Domain of Attraction

1. Domain of Attraction • Remarks on the domain of attraction Complete (total) domain of attraction Estimate of Domain of attraction : Lemma : The complete domain of attraction is an open, invariant set. Its boundary is formed by trajectories

2. might be positive  could escape from Consider Let be such that and Is in ? What is a good ? Consider

3. Example Ex:

4. Example (Continued)

5. (i) (ii) (iii) (iv) Zubov’s Theorem

6. Example Ex:

7. Example (Continued) Solution:  

8. Example (Continued)

10. bounded bounded ? Advanced Stability Theory

11. (1) Stability of time varying systems • Stability of time varying systems f is piecewise continuous in t and Lipschitz in x. Origin of time varying : (i) parameters change in time. (ii) investigation of stability of trajectories of time invariant system.

12. Stability • Definition of stability

13. Example Ex: Then Hence Then

14. Example (Continued)

15. Example (Continued)

16. Example (Continued) There is another class of systems where the same is true – periodic system. Like Reason : it is always possible to find

17. Positive definite function • Positive definite function Definition:

18. positive definite decrescent Decrescent Thoerem:

19. Decrescent (Continued) Proof : see Nonlinear systems analysis p.d, radially unbounded, not decrescent Ex: not l.p.d, not decrescent p.d, decrescent, radially unbounded p.d, not decrescent, not radially unbounded Finally

20. Stability theorem • Stability theorem Thoerem:

21. Stability theorem (Continued)

22. Mathieu eq. decrescent positive definite Example Ex: Thus is uniformly stable.

23. Theorem • Remark : LaSalle’s theorem does not work in general for time-varying system. But for periodic systems they work. So (uniformly) asymptotically stable. Theorem Suppose is a continuously differentiable p.d.f and radially unbounded with Define Suppose , and that contains no nontrivial trajectories. Under these conditions, the equilibrium point 0 is globally asymptotically stable.

24. Example Ex:

25. Example (Continued)

26. Instability Theorem (Chetaev) • Instability Theorem (Chetaev)

27. Linear time-varying systems and linearization • Linear time-varying systems and linearization

28. Example Ex:

29. Theorem Theorem: Proof : See Nonlinear systems analysis

30. Lyapunov function approach • Lyapunov function approach

31. Theorem Theorem: Proof : See Nonlinear systems analysis

32. Converse (Inverse) Theorem & Invariance Theorem • Converse (Inverse) Theorem • i) if stable • ii) (uniformly asymptotically exponentially) stable • Invariance Theorem

33. Theorem Theorem : Proof : See Ch 4.3 of Nonlinear Systems