Revision:

1. Revision: Statement of Lyapunov matrix lemma? Statement of LaSalle’s invariance principle?

2. Lecture 7 Class K and KL functions Comparison lemma

3. Recommended reading • Khalil Chapter 3 (2nd edition)

4. Outline: • Class K, K and KL functions • Properties of the above functions • Comparison lemma • Summary

5. 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).

6. 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)=

7. Examples Class K: Class K:

8. Class K? A B D C

9. 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.

10. Class K ? A C D B

11. 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)

12. Properties of K functions

13. Property 1: • K functions are one-to-one and, hence, they are globally invertible!  K-1 • Moreover, we also have that  K-1 K

14. Property 2: • For arbitrary 1,2 K we have that 1,2 K12 K • We can take an arbitrary number of compositions of K functions and we still obtain a K function

15. 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

16. Property 4: • Suppose that a function V is: • positive definite; • radially unbounded Then, there exist 1,2 K

17. 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)

18. 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

19. Properties of K functions • If 1,2 K with domain [0,a), then • 1-1 K with domain [0,1(a)) • 12 K • (s1+s2) 1(2s1)+1(2s2),  s1,s2 a/2 • If V>0 then there exists r>0 such that

20. 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

21. Examples • Class KL functions:

22. 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.

23. 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)

24. 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

25. 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.

26. 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.

27. Next lecture: • Stability of time-varying systems. Homework: read Chapter 3 in Khalil

28. Thank you for your attention!