ModelingToolkit Integration

This is a quick intro based on the lotka volterra fishing problem. It is similar to the classical fishing example, but we will use ModelingToolkit.jl to model it.

Setup

We will use Corleone to define our optimal control problem.

using Corleone
using ModelingToolkit
using ModelingToolkit: t_nounits as t, D_nounits as D
using OrdinaryDiffEqTsit5
using ComponentArrays, ForwardDiff
using Optimization
using OptimizationMOI, Ipopt
using LuxCore, Random
using CairoMakie

We start by defining our system

@variables begin
    x(..) = 0.5, [tunable = false]
    y(..) = 0.7, [tunable = false]
    u(..) = 0.0, [bounds = (0.0, 1.0), input = true]
end
@constants begin
    c₁ = 0.4
    c₂ = 0.2
end
eqs = [
    D(x(t)) ~ x(t) - x(t) * y(t) - c₁ * u(t) * x(t)
    D(y(t)) ~ -y(t) + x(t) * y(t) - c₂ * u(t) * y(t)
]
costs = [
    Symbolics.Integral(t in (0.0, 12.0))(
        (x(t) - 1.0)^2 + (y(t) - 1.0)^2
    ),
]
@named lotka_volterra = System(eqs, t, costs = costs)

function plot_lotka(sol)
    f = Figure()
    ax = CairoMakie.Axis(f[1, 1])
    scatterlines!(ax, sol, vars = [1, 2, 3])
    f[1, 2] = Legend(f, ax, "States", framevisible = false)
    ax1 = CairoMakie.Axis(f[2, 1])
    stairs!(ax1, sol, vars = [4])
    f[2, 2] = Legend(f, ax1, "Controls", framevisible = false)
    return f
end
plot_lotka (generic function with 1 method)

Single Shooting

dynopt = CorleoneDynamicOptProblem(
    lotka_volterra, [],
    u(t) => 0.0:0.1:11.9,
    algorithm = Tsit5(),
)

optprob = OptimizationProblem(dynopt, AutoForwardDiff(), Val(:ComponentArrays))

sol = solve(
    optprob, Ipopt.Optimizer(), max_iter = 1000, tol = 5.0e-6,
    hessian_approximation = "limited-memory"
)


u_opt = ComponentVector(sol.u, optprob.f.f.ax)
opt_traj, _ = dynopt.layer(nothing, u_opt, LuxCore.initialstates(Random.default_rng(), dynopt.layer))

plot_lotka(opt_traj)

Multiple Shooting

dynopt = CorleoneDynamicOptProblem(
    lotka_volterra, [],
    u(t) => 0.0:0.1:11.9,
    algorithm = Tsit5(),
    shooting = [0.0, 3.0, 6.0, 9.0]
)

optprob = OptimizationProblem(dynopt, AutoForwardDiff(), Val(:ComponentArrays))

sol = solve(
    optprob, Ipopt.Optimizer(), max_iter = 1000, tol = 5.0e-6,
    hessian_approximation = "limited-memory"
)

u_opt = ComponentVector(sol.u, optprob.f.f.ax)
opt_traj, _ = dynopt.layer(nothing, u_opt, LuxCore.initialstates(Random.default_rng(), dynopt.layer))

plot_lotka(opt_traj)

Appendix

This example was built using these direct dependencies,
        using Pkg
        Pkg.status()
Status `~/_work/Corleone.jl/Corleone.jl/docs/Project.toml`
  [13f3f980] CairoMakie v0.15.13
  [b0b7db55] ComponentArrays v0.15.47
  [2751e0db] Corleone v0.0.7 `~/_work/Corleone.jl/Corleone.jl`
  [6590a4b1] CorleoneOED v0.0.8 `~/_work/Corleone.jl/Corleone.jl/lib/CorleoneOED`
  [e30172f5] Documenter v1.17.0
  [daee34ce] DocumenterCitations v1.5.0
  [d12716ef] DocumenterInterLinks v1.1.0
  [f6369f11] ForwardDiff v1.4.5
  [b6b21f68] Ipopt v1.15.0
  [682c06a0] JSON v1.7.1
  [2ddba703] Juniper v0.9.4
  [98b081ad] Literate v2.21.0
  [bb33d45b] LuxCore v1.5.3
  [961ee093] ModelingToolkit v11.40.0
  [d147904a] OptimalControlBenchmarks v0.0.3 `~/_work/Corleone.jl/Corleone.jl/lib/OptimalControlBenchmarks`
  [7f7a1694] Optimization v5.9.0
  [22f7324a] OptimizationLBFGSB v1.5.1
  [fd9f6733] OptimizationMOI v1.4.1
  [b1df2697] OrdinaryDiffEqTsit5 v2.1.4
  [2efcf032] SymbolicIndexingInterface v0.3.55
  [ddb6d928] YAML v0.4.16
  [9a3f8284] Random v1.11.0
and using this machine and Julia version.
        using InteractiveUtils
        InteractiveUtils.versioninfo()
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: 128 × AMD EPYC 9354 32-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_CPU_THREADS = 4
  JULIA_NUM_PRECOMPILE_TASKS = 4
A more complete overview of all dependencies and their versions is also provided.
        using Pkg
        Pkg.status(; mode = PKGMODE_MANIFEST)
Status `~/_work/Corleone.jl/Corleone.jl/docs/Manifest.toml`
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  [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
  [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
  [f50d1b31] Rmath_jll v0.5.2+0
  [319450e9] SPRAL_jll v2025.9.18+0
⌅ [02c8fc9c] XML2_jll v2.13.9+0
  [ffd25f8a] XZ_jll v5.8.3+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+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
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [9a68df92] isoband_jll v0.2.3+0
  [a4ae2306] libaom_jll v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [8e53e030] libdrm_jll v2.4.134+0
  [f638f0a6] libfdk_aac_jll v2.0.4+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
  [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+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
  [9abbd945] Profile 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`

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