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.
Motivation
The guiding philosophy of OptimalControl.jl is to offer, to our knowledge, the only Julia package that unifies both direct and indirect methods for optimal control within a single, coherent framework. This fills a gap in a landscape where existing tools are fragmented across programming languages and paradigms, and are usually restricted to a single family of methods. The package provides a domain-specific language that closely matches mathematical notation, together with multiple discretization schemes and shooting methods, and planned support for homotopy continuation methods. Its modeler–solver separation makes it agnostic to the underlying NLP modeling backend and optimization solver, and enables seamless execution on both CPU and GPU with minimal user intervention. Combined with an ecosystem of domain-specific applications, tutorials, and benchmarking tools, this design targets researchers and engineers working in optimal control, control theorists developing new algorithms, and students learning the field through interactive tutorials.
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 # Ipopt solver extension (needed for solve)
using Plots # plotting extension (needed for plot)
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)
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.
Testing
OptimalControl.jl is the umbrella package of a multi-repository ecosystem, and testing is organized in layers. Each sub-package (CTBase, CTParser, CTModels, CTDirect, CTSolvers, CTFlows) has its own test suite combining unit tests, integration tests, and code-quality checks (e.g. with Aqua.jl). At the umbrella level, OptimalControl.jl adds strong end-to-end integration tests that solve complete optimal control problems with both direct and indirect methods. Continuous integration runs on Linux, macOS, and Windows, on both CPU and GPU (via a self-hosted CUDA runner), through reusable workflows centralized in CTActions. Code coverage is tracked on Codecov, downstream packages are guarded against regressions through dedicated breakage tests, and beta versions are distributed during development via a local registry, ct-registry. Part of the test code is written with the help of AI agents, always under human review.
Reproducibility
Download the exact environment used to build these docs:
📦 Project.toml - Package dependencies
📋 Manifest.toml - Full dependency tree
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: 20 × Intel(R) Xeon(R) CPU E5-2640 v4 @ 2.40GHz
WORD_SIZE: 64
LLVM: libLLVM-18.1.7 (ORCJIT, broadwell)
GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 20 virtual cores)
Environment:
JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
JULIA_DEPOT_PATH = /projects/ctb/gha-runner/julia_depot📦 Package status
Status `/projects/ctb/gha-runner/_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 `/projects/ctb/gha-runner/_work/OptimalControl.jl/OptimalControl.jl/docs/Manifest.toml`
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[a38c48d9] CoreMath_jll v0.1.0+0
[ee1fde0b] Dbus_jll v1.16.2+0
⌅ [5ae413db] EarCut_jll v2.2.4+0
[2702e6a9] EpollShim_jll v0.0.20230411+1
[2e619515] Expat_jll v2.8.3+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
[a3f928ae] Fontconfig_jll v2.17.1+0
[d7e528f0] FreeType2_jll v2.14.3+1
[559328eb] FriBidi_jll v1.0.17+0
[0656b61e] GLFW_jll v3.5.1+0
[d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
[61579ee1] Ghostscript_jll v9.55.1+0
⌅ [59f7168a] Giflib_jll v5.2.3+0
[020c3dae] Git_LFS_jll v3.7.1+0
[f8c6e375] Git_jll v2.55.0+0
[7746bdde] Glib_jll v2.88.3+0
[3b182d85] Graphite2_jll v1.3.16+0
[017b0a0e] HSL_jll v4.0.8+0
[2e76f6c2] HarfBuzz_jll v100.14003.0+0
⌅ [8fd58aa0] HiGHS_jll v1.15.0+0
[e33a78d0] Hwloc_jll v2.14.0+0
[905a6f67] Imath_jll v3.2.2+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
⌅ [9cc047cb] Ipopt_jll v300.1400.1902+0
[aacddb02] JpegTurbo_jll v3.2.0+1
[9c1d0b0a] JuliaNVTXCallbacks_jll v0.2.1+0
[0e6b36f8] KNITRO_jll v16.0.0
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.1.0+0
[dad2f222] LLVMExtra_jll v0.0.47+0
[1d63c593] LLVMOpenMP_jll v22.1.7+0
[ad6e5548] LibTracyClient_jll v0.13.1+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[d00139f3] METIS_jll v5.1.3+0
[856f044c] MKL_jll v2025.2.0+0
[d7ed1dd3] MUMPS_seq_jll v500.900.100+0
[ef6e0fe3] NVPTX_LLVM_Backend_jll v22.1.7+1
[e98f9f5b] NVTX_jll v3.2.2+0
[c7aee132] NodeJS_20_jll v20.12.2+0
[e7412a2a] Ogg_jll v1.3.6+0
[656ef2d0] OpenBLAS32_jll v0.3.34+0
[6cdc7f73] OpenBLASConsistentFPCSR_jll v0.3.34+0
[18a262bb] OpenEXR_jll v3.4.14+0
[9bd350c2] OpenSSH_jll v10.5.1+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.58.2+0
[30392449] Pixman_jll v0.46.4+0
[c0090381] Qt6Base_jll v6.10.2+2
[629bc702] Qt6Declarative_jll v6.10.2+2
[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[f50d1b31] Rmath_jll v0.5.2+0
[319450e9] SPRAL_jll v2025.9.18+1
[396d5378] Uno_jll v2.8.0+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
⌅ [02c8fc9c] XML2_jll v2.13.9+0
[ffd25f8a] XZ_jll v5.8.3+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.13+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[a51aa0fd] Xorg_libXi_jll v1.8.4+0
[d1454406] Xorg_libXinerama_jll v1.1.7+0
[ec84b674] Xorg_libXrandr_jll v1.5.6+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.2.0+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
[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
[9a68df92] isoband_jll v0.2.3+0
[a4ae2306] libaom_jll v3.14.1+0
[0ac62f75] libass_jll v0.17.5+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
[075b6546] libsixel_jll v1.10.5+0
[9a156e7d] libva_jll v2.23.0+0
[f27f6e37] libvorbis_jll v1.3.8+0
[c5f90fcd] libwebp_jll v1.6.0+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.3.0+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
[8bf52ea8] CRC32c 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
[1a1011a3] SharedArrays 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`