The main goal of the course is to introduce students to the theory of optimal control for differential systems. We also intend to discuss specific problems which arise from real world applications in order to illustrate this remarkable theory.

**The learning outcomes of the course:**

The students will learn some basic notions and results in control theory, which are very useful for applied mathematicians. Even more, they will learn how to use these tools in solving specific problems.

**More detailed display of contents (week-by-week):**

Weeks 1-2: Linear and nonlinear differential systems (existence of solutions, continuous dependence of solutions on data, stability)

Week 3: Observability of linear autonomous systems (definition, observability matrix, necessary and sufficient conditions for observability)

Week 4: Observability of linear time varying systems (definition, observability matrix, numerical algorithms for observability)

Week 5: Input identification for linear systems (definition, the rank condition in case of autonomous systems, examples)

Week 6: Controllability of linear systems (definition, controllability of autonomous systems, controllability matrix, Kalman’s rank condition, the case of time varying systems)

Week 7: Controllability of perturbed systems (perturbations of the control matrix, nonlinear autonomous systems, time varying systems)

Week 8: Stabilizability (definition, state feedback, output feedback, applications)

Week 9: General optimal control theory (Meyer’s problem, Pontryagin’s minimum principle, examples)

Weeks 10-11: Linear quadratic regulator theory (introduction, the Riccati equation, perturbed regulators, applications)

Week 12: Time optimal control (the general problem, linear systems, bang-bang control, applications)

**Books:**

1. N.U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, 2006.

2. E.B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley, 1967.

**Office hours:** by appointment

*Gheorghe Morosanu*

Zrinyi u. 14, Third Floor, Office 310

morosanug [at] ceu.hu

for differential systems. We also intend to discuss specific problems which arise

from real world applications in order to illustrate this remarkable theory.

9. The learning outcomes of the:

The students will learn some basic notions and results in control theory, which are

very useful for applied mathematicians. Even more, they will learn how to use these

tools in solving specific problems.

10. More detailed display of contents.

Week 1: Linear differential systems (existence of solutions, variation of constants

formula, continuous dependence of solutions on data, stability)

Week 2: Nonlinear differential systems (local and global existence of solutions,

continuous dependence on data, differential inclusions)

Week 3: Observability of linear autonomous systems (definition, observability matrix,

necessary an sufficient conditions for observability)

Week 4: Observability of linear time varying systems (definition, observability matrix,

numerical algorithms for observability)

Week 5: Input identification for linear systems (definition, the rank condition in case

of autonomous systems, examples)

Week 6: Controllability of linear systems (definition, controllability of autonomous

systems, controllability matrix, Kalman’s rank condition, the case of time varying

systems)

Week 7: Controllability of perturbed systems (perturbations of the control matrix,

nonlinear autonomous systems, time varying systems)

Week 8: Stabilizability (definition, state feedback, output feedback, applications)

Week 9: General optimal control theory (Meyer’s problem, Pontryagin’s Minimum

Principle, examples)

Weeks 10-11: Linear quadratic regulator theory (introduction, the Riccati equation,

perturbed regulators, applications)

Week 12: Time optimal control (the general problem, linear systems, bang-bang

control, applications)

Books:

1. N.U. Ahmed, Dynamic Systems and Control with Applications, World Scientific,

2006.

2. E.B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley,

1967.

Office hours: by appointment

Gheorghe MoroBrief introduction to the course:

Basic principles and methods concerning dynamic control systems are discussed.

The main concepts (observability, controllability, stabilizability, optimality

conditions, etc.) are addressed, with special emphasis on linear systems and

quadratic functionals. Many applications are discussed in detail.

The course is designed for students oriented to Applied Mathematics.

8. The goals of the course:

The main goal of the course is to introduce students to the theory of optimal control

for differential systems. We also intend to discuss specific problems which arise

from real world applications in order to illustrate this remarkable theory.

9. The learning outcomes of the:

The students will learn some basic notions and results in control theory, which are

very useful for applied mathematicians. Even more, they will learn how to use these

tools in solving specific problems.

10. More detailed display of contents.

Week 1: Linear differential systems (existence of solutions, variation of constants

formula, continuous dependence of solutions on data, stability)

Week 2: Nonlinear differential systems (local and global existence of solutions,

continuous dependence on data, differential inclusions)

Week 3: Observability of linear autonomous systems (definition, observability matrix,

necessary an sufficient conditions for observability)

Week 4: Observability of linear time varying systems (definition, observability matrix,

numerical algorithms for observability)

Week 5: Input identification for linear systems (definition, the rank condition in case

of autonomous systems, examples)

Week 6: Controllability of linear systems (definition, controllability of autonomous

systems, controllability matrix, Kalman’s rank condition, the case of time varying

systems)

Week 7: Controllability of perturbed systems (perturbations of the control matrix,

nonlinear autonomous systems, time varying systems)

Week 8: Stabilizability (definition, state feedback, output feedback, applications)

Week 9: General optimal control theory (Meyer’s problem, Pontryagin’s Minimum

Principle, examples)

Weeks 10-11: Linear quadratic regulator theory (introduction, the Riccati equation,

perturbed regulators, applications)

Week 12: Time optimal control (the general problem, linear systems, bang-bang

control, applications)

Books:

1. N.U. Ahmed, Dynamic Systems and Control with Applications, World Scientific,

2006.

2. E.B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley,

1967.

Office hours: by appointment

Gheorghe Morosanu

Basic principles and methods concerning dynamic control systems are discussed.

The main concepts (observability, controllability, stabilizability, optimality

conditions, etc.) are addressed, with special emphasis on linear systems and

quadratic functionals. Many applications are discussed in detail.

The course is designed for students oriented to Applied Mathematics.

8. The goals of the course:

The main goal of the course is to introduce students to the theory of optimal control

for differential systems. We also intend to discuss specific problems which arise

from real world applications in order to illustrate this remarkable theory.

9. The learning outcomes of the:

The students will learn some basic notions and results in control theory, which are

very useful for applied mathematicians. Even more, they will learn how to use these

tools in solving specific problems.

10. More detailed display of contents.

Week 1: Linear differential systems (existence of solutions, variation of constants

formula, continuous dependence of solutions on data, stability)

Week 2: Nonlinear differential systems (local and global existence of solutions,

continuous dependence on data, differential inclusions)

Week 3: Observability of linear autonomous systems (definition, observability matrix,

necessary an sufficient conditions for observability)

Week 4: Observability of linear time varying systems (definition, observability matrix,

numerical algorithms for observability)

Week 5: Input identification for linear systems (definition, the rank condition in case

of autonomous systems, examples)

Week 6: Controllability of linear systems (definition, controllability of autonomous

systems, controllability matrix, Kalman’s rank condition, the case of time varying

systems)

Week 7: Controllability of perturbed systems (perturbations of the control matrix,

nonlinear autonomous systems, time varying systems)

Week 8: Stabilizability (definition, state feedback, output feedback, applications)

Week 9: General optimal control theory (Meyer’s problem, Pontryagin’s Minimum

Principle, examples)

Weeks 10-11: Linear quadratic regulator theory (introduction, the Riccati equation,

perturbed regulators, applications)

Week 12: Time optimal control (the general problem, linear systems, bang-bang

control, applications)

Books:

1. N.U. Ahmed, Dynamic Systems and Control with Applications, World Scientific,

2006.

2. E.B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley,

1967.

Office hours: by appointment

Gheorghe Morosanu