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Linear Algebra

1. Introduction to Linear Algebra
2. Matrices
3. Matrix factorizations
- 3.3. Singular Value Decomposition
4. Linear Systems
5. Spectral decomposition
- 5.1. Spectral decomposition of symmetric matrices
- 5.2. Sensitivity of spectral decomposition
6. Cayley-Hamilton theorem

Multivariable Calculus

7. Introduction to multi-variable calculus

Differential Geometry

8. Introduction to Differential Geometry

Vector and Tensor Algebra and Calculus

9. Tensor Algebra
10. Tensor Calculus in Euclidean Spaces
11. Tensor Invariants
12. Unitary and rotation tensors
13. Isotropic Tensors
14. Time derivative of integrals over moving domains
15. Calculus identities

Functional Analysis

16. Introduction to Functional Analysis
17. Dirac’s delta

Complex Calculus

18. Complex Analysis
19. Laplace Transform
- 19.1. Definition and Properties
- 19.2. Applications of Laplace Transform
20. Fourier Transforms

Calculus of Variations

21. Introduction to Calculus of Variations

Ordinary Differential Equations

22. Introduction to Ordinary Differential Equations
23. Linear systems

System Theory

24. Introduction to System Theory
25. Linear Time-Invariant Systems
26. LTI system response
27. LTI: stability and feedback
28. Observability and Detectability
29. Reachability and Controllability
30. Lyapunov equation

Control Theory

31. Introduction to control methods
- 31.1. Frequency domain control
- 31.2. Optimal control

Partial Differential Equations

32. Introduction to Partial Differential Equations
33. Elliptic equations
34. Parabolic equations
35. Hyperbolic equations
36. Navier-Cauchy equations
37. Navier-Stokes equations
38. Arbitrary Lagrangian-Eulerian description

Numerical Methods for PDEs

39. Introduction to numerical methods for PDEs
40. Finite Element Method
- 40.1. 1-dimensional Poisson equation
41. Finite Volume Method
42. Boundary Element Method

Boundary Methods for PDEs

43. Green’s function method

Optimization

44. Optimization

Reinforcement Learning

45. Introduction to Reinforcement Learning
46. Markov Processes
47. Methods of solution of MPD: DP and LP
48. Methods of solution of MPD: RL
49. Large or Continuous MDPs

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Lyapunov equation

30. Lyapunov equation#

Examples.

Controllability Gramian
Observability Gramian
In optimal control
Variance of the state in linear system forced by white Gaussian noise
…

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