Ozone (O3): (O)pen-source (O)DE and (O)ptimal control solver

Ozone is an open-source tool for solving ordinary differential equations (ODEs) and optimal control problems. It is designed to enable the integration of ODEs in gradient-based multidisciplinary design optimization (MDO) problems, where the ODE is a single component in the larger model and derivatives of the ODE integration process are required. Ozone can also be used for solving optimal control problems with direct transcription or indirect approaches. It is built on top of the OpenMDAO framework, which is described and documented here.

An ODE is of the form

\[\frac{\partial \textbf y}{\partial t} = \textbf f(t, \textbf y, \textbf x) , \qquad \textbf y(t_0) = \textbf y_0\]

where \(\textbf y\) is the vector of state variables (the variable being integrated), \(t\) is time (the independent variable), \(\textbf x\) is the vector of parameters (an input to the ODE), \(\textbf f\) is the ODE function, and \(\textbf y_0\) is the vector of initial conditions.

Ozone provides a large library of Runge–Kutta and linear multistep methods thanks to its use of a unified formulation called general linear methods. The list of methods can be found below. There are also 3 formulations for solving the ODE: time-marching, system-based (formulate the ODE equations as a nonlinear system), optimizer-based (formulate the ODE state variables and equations as design variables and constraints in an optimization problem).

Documentation

• Getting started
  • A simple ODE example
  • A simple optimal control example
• API
  • ODEFunction
  • ODEIntegrator
• Interacting with the integrator group
  • Specifying times
  • Specifying initial conditions
  • Specifying parameters
• Integration methods
  • List of methods
  • Convergence order
  • Manually checking convergence order
• Formulations
  • Comparison between formulations
• Examples library
  • Simple homogeneous ODE
  • Simple linear ODE
  • Simple nonlinear ODE
  • Projectile dynamics ODE
  • 2-D orbit ODE
  • 3-D orbit ODE

Indices and tables

• Index
• Module Index
• Search Page