From 1d2fcc357663f7315ee2f32659e557f6a0857c58 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sat, 7 Feb 2026 23:25:26 +0100
Subject: [PATCH 01/24] test implement

---
 ext/Descriptions/brachistochrone.md           |  55 ++++++++
 ext/JuMPModels/brachistochrone.jl             | 127 ++++++++++++++++++
 ext/MetaData/brachistochrone.jl               |  12 ++
 ext/OptimalControlModels/brachistochrone.jl   |  77 +++++++++++
 .../brachistochrone_s.jl                      |  95 +++++++++++++
 src/OptimalControlProblems.jl                 |  11 ++
 6 files changed, 377 insertions(+)
 create mode 100644 ext/Descriptions/brachistochrone.md
 create mode 100644 ext/JuMPModels/brachistochrone.jl
 create mode 100644 ext/MetaData/brachistochrone.jl
 create mode 100644 ext/OptimalControlModels/brachistochrone.jl
 create mode 100644 ext/OptimalControlModels_s/brachistochrone_s.jl

diff --git a/ext/Descriptions/brachistochrone.md b/ext/Descriptions/brachistochrone.md
new file mode 100644
index 00000000..da388cf9
--- /dev/null
+++ b/ext/Descriptions/brachistochrone.md
@@ -0,0 +1,55 @@
+The **Brachistochrone problem** is a classical benchmark in the history of calculus of variations and optimal control.  
+It seeks the shape of a curve (or wire) connecting two points $A$ and $B$ within a vertical plane, such that a bead sliding frictionlessly under the influence of uniform gravity travels from $A$ to $B$ in the shortest possible time.  
+Originating from the challenge posed by Johann Bernoulli in 1696 [Bernoulli 1696](https://doi.org/10.1002/andp.19163551307), it marks the birth of modern optimal control theory.  
+The problem involves nonlinear dynamics where the state includes position and velocity, and the control is the path's angle, with the final time $t_f$ being a decision variable to be minimised.
+
+### Mathematical formulation
+
+The problem can be stated as a free final time optimal control problem:
+
+```math
+\begin{aligned}
+\min_{u(\cdot), t_f} \quad & J = t_f = \int_0^{t_f} 1 \,\mathrm{d}t \\[1em]
+\text{s.t.} \quad & \dot{x}(t) = v(t) \sin u(t), \\[0.5em]
+& \dot{y}(t) = v(t) \cos u(t), \\[0.5em]
+& \dot{v}(t) = g \cos u(t), \\[0.5em]
+& x(0) = x_0, \; y(0) = y_0, \; v(0) = v_0, \\[0.5em]
+& x(t_f) = x_f, \; y(t_f) = y_f,
+\end{aligned}
+```
+
+where $u(t)$ represents the angle of the tangent to the curve with respect to the vertical axis, and $g$ is the gravitational acceleration.
+
+### Qualitative behaviour
+
+This problem is a classic example of **minimum time control** with nonlinear dynamics.  
+Unlike the shortest path problem (a straight line), the optimal trajectory balances the need to minimize path length with the need to maximize velocity.
+
+The analytical solution to this problem is a **cycloid**—the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping.  
+Specifically:
+
+- **Energy Conservation**: Since the system is conservative and frictionless, the speed at any height $h$ is given by $v = \sqrt{2gh}$ (assuming start from rest). This implies that maximizing vertical drop early in the trajectory increases velocity for the remainder of the path.
+- **Concavity**: The optimal curve is concave up. It starts steeply (vertical tangent if $v_0=0$) to gain speed quickly, then flattens out near the bottom before potentially rising again to reach the target.
+- **Singularity**: At the initial point, if $v_0=0$, the control is theoretically singular as the bead has no velocity to direct. Numerical solvers often require a small non-zero initial velocity or a robust initial guess to handle this start.
+
+The control $u(t)$ is continuous and varies smoothly to trace the cycloidal arc.
+
+### Characteristics
+
+- **Nonlinear dynamics** involving trigonometric functions of the control.
+- **Free final time** problem (time-optimal).
+- Analytical solution is a **Cycloid**.
+- Historically significant as the first problem solved using techniques that evolved into the Calculus of Variations.
+
+### References
+
+- Bernoulli, J. (1696). *Problema novum ad cujus solutionem Mathematici invitantur*.  
+  Acta Eruditorum.  
+  The original publication posing the challenge to the mathematicians of Europe, solved by Newton, Leibniz, L'Hôpital, and the Bernoulli brothers.
+
+- Sussmann, H. J., & Willems, J. C. (1997). *300 years of optimal control: from the brachystochrone to the maximum principle*.  
+  IEEE Control Systems Magazine. [doi.org/10.1109/37.588108](https://doi.org/10.1109/37.588108)  
+  A comprehensive historical review linking the classical Brachistochrone problem to modern optimal control theory and Pontryagin's Maximum Principle.
+
+- Dymos Examples: Brachistochrone. [OpenMDAO/Dymos](https://openmdao.github.io/dymos/examples/brachistochrone/brachistochrone.html)  
