan (2006), and Goodwin and Sin (1984). If the original openloopsystem is controllable and observable, then a feedback control systemcan be designed to meet desired performance. If the system has certain