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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. might be positive. . could escape from.
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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
might be positive could escape from Consider Let be such that and Is in ? What is a good ? Consider
Example Ex:
(i) (ii) (iii) (iv) Zubov’s Theorem
Example Ex:
Example (Continued) Solution:
bounded bounded ? Advanced Stability Theory
(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.
Stability • Definition of stability
Example Ex: Then Hence Then
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 • Positive definite function Definition:
positive definite decrescent Decrescent Thoerem:
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
Stability theorem • Stability theorem Thoerem:
Mathieu eq. decrescent positive definite Example Ex: 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:
Instability Theorem (Chetaev) • Instability Theorem (Chetaev)
Linear time-varying systems and linearization • Linear time-varying systems and linearization
Example Ex:
Theorem Theorem: Proof : See Nonlinear systems analysis
Lyapunov function approach • Lyapunov function approach
Theorem Theorem: Proof : See Nonlinear systems analysis
Converse (Inverse) Theorem & Invariance Theorem • Converse (Inverse) Theorem • i) if stable • ii) (uniformly asymptotically exponentially) stable • Invariance Theorem
Theorem Theorem : Proof : See Ch 4.3 of Nonlinear Systems