Applied Optimal Control(06）-Constrained Optimal Control (Pontryagin's minimum principle)

Keypoints:

• Pontryagin’s minimum principle: optimal control must minimize the Hamiltonian
• Minimum time problems($u$ is bounded):
  • $\nabla_u H$ does not contain $u$, i.e $H$ is linear in $u$. So, check the sign of $\nabla_u H$ and then determine $u$.
  • bang-bang control: $u$ is either $u_{min}$ or $u_{max}$.
• Minimum control effort
• Singular control: $\nabla_u H$ is zero at some point in the state space, which provides no information about the optimal control. The solution is to differentiate $\nabla_u H$ with respect to time enough times until $u$ appears.
• General constraints:
  • Necessary conditions in general constraints.
  • Inequality constraints on state only: differentiate the constraint enough times until $u$ appears.
• Corner conditions: smoothness of the optimal control at the corner points.

Based on the lecture notes by Dr. Marin Kobilarov on “Applied Optimal Control (2021 Fall) at Johns Hopkins University