+  The numerical implementation serving as the basis for the current benchmark formulation.
\ No newline at end of file
diff --git a/ext/JuMPModels/brachistochrone.jl b/ext/JuMPModels/brachistochrone.jl
new file mode 100644
index 00000000..6678cc7c
--- /dev/null
+++ b/ext/JuMPModels/brachistochrone.jl
@@ -0,0 +1,127 @@
+"""
+$(TYPEDSIGNATURES)
+
+Constructs a JuMP model representing the **Brachistochrone optimal control problem**.
+The objective is to minimize the final time `tf` to travel between two points under gravity.
+The problem uses a direct transcription (trapezoidal rule) manually implemented in JuMP.
+
+# Arguments
+
+- `::JuMPBackend`: Placeholder type to specify the JuMP backend or solver interface.
+- `grid_size::Int=50`: (Keyword) Number of discretisation steps for the time grid.
+
+# Returns
+
+- `model::JuMP.Model`: A JuMP model containing the decision variables (including final time), dynamics constraints, and boundary conditions.
+
+# Example
+
+```julia-repl
+julia> using OptimalControlProblems
+julia> using JuMP
+julia> model = OptimalControlProblems.brachistochrone(JuMPBackend(); N=100)
+```
+
+# References
+
+- Dymos Brachistochrone: [https://openmdao.github.io/dymos/examples/brachistochrone/brachistochrone.html](https://openmdao.github.io/dymos/examples/brachistochrone/brachistochrone.html)
+"""
+function OptimalControlProblems.brachistochrone(
+    ::JuMPBackend,
+    args...;
+    grid_size::Int=grid_size_data(Val(:brachistochrone)),
+    parameters::Union{Nothing,NamedTuple}=nothing,
+    kwargs...,
+)
+
+    # parameters
+    params = parameters_data(Val(:brachistochrone), parameters)
+    g  = params[:g]
+    t0 = params[:t0]
+    x0 = params[:x0]
+    y0 = params[:y0]
+    v0 = params[:v0]
+    xf = params[:xf]
+    yf = params[:yf]
+
+    # model
+    model = JuMP.Model(args...; kwargs...)
+
+    # N = grid_size
+    @expression(model, N, grid_size)
+
+    @variables(
+        model,
+        begin
+            0.1 <= tf <= 20.0, (start = 2.0)
+            x[0:N]
+            y[0:N]
+            v[0:N]
+            u[0:N]
+        end
+    )
+
+
+    model[:time_grid] = () -> range(t0, value(tf), N+1)
+    model[:state_components] = ["x", "y", "v"]
+    model[:control_components] = ["u"]
+    model[:variable_components] = ["tf"]
+
+    for i in 0:N
+        alpha = i / N
+        set_start_value(x[i], x0 + alpha * (xf - x0))
+        set_start_value(y[i], y0 + alpha * (yf - y0))
+        set_start_value(v[i], v0 + alpha * 10.0) # Estimate speed
+        set_start_value(u[i], 1.57) # ~90 degrees
+    end
+
+    # boundary constraints
+    @constraints(
+        model,
+        begin
+            # Start
+            x[0] == x0
+            y[0] == y0
+            v[0] == v0
+            
+            # End
+            x[N] == xf
+            y[N] == yf
+            # v[N] is free
+        end
+    )
+
+    # dynamics
+    @expressions(
+        model,
+        begin
+            # Time step is variable: dt = (tf - t0) / N
+            dt, (tf - t0) / N
+
+            # Dynamics expressions (Dymos formulation)
+            # dx/dt = v * sin(u)
+            dx[i = 0:N], v[i] * sin(u[i])
+            
+            # dy/dt = v * cos(u)
+            dy[i = 0:N], v[i] * cos(u[i])
+            
+            # dv/dt = g * cos(u)
+            dv[i = 0:N], g * cos(u[i])
+        end
+    )
+
+    # Trapezoidal rule integration (Implicit)
+    @constraints(
+        model,
+        begin
+            ∂x[i = 1:N], x[i] == x[i - 1] + 0.5 * dt * (dx[i] + dx[i - 1])
+            ∂y[i = 1:N], y[i] == y[i - 1] + 0.5 * dt * (dy[i] + dy[i - 1])
+            ∂v[i = 1:N], v[i] == v[i - 1] + 0.5 * dt * (dv[i] + dv[i - 1])
+        end
+    )
+
+    # objective: Minimize final time
+    @objective(model, Min, tf)
+
+    return model
+end
\ No newline at end of file
diff --git a/ext/MetaData/brachistochrone.jl b/ext/MetaData/brachistochrone.jl
new file mode 100644
index 00000000..17d700d0
--- /dev/null
+++ b/ext/MetaData/brachistochrone.jl
@@ -0,0 +1,12 @@
+brachistochrone_meta = OrderedDict(
+    :grid_size => 500,                 
+    :parameters => (
+        g  = 9.80665,                 
+        t0 = 0.0,                      
+        x0 = 0.0,                      
+        y0 = 10.0,                     
+        v0 = 0.0,                    
+        xf = 10.0,                    
+        yf = 5.0,                     
+    ),
+)
\ No newline at end of file
diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
new file mode 100644
index 00000000..fd71ea1b
--- /dev/null
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -0,0 +1,77 @@
+"""
+$(TYPEDSIGNATURES)
+
+Constructs the **Brachistochrone** optimal control problem.
+The goal is to move from point A to point B in minimum time under gravity.
+
