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	

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

II. Full status faculty members
Name and surname	Scientific	Didactic	Chair	Type	
	Degree	Degree		C	S	L	
 Mureşan Marian	Dr.	Prof.	Analysis and optimization 	*	*		

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

III. Course objectives
The aim of this course is to introduce the students to basic notions and results to classical and modern optimal control problems including the Pontryagin maximum principle.  In the second part of this course we discuss modern approaches to the topic by the methods of nonsmooth analysis. Applications are introduced to enlighten concepts and limits of the results.	

IV. Course contents
Introduction and motivation: basic concepts, mathematical formulation of the control problem, controllability, optimal control, example. Controllability: some simple general cases, the linear case, controllability for nonlinear autonomous systems, special controls. Linear autonomous time-optimal control problems: summary of results, the existence of a time-optimal control, extremal controls, the bang-bang principle, normality and uniqueness of the optimal control, applications, the converse of the maximum principle, extensions to more  general problems. Existence theorems for optimal control problems: three examples, basic approach to existence proofs, existence for special control classes, existence under convexity assumptions, existence for systems linear in state, applications. Necessary conditions for optimal control (the Pontryagin maximum principle, PMP): the PMP for autonomous systems, applying the PMP, a dynamic system approach to the proof of PMP, the PMP for more complicated problems.  Multifunctions: elementary properties, concepts of measurability to multifunctions, selections, functions of two variables and measurability, the implicit function theorem. Differential inclusions. Relaxation. Variational principles: Ekeland variational principle, Borwein-Preiss variational principle. Elements of nonsmooth analysis: generalized gradients, calculus rules, cones. Existence of solutions to certain Mayer and Bolza problems: a Mayer problem, the set of trajectories, the existence of solutions. Necessary conditions to certain nonlinear problems: controllability, the maximum principle. 	

V. Bibliography 
1. Cesari, L., Optimization-Theory and Applications. Problems with ordinary differential equations, Springer-Verlag, New York, 1983.
2. Clarke, F., Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990.
3. Clarke, F., Ledyaev, Yu., Stern, R., and Wolenski, P., Nonsmooth Analysis and Optimal Control, Springer-Verlag, New York, 1998. 
4. Macki, J., Strauss, A., Introduction to Optimal Control Theory, Springer-Verlag, New York, 1982. 
5. Mureşan, M., Introducere in Control Optimal, Risoprint, Cluj-Napoca, 1999.
6. Mureşan, M., Analiză Nenetedă şi Aplicaţii, Risoprint, Cluj-Napoca, 2001.
7. Strauss, A., An Introduction to Optimal Control Theory, Springer-Verlag, Berlin, 1968.	

VI. Thematic of didactic activities per weeks
Lectures

1. Introduction and motivation   [4, Chap. 1] [1, Chap. 6]
       1.1. Basic concepts  
       1.2. Mathematical formulation of the control problem 
       1.3. Controllability 
       1.4. Optimal control, example 
        1.5 Exercises 
2. Controllability   [4, Chap. 2]
       2.1. Some simple general cases 
       2.2. The linear case 
       2.3. Controllability for nonlinear autonomous systems
       2.4. Special controls
       2.5. Exercises 
3-4. Linear autonomous time-optimal control problems [4, Chap. 3]
       3.1. Summary of results 
       3.2. The existence of a time-optimal control; extremal controls; the bang-bang principle  
       3.3. Normality and uniqueness of the optimal control 
       4.1. Applications  
       4.2. The converse of the maximum principle  
       4.3. Extensions to more general problems 
       4.4. Exercises 
5. Existence theorems for optimal control problems [4, Chap. 4]
       5.1. Three examples  
       5.2. Basic approach to existence proofs   
       5.3. Existence for special classes   
       5.4. Existence under convexity assumptions  
       5.5. Existence for systems linear in state   
       5.6. Applications  
       5.7. Exercises 
 6. Necessary conditions for optimal control (the Pontryagin maximum principle) [4, Chap. 5]
       6.1. The PMP for autonomous systems   
       6.2. Applying the PMP    
       6.3. Exercises
7. More on the Pontryagin maximum principle [4, chap. 5]
       7.1. A dynamic system approach to the proof of PMP    
       7.2. The PMP for more complicated problems   
       7.3. Exercises   
8-9. Multifunctions [5, Chap. 2] [6, Chap. 2] [1, Chap. 8]
       8.1. Elementary properties   
       8.2. Concepts of measurability to multifunctions    
       9.1. Selections   
       9.2. Functions of two variables and measurability   
       9.3. The implicit function theorem
       9.4. Exercises
10. Differential inclusions  [5, Chap. 2]  [6, Chap. 3] 
       10.1. Differential inclusions  
       10.2. Relaxation 
       10.3. Exercises
11. Elements of nonsmooth analysis [5, Chap. 5] [6, Chap. 5] [3, Chap. 2]
       11.1. Generalized gradients
       11.2. Calculus rules
       11.3. Cones
       11.4. Exercises 
12. Variational principles  [5, Chap. 6] [6, Chap. 2]  
       12.1. Ekeland variational principle  
       12.2. Borwein-Preiss variational principle 
       12.3. Exercises
13. Existence of solutions to certain Mayer and Bolza problems  [5, Chap. 7] [1, § 1.7-1.9]   
       13.1. A Mayer problem 
       13.2. The set of trajectories 
       13.3. The existence of solutions   
       13.4. Exercises 
14. Necessary conditions to certain nonlinear problems  [5, Chap. 8] [1, Chap. 7] 
       14.1. Controllability   
       14.2. The maximum principle   
       14.3. Exercises 

Seminars

1. Stabilization of a material point moving on a straight line under a limited external force
2. Stabilization of a material point under an elastic force and a limited external force
3. Minimum time stabilization of a reentry vehicle
4. Soft landing on the Moon
5. Three more problems on stabilization of a material point moving on a straight line 
6. Optimal economic growth
7. A problem of length and area and a problem on the shape of hanging rope
8. The navigation problem
9. Lower continuous multifunctions, continuous selection
10. Topological properties of solutions to differential inclusions
11. Nonsmooth calculus rules applied to several function
12. Variational principles in Hilbert spaces
13. Necessary conditions for state constrained problems: convex velocity case
14. Free time problems with state constrains	

VII. Didactic methods used
Lectures, presentations, conversations, exercises, individual study, homework assignments.	

VIII. Assessment
The activity ends with a written final. The exam subjects have theoretical questions from all the studied topics, and one problem, among the problems studied at the course or the seminar. There is an evaluation of the overall seminar activity, and the documentations prepared for the programming projects will be evaluated. The final grade is the mean of the grades mentioned above. The final grade = Homework 30% + Seminar Activity 20%+Written Exam 50%.	

IX. Additional references
1. Rockafellar, R., Wets, R., Variational Analysis, Springer-Verlag, Berlin, 1998.
2. Vinter, R., Optimal Control, Birkhäuser, Boston, 2000.	

	Date:	2007, April 16							Course responsible,
Prof. dr. Marian Mureşan