Stanford AA203 Optimal and Learning-Based Control | Spring 2026 | Lecture 3: Calculus of Variations

Finding local minima in complex optimization problems involves identifying active constraint boundaries to establish candidate points, a process that evolves from finite-dimensional vector calculus into the Calculus of Variations to solve for entire signal trajectories.

• Active constraints — Boundary conditions met at a local minimum [05:17].

• Inactive constraints — Restrictions remaining below zero, which are disregarded because they do not influence the local minimum [05:45].

• Lagrange multipliers — Variables assigned to constraints that must be non-negative; for inactive constraints, these values are set to zero [17:07].

• Optimality requirements — Known as KKT conditions, these rules define necessary criteria that any local minimum must fulfill to be valid [22:14].

• Functionals — Mathematical rules that accept an entire signal trajectory as input and return a single scalar value for evaluation [43:16].

• Norms — Methods used to define closeness or distance between different function signals [46:37].

• Fundamental Theorem — A requirement stating that the variation of a functional must be zero at a local minimum [59:56].

• Integration by parts — A technique applied to manipulate the mathematical expression, allowing for the isolation of the state trajectory variation [1:13:22].

• Euler equation — The final necessary condition for a function to minimize a functional, obtained after setting the variation to zero [1:22:44].

• How does the Euler equation assist in determining an optimal path?

• What distinguishes indirect optimization methods from direct ones?