+# Arguments
+- `::OptimalControlBackend`: The backend type.
+- `grid_size`: Number of time steps.
+- `parameters`: Optional parameter overrides.
+
+```julia-repl
+julia> using OptimalControlProblems
+
+julia> docp = OptimalControlProblems.brachistochrone(OptimalControlBackend(); N=100);
+```
+"""
+function OptimalControlProblems.brachistochrone(
+    ::OptimalControlBackend,
+    description::Symbol...;
+    grid_size::Int=grid_size_data(Val(:brachistochrone)),
+    parameters::Union{Nothing,NamedTuple}=nothing,
+    kwargs...,
+)
+    
+    params = parameters_data(Val(:brachistochrone), parameters)
+    g  = params[:g]
+    t0 = params[:t0]
+    x0 = params[:x0]
+    y0 = params[:y0]
+    v0 = params[:v0]
+    xf = params[:xf]
+    yf = params[:yf]
+
+    ocp = @def begin
+        tf ∈ R, variable
+        t ∈ [t0, tf], time
+
+        x ∈ R, state
+        y ∈ R, state
+        v ∈ R, state
+
+        u ∈ R, control
+
+        x(t0) == x0
+        y(t0) == y0
+        v(t0) == v0
+        
+        x(tf) == xf
+        y(tf) == yf
+
+        0.1 ≤ tf ≤ 20.0 
+
+        ẋ(t) == v(t) * sin(u(t))
+        ẏ(t) == v(t) * cos(u(t))
+        v̇(t) == g * cos(u(t))
+
+        tf → min
+    end
+
+ 
+    init = (
+        state = [5.0, 7.5, 5.0], 
+        control = 1.5, 
+        variable = 1.0
+    )
+
+    docp = direct_transcription(
+        ocp,
+        description...;
+        init = init,
+        grid_size = grid_size,
+        kwargs...
+    )
+
+    return docp
+end
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
new file mode 100644
index 00000000..a8ca2fe2
--- /dev/null
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -0,0 +1,95 @@
+"""
+$(TYPEDSIGNATURES)
+
+Constructs an **OptimalControl problem** representing the Brachistochrone problem using the OptimalControl backend.
+The function sets up the state and control variables, boundary conditions, dynamics, and the objective functional.
+It then performs direct transcription to generate a discrete optimal control problem (DOCP).
+
+# Arguments
+
+- `::OptimalControlBackend`: Placeholder type to specify the OptimalControl backend or solver interface.
+- `grid_size::Int=grid_size_data(:brachistochrone)`: (Keyword) Number of discretisation points for the direct transcription grid.
+- `parameters::Union{Nothing,NamedTuple}=nothing`: (Keyword) Custom parameters to override defaults.
+
+# Returns
+
+- `docp`: The direct optimal control problem object, representing the discretised problem.
+
+# Example
+
+```julia-repl
+julia> using OptimalControlProblems
+
+julia> docp = OptimalControlProblems.brachistochrone(OptimalControlBackend(); N=100);
+```
+
+# References
+
+- Dymos Brachistochrone: https://openmdao.github.io/dymos/examples/brachistochrone/brachistochrone.html
+"""
+function OptimalControlProblems.brachistochrone(
+    ::OptimalControlBackend,
+    description::Symbol...;
+    grid_size::Int=grid_size_data(Val(:brachistochrone)),
+    parameters::Union{Nothing,NamedTuple}=nothing,
+    kwargs...,
+)
+
+   
+    params = parameters_data(Val(:brachistochrone), parameters)
+    g  = params[:g]
+    t0 = params[:t0]
+    x0 = params[:x0]
+    y0 = params[:y0]
+    v0 = params[:v0]
+    xf = params[:xf]
+    yf = params[:yf]
+
+    # model
+    ocp = @def begin
+        
+        tf ∈ R, variable
+        t ∈ [t0, tf], time
+
+        x ∈ R, state
+        y ∈ R, state
+        v ∈ R, state
+
+        u ∈ R, control
+
+        x(t0) == x0
+        y(t0) == y0
+        v(t0) == v0
+        
+        x(tf) == xf
+        y(tf) == yf
+
+        0.1 ≤ tf ≤ 20.0
+
+        ∂(x)(t) == v(t) * sin(u(t))
+        ∂(y)(t) == v(t) * cos(u(t))
+        ∂(v)(t) == g * cos(u(t))
+
+        tf → min
+    end
+
+    # initial guess
+    init = (
+        state = [5.0, 7.5, 5.0], 
+        control = 1.57, 
+        variable = 2.0
+    )
+
+    # discretise the optimal control problem
+    docp = direct_transcription(
+        ocp,
+        description...;
+        lagrange_to_mayer=false,
+        init=init,
+        grid_size=grid_size,
+        disc_method=:trapeze,
+        kwargs...,
+    )
+
+    return docp
+end
\ No newline at end of file
diff --git a/src/OptimalControlProblems.jl b/src/OptimalControlProblems.jl
index 9729407d..2a27ad5c 100644
--- a/src/OptimalControlProblems.jl
+++ b/src/OptimalControlProblems.jl
@@ -72,6 +72,17 @@ function make_list_of_problems()
         :robot,
         :space_shuttle,
         :steering,
+        :beam,
+        :cart_pendulum,
+        :chain,
+        :dielectrophoretic_particle,
+        :double_oscillator,
+        :electric_vehicle,
+        :glider,
+        :jackson,
+        :robbins,
+        :rocket,
+        :vanderpol,
     ]
     list_of_problems = setdiff(list_of_problems, problems_to_exclude)
 

