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Revision:. Statement of Lyapunov matrix lemma? Statement of LaSalle’s invariance principle?. Lecture 7. Class K and KL functions Comparison lemma. Recommended reading. Khalil Chapter 3 (2 nd edition). Outline:. Class K, K and KL functions Properties of the above functions

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**Revision:**Statement of Lyapunov matrix lemma? Statement of LaSalle’s invariance principle?**Lecture 7**Class K and KL functions Comparison lemma**Recommended reading**• Khalil Chapter 3 (2nd edition)**Outline:**• Class K, K and KL functions • Properties of the above functions • Comparison lemma • Summary**Class K, K and KL functions**• Ubiquitous in stability analysis. • K and K classes of functions are used to upper and lower bound arbitrary nonlinear functions (e.g. positive definite V). • KL functions are used to bound the solutions of UGAS systems. • There are subtle and non-trivial relations between K and KL functions (e.g.comparison principle).**Class K and K functions**• Class K satisfy (we write K): • : [0,a) R 0 • is continuous • (0)=0 • is strictly increasing • Class K satisfy (we write K): • All of the above with a= • lims (s)=**Examples**Class K: Class K:**Class K?**A B D C**A relationship**• Note that the following holds: K K • The opposite does not hold in general ! • Example: 1(s)=arctan(s) and 2(s)=tan(s) satisfy 1,2 K and 1,2 K.**Class K ?**A C D B**Notation**• A composition of two functions 1,2 is denoted: • -1 denotes the inverse function of , that is: • Example: 1(s)=s2, 2(s)=arctan(s)**Property 1:**• K functions are one-to-one and, hence, they are globally invertible! K-1 • Moreover, we also have that K-1 K**Property 2:**• For arbitrary 1,2 K we have that 1,2 K12 K • We can take an arbitrary number of compositions of K functions and we still obtain a K function**Property 3:**• The following “weak triangle inequality” holds (s1+s2) (2s1)+(2s2) s1,s2 0 • Actually, given arbitrary , K, we have (s1+s2) (s1+(s1))+(s2+-1(s2)) s1,s2 0**Property 4:**• Suppose that a function V is: • positive definite; • radially unbounded Then, there exist 1,2 K**A consequence of Property 4**• We can fit the set c:={x: V(x) c} in the ball of radius 1-1(c) V(x) c |x| 1-1(c) • We can fit the ball of radius r in the set c, c=2(r): |x| r V(x) 2(r)**Graphical interpretation**2(|x|) 1(|x|) V(x) c c 1-1(c) r |x| r V(x) c |x| 1-1(c) V(x) 2(r)=c**Properties of K functions**• If 1,2 K with domain [0,a), then • 1-1 K with domain [0,1(a)) • 12 K • (s1+s2) 1(2s1)+1(2s2), s1,s2 a/2 • If V>0 then there exists r>0 such that**Class KL**• Class KL satisfy (we write KL): • : [0,a) X R 0 R 0 • is continuous. • s (0,a), (s,.) is decreasing to zero • t 0, (.,t) K NOTE: Sometimes we also require • strictly decreases in the second argument**Examples**• Class KL functions:**A property of KL, K functions**• Suppose KL, 1,2 K, then we have that 1( (2(s),t) ) KL • A similar statement holds for K functions but the domains need to be appropriately restricted. • The above property is used in the next lecture.**Another property of K functions**• Given an arbitrary K, there exists K that is locally Lipschitz and such that (s) (s) s [0,a) (s) (s)**An application to differential equations**• Let K be locally Lipschitz. Consider the following differential equation Then, its solution u(s,t) satisfies u(s,t) KL**A comparison principle**• Let K be locally Lipschitz and consider the differential inequality: Let the solution of the following equation Be denoted as (u0,t) KL. Then, we have V(t) (V0,t) t 0.**Summary**• Class K, K and KL functions are extremely important tools in stability analysis of nonlinear systems. • K functions are globally invertible. • The properties of these functions that we discussed are used in various proofs. • The comparison lemma gives us a bound on solutions of a differential inequality via an auxiliary differential equation.**Next lecture:**• Stability of time-varying systems. Homework: read Chapter 3 in Khalil

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