Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Matematica
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MMA1006	Optimal Control Theory	

Semester	Hours: C+S+L	Category	Status	
3	2+2+0	speciality	optional	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
MURESAN Marian	Ph.D.	Prof.	Mathematical Analysis and Optimization	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
Introducing students to the field of variational calculus and optimal control: formulating a problem on variational calculus and optimal control, its study and solving. 	

IV. Course contents
1. Introduction 
1.1. Variational calculus. Problems and mathematical framework.  
1.2. Optimal control. Problems and mathematical framework. 
2. Variational calculus. 
2.1. Necessary conditions: Euler-Lagrange equation, Weierstrass, Legendre, Erdman, and  Jacobi conditions; conditions involving the Gateaux derivative, transversality condition. 
2.2. The existence theorem of Tonelli. 
2.3. Lavrentiev phenomenon.
2.4. Sufficiency conditions of Weierstrass and of Hamilton-Jacobi type.
3. Controlul optimal of linear systems and linear-quadratic systems.  
3.1. Bang-bang theorems
3.2. Controlability and observability. Kalman theorem.
3.3. Maximum principle.  
3.4. Synthesis. 
3.5. Duality 
4. Applications to economy and engeenering. 	

V. Bibliography
1. Cesari, L., Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Springer, New-York, 1983. 
2. Clarke, F. H., Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990. 
3. Hestenes, M. R., Calculus of Variations and Optimal Control Theory, Wiley, New-York, 1966. 
4. Lee, E. B., Markus, L., Foundations of Optimal Control Theory, Wiley, New-York, 1967. 
5. Loewen, P. D., Optimal Control and Nonsmooth Analysis, AMS, Providence, 1993.
6. Muresan, M., Introducere in control optimal, Risoprint, Cluj-Napoca, 1999.
7. Muresan, M. A Concret Approach to Classical Analysis, Springer, New York, 2008.
8. Vinter, R. B., Optimal Control, Notes, 75p.

VI. Thematic of didactic activities per weeks

VII. Didactic methods used

VIII. Assessment
Written and oral exam.	

IX. Additional references

	Dean,							Chair head,

	Date,							Course responsible,
								Prof. MURESAN Marian, Ph.D.