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Control & dynamics
6 Jul 2023

Applied Optimal Control(02) - Constrained Optimization Basics(Hamiltonian)

Keypoints:

  • Equality constraints
    • Differentiate the constraint equation to get the relationship between $\dot{x}$ and $\dot{u}$ for first-order necessary condition
    • Lagrangian multiplier: $\lambda$ is the costate variable. It’s to transform the constrained optimization problem to unconstrained optimization problem.
      • Hamiltonian: $H(x,u,\lambda) = L(x,u) + \lambda^T f(x,u)$
      • $dL = 0$ equals to $dH = 0$. With $dH = 0$, we can get the first-order necessary condition and solve the state.
      • General optimization setting.
    • Sufficient condition in practice: express $dy$ in terms of $du$, then substitute $du$ into $dL = 0$ to get the second-order necessary condition to get a less strict condition.
  • Inequality constraints

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

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