From 949511361fa5086396686c40882614de4a8d6b44 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sat, 7 Feb 2026 23:33:33 +0100
Subject: [PATCH 02/24] fix bug

---
 ext/JuMPModels/brachistochrone.jl               | 2 +-
 ext/OptimalControlModels/brachistochrone.jl     | 2 +-
 ext/OptimalControlModels_s/brachistochrone_s.jl | 2 +-
 3 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/ext/JuMPModels/brachistochrone.jl b/ext/JuMPModels/brachistochrone.jl
index 6678cc7c..ea646624 100644
--- a/ext/JuMPModels/brachistochrone.jl
+++ b/ext/JuMPModels/brachistochrone.jl
@@ -29,7 +29,7 @@ julia> model = OptimalControlProblems.brachistochrone(JuMPBackend(); N=100)
 function OptimalControlProblems.brachistochrone(
     ::JuMPBackend,
     args...;
-    grid_size::Int=grid_size_data(Val(:brachistochrone)),
+    grid_size::Int=grid_size_data(:brachistochrone),
     parameters::Union{Nothing,NamedTuple}=nothing,
     kwargs...,
 )
diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index fd71ea1b..4d319c25 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -18,7 +18,7 @@ julia> docp = OptimalControlProblems.brachistochrone(OptimalControlBackend(); N=
 function OptimalControlProblems.brachistochrone(
     ::OptimalControlBackend,
     description::Symbol...;
-    grid_size::Int=grid_size_data(Val(:brachistochrone)),
+    grid_size::Int=grid_size_data(:brachistochrone),
     parameters::Union{Nothing,NamedTuple}=nothing,
     kwargs...,
 )
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index a8ca2fe2..41de40d0 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -30,7 +30,7 @@ julia> docp = OptimalControlProblems.brachistochrone(OptimalControlBackend(); N=
 function OptimalControlProblems.brachistochrone(
     ::OptimalControlBackend,
     description::Symbol...;
-    grid_size::Int=grid_size_data(Val(:brachistochrone)),
+    grid_size::Int=grid_size_data(:brachistochrone),
     parameters::Union{Nothing,NamedTuple}=nothing,
     kwargs...,
 )

