# MA3H7 Control Theory

**Lecturer: **Tim Sullivan

**Term(s): **Term 2

**Status for Mathematics students: **List A

**Commitment: **30 one hour lectures

**Assessment: **100% 3-hour written examination

**Formal registration prerequisites: **None

**Assumed knowledge:**

- MA106 Linear Algebra
- MA133 Differential Equations or MA113 Differential Equations A
- MA244 Analysis III or MA258 Mathematical Analysis III

**Useful background:**

- ST112 Probability B
- MA254 Theory of ODEs
- MA259 Multivariable Calculus

**Synergies: **

- MA254 Theory of ODEs
- MA261 Differential Equations: Modelling and Numerics
- MA4K0 Introduction to Uncertainty Quantification
- MA4M2 Mathematics of Inverse Problems

**Content: **Will include the study of controllability, stabilization, observability, filtering and optimal control. Furthermore connections between these concepts will also be studied. Both linear and nonlinear systems will be considered. The module will comprise six chapters. The necessary background material in linear algebra, differential equations and probability will be developed as part of the course.

- Introduction to key concepts
- Background material
- Controllability
- Stabilization
- Observability and filtering
- Optimal control

**Aims: **The aim of the module is to show how, as a result of extensive interests of mathematicians, control theory has developed from being a theoretical basis for control engineering into a versatile and active branch of applied mathematics.

**Objectives: **By the end of the module the student should be able to:

- Explain and exploit role of controllability matrix in linear control systems
- Explain and exploit stabilization for linear control systems
- Derive and analyse the Kalman filter
- Understand linear ODEs and stability theory
- Understand and manipulate Gaussian probability distributions
- Understand basic variational calculus for constrained minimization in Hilbert space

**Books:
**E. D. Sontag,

*Mathematical Control Theory: Deterministic Finite Dimensional Systems,*Texts in Applied Mathematics No 6, Springer Verlag, 1990.

J. Zabczyk,

*Mathematical Control Theory: An Introduction*, Birkhauser, 1992.