MAE 270C

Optimal Control MAE 270C (Spring 2026 @ UCLA)

A rigorous introduction to optimal control theory, focusing on the mathematical foundations and engineering applications of decision-making over dynamical systems. Topics include the calculus of variations, Pontryagin’s Maximum Principle, and the Hamilton–Jacobi–Bellman framework, with an emphasis on both necessary and sufficient conditions for optimality. Students explore analytical and geometric perspectives on optimal trajectories and controls, including constrained problems and special structures such as bang-bang and singular controls. The course also introduces linear-quadratic regulator (LQR) problems and highlights connections to practical applications in mechanical, chemical, and electrical engineering systems.

Instructor's bio: Shahriar Talebi is an Assistant Professor in the Department of Mechanical & Aerospace Engineering at the UCLA Samueli School of Engineering (since July 2025). Before UCLA, he was a Postdoctoral Research Fellow at Harvard University and a contributor to the NSF AI Institute in Dynamic Systems (Dynamics AI). He received a Ph.D. in control theory from the University of Washington in 2023—focusing on constrained decision-making and control in complex systems—and an M.Sc. in Mathematics (differential geometry) from UW.

Course Logistics

  • Cross-listed as: MECH&AE-M270C / CH ENGR-M280C / EC ENGR-M240C
  • Instructor: Dr. Shahriar Talebi, s.talebi@ucla.edu, office: Eng IV, 38-137F
  • Lectures: Tuesdays & Thursdays, 12:00pm–1:50pm
  • Class Location: Boelter Hall 5252
  • Office Hours: Tuesdays & Thursdays 2:00pm–3:00pm (or by arrangement)

Main Textbook

  1. D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction, Princeton University Press, 2012. ISBN 978-0-691-15187-8. (We use the Main Version of the preliminary copy released by the author.)

Additional Resources

Learning Objectives

By the end of this course, students will be able to formulate and analyze optimal control problems arising in engineering systems, distinguishing between path and point optimization frameworks. They will develop a strong foundation in the calculus of variations, including deriving and interpreting the Euler–Lagrange equations, applying Hamiltonian and Legendre transformations, and assessing optimality through second-order conditions and conjugate points. Students will gain proficiency in applying Pontryagin’s Maximum Principle to characterize optimal controls, including bang-bang and singular solutions, and understanding their geometric and algebraic structure. They will also learn to use dynamic programming and the Hamilton–Jacobi–Bellman equation to obtain sufficient conditions for optimality and interpret value functions. By integrating these analytical tools, students will be equipped to solve and evaluate linear-quadratic regulator (LQR) problems and related optimal control applications, and to communicate their findings effectively through written and oral presentations.

Topics

  • Introduction (1 week) — The goals of the course; path optimization vs. point optimization; basic facts from finite-dimensional optimization.
  • Calculus of variations (3 weeks) — Examples of variational problems; Euler–Lagrange equation; Hamiltonian formalism and Legendre transformation; mechanical interpretation; constraints; second variation and Legendre’s necessary condition; weak and strong extrema; conjugate points and sufficient conditions.
  • The maximum principle (4 weeks) — Statement of the optimal control problem; variational argument and preview of the maximum principle; statement and proof of the maximum principle; relation to Lie brackets; bang-bang and singular optimal controls.
  • Hamilton–Jacobi–Bellman equation (1 week) — Dynamic programming; sufficient conditions for optimality; viscosity solutions of the HJB equation.
  • LQR problems and other topics (time permitting)

Lecture Notes — Table of Contents

Table of Contents
Lecture Topic Notes
1–2 The optimal control problem: dynamical systems, admissible inputs, and the cost functional (terminal + running cost) PDF
2 Review of finite-dimensional optimization: first- and second-order necessary & sufficient conditions PDF
3 Constrained optimization: regular points, the Lagrangian, and KKT-type conditions PDF
4 The basic variational problem: strong vs. weak local minima; first and second variations (Gâteaux derivatives) PDF
5 The Euler–Lagrange equation; the Hamiltonian and momentum via special cases PDF
6 The fixed end-point variational problem: derivation of the Euler–Lagrange equation and the smoothness condition PDF
7 Boundary conditions for variational problems; the Hamiltonian canonical form PDF
8 Integral-constrained variational problems (isoperimetric problems) PDF
9 Non-integral (holonomic) constrained variational problems PDF
10 Second-order sufficient conditions: conjugate points and the Riccati equation PDF
11 Piecewise-C1 extremals: Weierstrass–Erdmann corner conditions and the Weierstrass excess condition PDF
12–13 Statement of the optimal control problem and the variational argument previewing the maximum principle PDF
14 The fixed-time, free-endpoint problem: deriving first-order necessary conditions via the Hamiltonian PDF
15 Pontryagin’s Maximum Principle: free-time fixed-endpoint and free-time variable-endpoint optimal control PDF
15 (reference) Reference table: Pontryagin’s Minimum/Maximum Principle across problem types (after Athans & Falb) PDF
16 (slides) Efficient recursion and dynamic programming (slides) PDF
16–17 Applying the maximum principle: minimum-time optimal control, bang-bang control, and switching curves PDF

Grading

  • Reading Assignments — 0% (weekly)
  • Class and Discussion Participation — 10%
  • Midterm Exam — 40% (around week 7)
  • Final Project and Presentation — 50% (10-minute presentation due the last week; full submission due finals week)