Optimal Control

Instructor: Dr. Gregory FRANCOIS, Senior Research Associate & Lecturer, LA EPFL, ME C2 401, tél: +41 21 6933844

Objectives: The objective is to familiarize the student with the theoretical and numerical issues associated with nonlinear programming and optimal control problems. This course, on the one hand, provides the basic analysis tools for future research, while on the other hand, it provides insight into numerical techniques so that they can be effectively used for solving practical problems.

Content: Slides

  • Nonlinear Programming – Existence of a solution. Convex optimization. KKT necessary and sufficient conditions of optimality. Interpretation of the Lagrange multipliers. Interior-point and SQP methods.
  • Calculus of Variations – Existence of a solution. Euler’s equation. Weierstrass’ necessary conditions. Sufficient conditions. Method of Lagrange multipliers for constrained extrema.
  • Optimal Control – Existence of an optimal control. Euler-Lagrange equations. Sufficient conditions. Interpretation of the adjoint variables. Hamilton-Jacobi-Bellman (HJB) equation. Problems with terminal constraints. Problems with input path constraints. Pontryagin maximum principle (PMP). Problems with state path constraints. Indirected numerical methods. Approximate solution methods (sequential and simultaneous approaches).
  • Applications – Parameter estimation. NCO Tracking.

Pedagogical vehicles: Ex-cathedra lectures . Research or Take-home project. Homework problem sets.

Nonlinear Programming: Lecture notes (Lectures #1 to #8)

Homework Problem Sets:


  • M. S. Bazaraa, H. D. Sherali, and C. M. Shetty. Nonlinear Programming: Theory and Algorithms. John Wiley and Sons, New York, 2nd Ed., 1993.
  • D. P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont (MA), 2nd Ed., 1999.
  • D. G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley, Reading (MA), 2nd Ed., 1984.

Calculus of Variations: Lecture notes (Lectures #9 to #16) (lagrange.m, dlagrange.m)

Homework Problem Sets:


    • M. I. Kamien, N. L. Schwartz. Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. (Part I) North-Holland, Advanced Textbooks in Economics, Volume 31, 2nd Ed., Amsterdam (The Netherlands), 1991.
    • D. E. Kirk. Optimal Control Theory: An Introduction. (Chapter 4) Prentice-Hall, Englewood Cliffs, 1970.
    • J. L. Troutman. Variational Calculus and Optimal Control: Optimization with Elementary Convexity. Undergraduate Texts in Mathematics, Springer, New York (NY), 2nd Ed., 1995.

Optimal Control: Lecture notes (Lectures #17 to #28) – Example (powerpoint, movie)


Homework Problem Sets:

Overview Papers:


    • A. E. Bryson Jr., Y.-C. Ho. Applied Optimal Control: Optimization, Estimation and Control. Hemisphere Publishing Corporation, Washington D.C., 1975.
    • D. E. Kirk. Optimal Control Theory: An Introduction. (Chapters 5, 6) Prentice-Hall, Englewood Cliffs, 1970.
    • L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko. The Mathematical Theory of Optimal Processes. Pergamon Press, New York, 1964.
    • K. L. Teo, C. J. Goh, K. H. Wong. A Unified Computational Approach to Optimal Control Problems. John Wiley & Sons, Pitman Monographs and Surveys in Pure and Applied Mathematics, New York (NY), 1991.

Appendices: Notations, Real Analysis, Convex Analysis, Linear Spaces, Ordinary Differential Equations, References


      • S. Boyd, and L. Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge (UK), 2nd Ed., 2006. (On-line version.)
      • H. K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River (NJ), 3rd Ed., 2002
      • E. Kreyszig. Introductory Functional Analysis with Applications John Wiley & Sons, New York, 1978.
      • R. T. Rockafellar. Convex Analysis. Princeton University Press, Princeton (NJ), 1970.
      • W. Rudin. Functional Analysis. McGraw Hill, New York (NY), 2nd Ed., 1991.
      • W. Rudin. Principles of Mathematical Analysis. McGraw Hill, New York (NY), 3rd Ed., 1976.
      • H. H. Sohrab. Basic Real Analysis. Birkhauser, Boston (MA), 2003.

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