OptimalControl.jl
The OptimalControl.jl package is the root package of the control-toolbox ecosystem. The control-toolbox ecosystem gathers Julia packages for mathematical control and applications. It aims to provide tools to model and solve optimal control problems with ordinary differential equations by direct and indirect methods, both on CPU and GPU.
Installation
See Installation — one package to add, plus a handful of optional pieces (a solver, plotting, flows, saving) loaded only when you use them.
Basic usage
Let us model, solve and plot a simple optimal control problem.
julia
using OptimalControl
using NLPModelsIpopt
using Plots
ocp = @def begin
t ∈ [0, 1], time
x ∈ R², state
u ∈ R, control
x(0) == [-1, 0]
x(1) == [0, 0]
ẋ(t) == [x₂(t), u(t)]
∫( 0.5u(t)^2 ) → min
end
sol = solve(ocp)
plot(sol; layout=:group)
For more details, see the energy minimisation example.
The
@defmacro defines the problem. See the abstract syntax guide.The
solvefunction has many options. See the solve overview.The
plotfunction is flexible. See the plot guide.
Where to go next
| Section | What's there |
|---|---|
| Getting started | Install, solve your first problem, then a longer guided tour covering both the direct and indirect methods. |
| Modelling | The @def DSL, the macro-free functional API, control-free (parameter estimation) problems, and problem introspection. |
| Solve (direct) | Discretize-and-transcribe methods: choosing a strategy, initial guesses, options, explicit mode, GPU. |
| Results | Read a Solution — trajectories, costate, duals, status — plot it, save it, reload it. |
| Flows (indirect) | The Pontryagin Maximum Principle as code: build, integrate, and shoot with Hamiltonian flows. |
| Geometry | The Lie-theoretic tools (Lift, ad, Poisson, @Lie) behind singular-control problems. |
| Examples | Six worked problems, direct and indirect, from energy minimisation to state constraints. |
| API reference | Every re-exported symbol, organised by theme. |
| Migrating to v2.1 | What changed since v2.0 and how to update your code. |
Mathematical formulation
Optimal control problems are stated in Bolza form — a cost functional combining a terminal (Mayer) and an integral (Lagrange) term, subject to dynamics and box/path/boundary constraints, with optionally free times and extra optimisation variables. See Formulation for the full mathematical setting.
Citing us
If you use OptimalControl.jl in your work, please cite us:
Caillau, J.-B., Cots, O., Gergaud, J., Martinon, P., & Sed, S. OptimalControl.jl: a Julia package to model and solve optimal control problems with ODE's. doi.org/10.5281/zenodo.13336563
or in bibtex format:
bibtex
@software{OptimalControl_jl,
author = {Caillau, Jean-Baptiste and Cots, Olivier and Gergaud, Joseph and Martinon, Pierre and Sed, Sophia},
doi = {10.5281/zenodo.13336563},
license = {MIT},
title = {{OptimalControl.jl: a Julia package to model and solve optimal control problems with ODE's}},
url = {https://control-toolbox.org/OptimalControl.jl}
}Contributing
If you think you found a bug or if you have a feature request / suggestion, feel free to open an issue. Before opening a pull request, please start an issue or a discussion on the topic.
Contributions are welcomed, check out how to contribute to a Github project. If it is your first contribution, you can also check this first contribution tutorial. You can find first good issues (if any 🙂) here. You may find other packages to contribute to at the control-toolbox organization.
If you want to ask a question, feel free to start a discussion here. This forum is for general discussion about this repository and the control-toolbox organization.
Note
If you want to add an application or a package to the control-toolbox ecosystem, please follow this set up tutorial.
Reproducibility
You can download the exact environment used to build this documentation:
📦 Project.toml - Package dependencies
📋 Manifest.toml - Complete dependency tree with versions
ℹ️ Version info
Julia Version 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 UTC)
Build Info:
Official https://julialang.org release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 4 × AMD EPYC 9V74 80-Core Processor
WORD_SIZE: 64
LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)
Environment:
JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager📦 Package status
Status `~/work/OptimalControl.jl/OptimalControl.jl/docs/Project.toml`
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`📚 Complete manifest
Status `~/work/OptimalControl.jl/OptimalControl.jl/docs/Manifest.toml`
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[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
[35661453] Xorg_xkbcomp_jll v1.4.7+0
[33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[3161d3a3] Zstd_jll v1.5.7+1
[1e29f10c] demumble_jll v1.3.0+0
[35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
[a4ae2306] libaom_jll v3.14.1+0
[0ac62f75] libass_jll v0.17.4+0
[1183f4f0] libdecor_jll v0.2.2+0
[8e53e030] libdrm_jll v2.4.134+0
[2db6ffa8] libevdev_jll v1.13.4+0
[f638f0a6] libfdk_aac_jll v2.0.4+0
[36db933b] libinput_jll v1.28.1+0
[b53b4c65] libpng_jll v1.6.58+0
[9a156e7d] libva_jll v2.23.0+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.3.0+0
[8b5cbfcf] tree_sitter_gcn_jll v0.1.0+0
[44208993] tree_sitter_llvm_jll v1.1.0+0
[71e3f6e6] tree_sitter_ptx_jll v0.1.0+0
[f0e86581] tree_sitter_spirv_jll v0.1.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.13.0+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.12.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.12.1
[de0858da] Printf v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.12.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.3.1+2
[deac9b47] LibCURL_jll v8.15.0+0
[e37daf67] LibGit2_jll v1.9.0+0
[29816b5a] LibSSH2_jll v1.11.3+1
[14a3606d] MozillaCACerts_jll v2025.11.4
[4536629a] OpenBLAS_jll v0.3.29+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.6+0
[efcefdf7] PCRE2_jll v10.44.0+1
[bea87d4a] SuiteSparse_jll v7.8.3+2
[83775a58] Zlib_jll v1.3.1+2
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.64.0+1
[3f19e933] p7zip_jll v17.7.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`