From ee2036d3e9e5a3ecd24d6d4e83c4db34282a45cf Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sat, 7 Feb 2026 23:38:52 +0100
Subject: [PATCH 03/24] bug 2

---
 ext/OptimalControlModels_s/brachistochrone_s.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 41de40d0..ad448b26 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -27,7 +27,7 @@ julia> docp = OptimalControlProblems.brachistochrone(OptimalControlBackend(); N=
 
 - Dymos Brachistochrone: https://openmdao.github.io/dymos/examples/brachistochrone/brachistochrone.html
 """
-function OptimalControlProblems.brachistochrone(
+function OptimalControlProblems.brachistochrone_s(
     ::OptimalControlBackend,
     description::Symbol...;
     grid_size::Int=grid_size_data(:brachistochrone),

From e2e1290453e27948fbeeb7019518193e1e4884d4 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sat, 7 Feb 2026 23:43:51 +0100
Subject: [PATCH 04/24] BUG 3

---
 ext/JuMPModels/brachistochrone.jl               | 2 +-
 ext/OptimalControlModels/brachistochrone.jl     | 2 +-
 ext/OptimalControlModels_s/brachistochrone_s.jl | 2 +-
 3 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/ext/JuMPModels/brachistochrone.jl b/ext/JuMPModels/brachistochrone.jl
index ea646624..a68f7293 100644
--- a/ext/JuMPModels/brachistochrone.jl
+++ b/ext/JuMPModels/brachistochrone.jl
@@ -35,7 +35,7 @@ function OptimalControlProblems.brachistochrone(
 )
 
     # parameters
-    params = parameters_data(Val(:brachistochrone), parameters)
+    params = parameters_data(:brachistochrone, parameters)
     g  = params[:g]
     t0 = params[:t0]
     x0 = params[:x0]
diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 4d319c25..25110d04 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -23,7 +23,7 @@ function OptimalControlProblems.brachistochrone(
     kwargs...,
 )
     
-    params = parameters_data(Val(:brachistochrone), parameters)
+    params = parameters_data(:brachistochrone, parameters)
     g  = params[:g]
     t0 = params[:t0]
     x0 = params[:x0]
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index ad448b26..dd706690 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -36,7 +36,7 @@ function OptimalControlProblems.brachistochrone_s(
 )
 
    
-    params = parameters_data(Val(:brachistochrone), parameters)
+    params = parameters_data(:brachistochrone, parameters)
     g  = params[:g]
     t0 = params[:t0]
     x0 = params[:x0]

From c9366b2ffc4a556873dc2b3343c7d02c4300674f Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sat, 7 Feb 2026 23:54:22 +0100
Subject: [PATCH 05/24] bug 4

---
 ext/JuMPModels/brachistochrone.jl               |  6 ++++++
 ext/OptimalControlModels/brachistochrone.jl     | 13 +++++++------
 ext/OptimalControlModels_s/brachistochrone_s.jl | 11 ++++++-----
 3 files changed, 19 insertions(+), 11 deletions(-)

diff --git a/ext/JuMPModels/brachistochrone.jl b/ext/JuMPModels/brachistochrone.jl
index a68f7293..cfde278e 100644
--- a/ext/JuMPModels/brachistochrone.jl
+++ b/ext/JuMPModels/brachistochrone.jl
@@ -64,6 +64,12 @@ function OptimalControlProblems.brachistochrone(
 
     model[:time_grid] = () -> range(t0, value(tf), N+1)
     model[:state_components] = ["x", "y", "v"]
+    
+    # --- CORRECTION ICI ---
+    # Il faut déclarer la liste des costates (vide ici) pour satisfaire les tests
+    model[:costate_components] = String[]
+    # ----------------------
+    
     model[:control_components] = ["u"]
     model[:variable_components] = ["tf"]
 
diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 25110d04..ab1778fb 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -36,9 +36,9 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x ∈ R, state
-        y ∈ R, state
-        v ∈ R, state
+        
+        [x, y, v] ∈ R³, state
+        # ----------------------
 
         u ∈ R, control
 
@@ -51,8 +51,9 @@ function OptimalControlProblems.brachistochrone(
 
         0.1 ≤ tf ≤ 20.0 
 
-        ẋ(t) == v(t) * sin(u(t))
-        ẏ(t) == v(t) * cos(u(t))
+        
+        ẋ(t) == v(t) * sin(u(t))
+        ẏ(t) == v(t) * cos(u(t))
         v̇(t) == g * cos(u(t))
 
         tf → min
@@ -74,4 +75,4 @@ function OptimalControlProblems.brachistochrone(
     )
 
     return docp
-end
+end
\ No newline at end of file
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index dd706690..c8710680 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -51,9 +51,9 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x ∈ R, state
-        y ∈ R, state
-        v ∈ R, state
+       
+        [x, y, v] ∈ R³, state
+        # ------------------
 
         u ∈ R, control
 
@@ -66,6 +66,7 @@ function OptimalControlProblems.brachistochrone_s(
 
         0.1 ≤ tf ≤ 20.0
 
+     
         ∂(x)(t) == v(t) * sin(u(t))
         ∂(y)(t) == v(t) * cos(u(t))
         ∂(v)(t) == g * cos(u(t))
@@ -73,14 +74,14 @@ function OptimalControlProblems.brachistochrone_s(
         tf → min
     end
 
-    # initial guess
+  
     init = (
         state = [5.0, 7.5, 5.0], 
         control = 1.57, 
         variable = 2.0
     )
 
-    # discretise the optimal control problem
+    
     docp = direct_transcription(
         ocp,
         description...;

From f5ddb9ac0c7ac94a87996c19fcff7bde77259cfd Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sun, 8 Feb 2026 00:01:42 +0100
Subject: [PATCH 06/24] bug 6

---
 ext/JuMPModels/brachistochrone.jl             |  2 --
 ext/OptimalControlModels/brachistochrone.jl   | 24 +++++++----------
 .../brachistochrone_s.jl                      | 27 +++++++++----------
 3 files changed, 22 insertions(+), 31 deletions(-)

diff --git a/ext/JuMPModels/brachistochrone.jl b/ext/JuMPModels/brachistochrone.jl
index cfde278e..7d56ff07 100644
--- a/ext/JuMPModels/brachistochrone.jl
+++ b/ext/JuMPModels/brachistochrone.jl
@@ -65,8 +65,6 @@ function OptimalControlProblems.brachistochrone(
     model[:time_grid] = () -> range(t0, value(tf), N+1)
     model[:state_components] = ["x", "y", "v"]
     
-    # --- CORRECTION ICI ---
-    # Il faut déclarer la liste des costates (vide ici) pour satisfaire les tests
     model[:costate_components] = String[]
     # ----------------------
     
diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index ab1778fb..a7b988ad 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -36,30 +36,26 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        
-        [x, y, v] ∈ R³, state
-        # ----------------------
+        q ∈ R³, state # q[1]=x, q[2]=y, q[3]=v
+        # ------------------------------------------
 
         u ∈ R, control
 
-        x(t0) == x0
-        y(t0) == y0
-        v(t0) == v0
+        q(t0) == [x0, y0, v0]
         
-        x(tf) == xf
-        y(tf) == yf
-
+    
+        q(tf)[1] == xf
+        q(tf)[2] == yf
+    
         0.1 ≤ tf ≤ 20.0 
 
-        
-        ẋ(t) == v(t) * sin(u(t))
-        ẏ(t) == v(t) * cos(u(t))
-        v̇(t) == g * cos(u(t))
+        q̇(t) == [ q(t)[3] * sin(u(t)), 
+                  q(t)[3] * cos(u(t)), 
+                  g * cos(u(t)) ]
 
         tf → min
     end
 
- 
     init = (
         state = [5.0, 7.5, 5.0], 
         control = 1.5, 
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index c8710680..82676868 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -35,7 +35,6 @@ function OptimalControlProblems.brachistochrone_s(
     kwargs...,
 )
 
