Optimal Control

In optimal control theory, we wish to solve problems of the form

\[\begin{split}\begin{aligned} \min_{\boldsymbol{u}} J &= \int_{t_0}^{t_f} L(t, \boldsymbol{x}, \boldsymbol{u}) \; dt + \eta(\boldsymbol{x}(t_f)) \\ \text{Subject to:}& \\ \dot{\boldsymbol{x}} &= \mathbf{f}(t, \boldsymbol{x}, \boldsymbol{u}) \\ 0 &= \boldsymbol{\phi}(t_0, \boldsymbol{x}(t_0)) \\ 0 &= \boldsymbol{\xi}(t_f, \boldsymbol{x}(t_f)) \\ \boldsymbol{g}_{lower} &\leq \boldsymbol{g}(\boldsymbol{x}, \boldsymbol{u}, t) \leq \boldsymbol{g}_{upper} \end{aligned}\end{split}\]

Setting up a Problem

Initialize a new Optimal Control Problem with

>>> import beluga
>>> ocp = beluga.OCP()
<beluga.problem.OCP object at 0x0...>

Define parameters in the Optimal Control Problem with

>>> ocp.<parameter>(value1, value2, ..., valuen)

for example

>>> ocp.independent('t', 's')

defines the independent variable \(t\) with units of \(s\) representing “seconds”. Info for available parameters are included in the OCP documentation.

beluga Examples

Examples are organized into folders for various disciplines. For more optimal control specific examples, see the following.

Pure control path-constrained:

• Aly Chan Totally Singular
• Financial Oscillator
• Goddard Rocket
• Hang Glider
• Hypersonic Nose Cone
• Mall’s Oscillator
• Moon Lander
• One-Loop Circuit
• Spacecraft Detumbling
• Space Shuttle

Pure state path-constrained:

• Bryson-Denham

Mixed state-control path-constrained:

• Rayleigh Oscillator

Staged vehicles and discontinuous functions:

• BFR

Custom function implementation:

• Zermelo’s Problem

Base Class Reference