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

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

Example Ex:

Example (Continued)

Zubov’s Theorem (i) (ii) (iii) (iv)

Example Ex:

Example (Continued) Solution:  

Example (Continued)

Advanced Stability Theory bounded bounded ?

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.

Stability Definition of stability

Example Ex: Then Hence Then

Example (Continued)

Example (Continued)

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

Positive definite function Definition:

Decrescent positive definite decrescent Thoerem:

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

Stability theorem Stability theorem Thoerem:

Stability theorem (Continued)

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

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.

Example Ex:

Example (Continued)

Instability Theorem (Chetaev)

Linear time-varying systems and linearization

Example Ex:

Theorem Theorem: Proof : See Nonlinear systems analysis

Lyapunov function approach

Theorem Theorem: Proof : See Nonlinear systems analysis

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

Theorem Theorem : Proof : See Ch 4.3 of Nonlinear Systems