### CM622: Control Theory

Prof.
Sorin Micu

1. **Introduction:** Optimal control and controllability. Exemples in finite and infinite dimension

2. **Control of the finite dimensional systems:** Necessary and sufficient conditions for controllability
Variational methods to find the control
.

3. **Controllability in one dimension with Fourier series:** Riesz basis. Biorthogonal sequences.
Ingham’s inequalities
Applications to the controllability of the wave and heat equation.

4. **Controllability of the wave and heat equation in several dimensions:**
Unique continuation principle. Approximate controllability.
Observability. Controllability.
Multipliers
Hilbert Uniqueness Method (HUM)

5. **Stabilizability of the wave equation:**
Problem of stabilizability for the wave equation.
Relation between stabilizability and controllability

**Bibliography**

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1. S.A. Avdonin and S.A. Ivanov, Families of exponentials.
The method of moments in controlability problems for distributed parameter systems,
Cambridge Univ. Press, 1995.

2. T. Cazenave and A. Haraux, *Introduction aux problemes d’evolution semi-lineaires*,
Mathematiques et Applications, 1, Ellipses, Paris, 1990.

3. E.B.Lee and L. Marcus, *Foundation of Optimal Control Theory*,
The SIAM Series in Applied Mathematics, John Wiley &Sons, 1967.

4. J.L. Lions, *Controlabilite exacte, perturbations et stabilisations de systemes distribues*,
Vol.1&2, Masson, RMA, Paris, 1988.

5. R. Young, *An Introduction to Nonharmonic Fourier Series**, Academic Press, 1980.*

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