-   
     params = parameters_data(:brachistochrone, parameters)
     g  = params[:g]
     t0 = params[:t0]
@@ -51,37 +50,35 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-       
-        [x, y, v] ∈ R³, state
-        # ------------------
+    
+        q ∈ R³, state
+        # ------------------------------------------
 
         u ∈ R, control
 
-        x(t0) == x0
-        y(t0) == y0
-        v(t0) == v0
+        q(t0) == [x0, y0, v0]
         
-        x(tf) == xf
-        y(tf) == yf
+        q(tf)[1] == xf
+        q(tf)[2] == yf
 
         0.1 ≤ tf ≤ 20.0
 
-     
-        ∂(x)(t) == v(t) * sin(u(t))
-        ∂(y)(t) == v(t) * cos(u(t))
-        ∂(v)(t) == g * cos(u(t))
+        
+        ∂(q)(t) == [ q(t)[3] * sin(u(t)), 
+                     q(t)[3] * cos(u(t)), 
+                     g * cos(u(t)) ]
 
         tf → min
     end
 
-  
+    # initial guess
     init = (
         state = [5.0, 7.5, 5.0], 
         control = 1.57, 
         variable = 2.0
     )
 
-    
+    # discretise the optimal control problem
     docp = direct_transcription(
         ocp,
         description...;

From 83748ba75932bdafd323f9ebe5e06d67550068a4 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sun, 8 Feb 2026 00:07:06 +0100
Subject: [PATCH 07/24] bug 7

---
 ext/OptimalControlModels/brachistochrone.jl     | 12 +++++-------
 ext/OptimalControlModels_s/brachistochrone_s.jl | 11 ++++-------
 2 files changed, 9 insertions(+), 14 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index a7b988ad..7fd5ba44 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -37,21 +37,19 @@ function OptimalControlProblems.brachistochrone(
         t ∈ [t0, tf], time
 
         q ∈ R³, state # q[1]=x, q[2]=y, q[3]=v
-        # ------------------------------------------
 
         u ∈ R, control
 
         q(t0) == [x0, y0, v0]
-        
-    
         q(tf)[1] == xf
         q(tf)[2] == yf
-    
+        # q(tf)[3] libre
+
         0.1 ≤ tf ≤ 20.0 
 
-        q̇(t) == [ q(t)[3] * sin(u(t)), 
-                  q(t)[3] * cos(u(t)), 
-                  g * cos(u(t)) ]
+        q̇(t)[1] == q(t)[3] * sin(u(t))  # dx/dt
+        q̇(t)[2] == q(t)[3] * cos(u(t))  # dy/dt
+        q̇(t)[3] == g * cos(u(t))        # dv/dt
 
         tf → min
     end
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 82676868..58c0d27b 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -50,9 +50,7 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-    
-        q ∈ R³, state
-        # ------------------------------------------
+        q ∈ R³, state 
 
         u ∈ R, control
 
@@ -63,10 +61,9 @@ function OptimalControlProblems.brachistochrone_s(
 
         0.1 ≤ tf ≤ 20.0
 
-        
-        ∂(q)(t) == [ q(t)[3] * sin(u(t)), 
-                     q(t)[3] * cos(u(t)), 
-                     g * cos(u(t)) ]
+        ∂(q)(t)[1] == q(t)[3] * sin(u(t))
+        ∂(q)(t)[2] == q(t)[3] * cos(u(t))
+        ∂(q)(t)[3] == g * cos(u(t))
 
         tf → min
     end

From b2e1fade83fead8a1a18b483c358c77dad9a4053 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sun, 8 Feb 2026 00:17:03 +0100
Subject: [PATCH 08/24] BUG 7

---
 ext/OptimalControlModels/brachistochrone.jl   | 20 +++++++++++--------
 .../brachistochrone_s.jl                      | 14 ++++++-------
 2 files changed, 19 insertions(+), 15 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 7fd5ba44..71bf3f41 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -36,20 +36,24 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        q ∈ R³, state # q[1]=x, q[2]=y, q[3]=v
+
+        z ∈ R³, state
 
         u ∈ R, control
 
-        q(t0) == [x0, y0, v0]
-        q(tf)[1] == xf
-        q(tf)[2] == yf
-        # q(tf)[3] libre
+
+        z(t0) == [x0, y0, v0]
+
+        z(tf)[1] == xf
+        z(tf)[2] == yf
+
 
