Revisiting Optimal Control Theory
Published:
Revisiting optimal control theory for reference to my future self.
Optimal Control Theory
Let the dynamics and the cost be given by
\[\dot{\boldsymbol{x}}(t) = \boldsymbol{f} (\boldsymbol{x}(t), \boldsymbol{u}(t), t)\] \[\mathcal{J}(\boldsymbol{u}) = h(\boldsymbol{x}(t_f), t_f) + \int_{t_0}^{t_f} g(\boldsymbol{x}(t), \boldsymbol{u}(t), t) dt\]The Hamiltonian is given by
\[\mathcal{H}(\boldsymbol{x}(t), \boldsymbol{u}(t), \boldsymbol{\lambda}(t), t) \triangleq g(\boldsymbol{x}(t), \boldsymbol{u}(t), t) + \boldsymbol{\lambda}^T(t) \boldsymbol{f} (\boldsymbol{x}(t), \boldsymbol{u}(t), t)\]