Stanford AA203 Optimal and Learning-Based Control | Spring 2026 | Lecture 5: Computational Methods

Optimal control analysis expands from unconstrained to constrained inputs by shifting from finding stationary points to identifying global minima of the Hamiltonian. This methodology yields different control behaviors—bang-bang or bang-off-bang—depending on the cost function, while practical solutions rely on numerical boundary value solvers.

• Control constraints — Standard indirect methods use a Hamiltonian to solve for optimal inputs, but restricted inputs require finding a global minimum rather than just a stationary point [02:10].

• Optimality principle — Pontryagin’s Minimum Principle enforces that the chosen control input must minimize the Hamiltonian across all allowed values, rather than just equating the gradient to zero [22:16].

• Bang-bang control — Problems minimizing time result in "bang-bang" profiles, where the system toggles exclusively between maximum acceleration and maximum braking [53:05].

• Bang-off-bang profiles — Minimizing fuel consumption forces a "bang-off-bang" pattern, where the system utilizes maximum effort only intermittently and coasts at zero input to save resources [1:04:56].

• Singular conditions — When the Hamiltonian's gradient does not yield a clear control input, the system enters a "singular arc," which cannot be solved using standard steps [51:19].

• Numerical solvers — Practical implementation uses two-point boundary value solvers, which iteratively refine guesses of state variables to match initial and final boundary requirements [1:11:03].

• What distinguishes a singular arc from other control conditions in optimal control theory?

• How do shooting methods differ from collocation methods when solving boundary value problems?