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Existence and Uniqueness Theorem (Local Theorem)

Existence and Uniqueness Theorem (Local Theorem). Existence and Uniqueness Theorem Local Theorem. Proof :. Existence and Uniqueness Theorem (proof). Existence and Uniqueness Theorem (proof). Existence and Uniqueness Theorem (proof). Existence and Uniqueness Theorem (Global Theorem).

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Existence and Uniqueness Theorem (Local Theorem)

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  1. Existence and Uniqueness Theorem (Local Theorem) • Existence and Uniqueness Theorem • Local Theorem Proof :

  2. Existence and Uniqueness Theorem (proof)

  3. Existence and Uniqueness Theorem (proof)

  4. Existence and Uniqueness Theorem (proof)

  5. Existence and Uniqueness Theorem (Global Theorem) • Global Theorem

  6. Continuous dependence(Case 1) • Continuous dependence on initial condition and system parameters • Case 1 Definition :

  7. Continuous dependence(Case 2) • Case 2 Definition :

  8. Continuous dependence(Case 2) Proof :

  9. Continuous dependence(Continued)

  10. Differentiability of solution and Sensitivity equations • Differentiability of solution and Sensitivity equations. • Scenario ?

  11. Calculation • Calculation

  12. Calculation(Continued)

  13. i Examples v i + Typical Non-linear Driving point Characteristic RESISTIVE ELEMENT C I V - Basic Oscillator Circuit

  14. Example (Continued)

  15. Example (Continued)

  16. 4. Lyapunov Stability • Motivation & Definition Assume that is such that the solution exists for all  0 is an equilibrium point. “  what will happen we start close to 0? ” • Continuously dependent on the initial condition such that Here is bounded. However what happens on infinite time interval?

  17. Lyapunov Stability (Continued) Ex: stable asymptotically stable unstable Stability : continuously dependent on initial condition at infinity.

  18. Methods of nonlinear analysis • Methods of nonlinear analysis • Qualitative method : finding properties of solution without actually finding the solution. • Quantitative method : concerned with explicitly finding closed forms of approximate or exact solution. • Computer(numerical) method : centered around developing numerical technique for solution on computer. Liapunov stability  most comprehensive idea

  19. Types of Stability • Stability • Lyapunov stability • Input-output stability • Orbital stability • Stability under permanently acting perturbation • Lagrange stability

  20. Ex : Stability Theorem • Stability Theorem

  21. Stability (Example) Ex:

  22. Asymptotic Stability Theorem: Let be an equilibrium point of . Let be a continuously differentiable positive definite function in and . Then if 0 is stable. If is locally negative definite in , then 0 is asymptotically stable. If , is positive definite, is negative definite and as ( is radially unbounded), then 0 is globally asymptotically stable.

  23. Proof for Asymptotic Stability Proof :

  24. Proof (Continued)

  25. Proof (Continued)

  26. Examples Ex : 1) 2)

  27. Examples 3)

  28. estimate of the region of attraction

  29. Examples Ex: (a)

  30. Examples (b)

  31. Examples

  32. Since the trajectories starting to the right of the hyperbola do not Reach the origin, the origin is not globally asymptotically stable Examples “ The trajectories can’t cross the branch of the hyperbola which lies in the first quadrant, in the direction toward the axes. ”

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