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\noindent Syllabus for Qualifying Exam, Spring 2000

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\noindent Madalena Chaves

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\noindent Topic 1: Control Theory

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\noindent 1.  Linear time-invariant finite-dimensional systems

- formula for solution

- continuous and discrete time

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\noindent 2. Controllability of linear systems

- time-invariant controllability rank condition

- Kalman controllability decomposition

- Hautus Lemma

- controllability under sampling

- computing controls; finite horizon least squares 

- time-varying controllability rank condition


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\noindent 3. Controllability of nonlinear systems

- local controllability

- accessibility rank condition

- reversible systems

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\noindent 4. Outputs

- observability of time-invariant linear systems

- sampled observability

- local observability

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\noindent 5. Feedback

- constant linear feedback; Pole-Shifting Theorem

- disturbance rejection and invariance

- stability and stabilizability

- control-Lyapunov functions

- observers and detectability; dynamic feedback

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\noindent 6. Linear Quadratic control

- LQ systems

- Deterministic Kalman filtering

- infinite time problems

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\noindent Topic 2: Numerical Analysis

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\noindent 1. Polynomial Approximation

- Lagrange interpolation

- Cubic Hermite interpolation

- Piecewise polynomial approximation

- Some error results

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\noindent 2. Numerical Quadrature

- (composite) Trapezoidal, Simpson's and midpoint rules

- Derivation and error formulas

- Basic results of Gaussian quadrature formulas

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\noindent 3. Numerical methods for ordinary differential equations

- Derivation and error estimates for one-step methods (e.g., Euler's method)

- Multistep methods (examples of explicit and implicit methods)

- Predictor-corrector methods

- Consistency, stability, and convergence of multistep methods

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\noindent 4. Finite Difference methods for partial differential equations

- Laplace's equation: 5 point difference scheme, error estimates using
       discrete maximum principle

- Simple difference approximations of time dependent equations \hfill\break
  (transport, heat, and wave equations)

- Error analysis by the maximum principle

- Von Neumann stability condition

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\noindent 5. Finite Element methods for elliptic partial differential equations

-  Standard variational formulation of boundary value problems with Dirichlet
or Neumann boundary conditions

-  Energy norm error estimates

-  Solution of the resulting matrix equations (existence and uniqueness
      of solutions, iterative schemes)



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