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

3. Example
Ex:

4. Example (Continued)

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

6. Example
Ex:

7. Example (Continued)
Solution:  

8. Example (Continued)

10. Advanced Stability Theory
bounded
bounded ?

11. Stability of time varying systems
(1)
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
Definition:

18. Decrescent
positive definite decrescent
Thoerem:

19. Decrescent (Continued)
Proof : see Nonlinear systems analysis
Ex:
p.d, radially unbounded,
not decrescent
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. Example Ex: Thus is uniformly stable. Mathieu eq. decrescent
positive definite
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)

27. Linear time-varying systems and linearization

28. Example
Ex:

29. Theorem
Theorem:
Proof : See Nonlinear systems analysis

30. Lyapunov function approach

31. Theorem
Theorem:
Proof : See Nonlinear systems analysis

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

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