JuDO Packages

The JuDO ecosystem consists of two main packages arranged in a layered architecture.

Macro / Function	Purpose
`DynModel(optimizer)`	Create a dynamic optimization model
`@phase(model, t)`	Declare the independent variable (time)
`@variable(model, bounds, DefinedOn(t))`	Declare a trajectory variable
`initial(x)`, `final(x)`	Access trajectory endpoints in constraints
`derivative(x)`	Declare the time derivative $\dot{x}(t)$
`@objective(model, sense, integral(expr))`	Lagrange (running) objective
`@objective(model, sense, final(expr))`	Mayer (terminal) objective
`JuDO.warmstart!(model, interpolant, x)`	Provide an initial guess trajectory
`JuDO.optimize!(model)`	Solve the problem
`dyn_value(model, x)`	Extract solution as a callable `t -> value`

DynOptInterface.jl

The mathematical abstraction layer (DOI).

DynOptInterface.jl (DOI) is to dynamic optimization what MathOptInterface (MOI) is to static mathematical programming. It defines a standard intermediate representation for Dynamic optimization Problems (DOPs) that solvers can attach to.

The general problem form DOI represents:

\min_{x,\,y(\cdot)} \quad f_o(x) + b_o\!\left(y(t_0), y(t_f), t_0, t_f, x\right) + \sum_i \int_{t_0^{(i)}}^{t_f^{(i)}} d_o\!\left(\dot y(t), y(t), t, x\right) \mathrm{d}t

subject to finite-dimensional constraints $f_c(x) \in S$ , boundary constraints $b_c(\cdot) \in B$ , and dynamic/path constraints $d_c(\dot y, y, t, x) \in D$ on each phase $[t_0^{(i)}, t_f^{(i)}]$ .

Key abstractions:

Type	Purpose
`Phase`	A time interval $[t_0, t_f]$ with variable boundaries
`DynamicVariable`	An optimizable trajectory $y(t)$
`Derivative`	Time derivative $\dot y(t)$
`Initial`, `Final`	Boundary-value operators
`Integral`, `MultiPhaseIntegral`	Lagrange cost terms
`Bolza`	Combined Mayer + Lagrange objective

Source Code Documentation

Current version: v0.3.0

Interesso.jl

The dynamic optimization solver.

Interesso.jl is a solver we develop that could attach to the DynOptInterface. It transcribes continuous-time DOPs into large-scale Nonlinear Programs (NLPs) using an Integrated Residuals Method (IRM) and dispatches them to Ipopt.

Apart from Interesso, we have also interfaced CTDirect.jl in OptimalControl toolbox under DOI, which demonstrates the capability of attaching existing solvers under DOI.

Features:

Multi-phase dynamic optimization
Explicit and implicit ODE dynamics
Free final time
NLP warm-starting via JuDO.warmstart!
Post-solve trajectory interpolants (callable solution functions)
Plots.jl extension for solution visualisation

Example problems included:

Example	Description
Cart-pole swing-up	Classic pendulum-on-cart control
Space shuttle reentry	6-state aerospace trajectory (benchmark: 34.14° crossrange)
Van der Pol oscillator	Nonlinear oscillator
Double integrator	LQR-type minimum-effort control
Bang-bang control	Switching control
Orbit raising	Orbital mechanics
Fuller's problem	Singular arc
Hyper-sensitive	Stiff boundary layer
Two-link robot arm	Robotics trajectory
Linear bicycle model	Vehicle dynamics

Source Code

Julia 1.12 or later is required.

CC BY-SA 4.0 Septimia Zenobia. Last modified: May 21, 2026. Website built with Franklin.jl and the Julia programming language.

JuDO Packages

Design philosophy

JuDO.jl

DynOptInterface.jl

Interesso.jl

Source Code