Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMA1006	Optimal Control Theory	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	3	2+1+0	speciality	optional	

Teaching Staff in Charge	Prof. MURESAN Marian, Ph.D., mmarian@math.ubbcluj.ro	
Aims	Introducing students to the field of variational calculus and optimal control: formulating a problem on variational calculus and optimal control, its study and solving. 	
Content	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. 	
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 Concrete Approach to Classical Analysis, Springer, New York, 2009.
8. Vinter, R. B., Optimal Control, Notes, 75p.
	
Assessment	Written and oral exam.