         0.1 ≤ tf ≤ 20.0 
 
-        q̇(t)[1] == q(t)[3] * sin(u(t))  # dx/dt
-        q̇(t)[2] == q(t)[3] * cos(u(t))  # dy/dt
-        q̇(t)[3] == g * cos(u(t))        # dv/dt
+ 
+        z'(t)[1] == z(t)[3] * sin(u(t))  # dx/dt
+        z'(t)[2] == z(t)[3] * cos(u(t))  # dy/dt
+        z'(t)[3] == g * cos(u(t))        # dv/dt
 
         tf → min
     end
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 58c0d27b..96912f62 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -50,20 +50,20 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        q ∈ R³, state 
+        z ∈ R³, state
 
         u ∈ R, control
 
-        q(t0) == [x0, y0, v0]
+        z(t0) == [x0, y0, v0]
         
-        q(tf)[1] == xf
-        q(tf)[2] == yf
+        z(tf)[1] == xf
+        z(tf)[2] == yf
 
         0.1 ≤ tf ≤ 20.0
 
-        ∂(q)(t)[1] == q(t)[3] * sin(u(t))
-        ∂(q)(t)[2] == q(t)[3] * cos(u(t))
-        ∂(q)(t)[3] == g * cos(u(t))
+        ∂(z)(t)[1] == z(t)[3] * sin(u(t))
+        ∂(z)(t)[2] == z(t)[3] * cos(u(t))
+        ∂(z)(t)[3] == g * cos(u(t))
 
         tf → min
     end

From ed3a8c32956fa4c74fd9e5e747571e7c2ae260af Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sun, 8 Feb 2026 10:56:06 +0100
Subject: [PATCH 09/24] fix bug 8

---
 ext/OptimalControlModels/brachistochrone.jl   | 19 ++++++-------------
 .../brachistochrone_s.jl                      | 13 +++++--------
 2 files changed, 11 insertions(+), 21 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 71bf3f41..1c9645c7 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -36,24 +36,17 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-
-        z ∈ R³, state
-
+        A ∈ R³, state
         u ∈ R, control
 
-
-        z(t0) == [x0, y0, v0]
-
-        z(tf)[1] == xf
-        z(tf)[2] == yf
-
+        A(t0) == [x0, y0, v0]
+        
+        A(tf)[1] == xf
+        A(tf)[2] == yf
 
         0.1 ≤ tf ≤ 20.0 
 
- 
-        z'(t)[1] == z(t)[3] * sin(u(t))  # dx/dt
-        z'(t)[2] == z(t)[3] * cos(u(t))  # dy/dt
-        z'(t)[3] == g * cos(u(t))        # dv/dt
+        A'(t) == [ A(t)[3] * sin(u(t)), A(t)[3] * cos(u(t)), g * cos(u(t)) ]
 
         tf → min
     end
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 96912f62..9d73bf87 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -50,20 +50,17 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        z ∈ R³, state
-
+        A ∈ R³, state
         u ∈ R, control
 
-        z(t0) == [x0, y0, v0]
+        A(t0) == [x0, y0, v0]
         
-        z(tf)[1] == xf
-        z(tf)[2] == yf
+        A(tf)[1] == xf
+        A(tf)[2] == yf
 
         0.1 ≤ tf ≤ 20.0
 
-        ∂(z)(t)[1] == z(t)[3] * sin(u(t))
-        ∂(z)(t)[2] == z(t)[3] * cos(u(t))
-        ∂(z)(t)[3] == g * cos(u(t))
+        ∂(A)(t) == [ A(t)[3] * sin(u(t)), A(t)[3] * cos(u(t)), g * cos(u(t)) ]
 
         tf → min
     end

From eaa014a7e8349e3e90325b2ec62c9bd613d2a889 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sun, 8 Feb 2026 11:01:30 +0100
Subject: [PATCH 10/24] fix bug 9

---
 ext/OptimalControlModels/brachistochrone.jl     | 12 ++++++------
 ext/OptimalControlModels_s/brachistochrone_s.jl | 10 +++++-----
 2 files changed, 11 insertions(+), 11 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 1c9645c7..f59519b7 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -36,17 +36,17 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        A ∈ R³, state
+        x ∈ R³, state 
         u ∈ R, control
 
-        A(t0) == [x0, y0, v0]
+        x(t0) == [x0, y0, v0]
+        
+        x₁(tf) == xf
+        x₂(tf) == yf
         
-        A(tf)[1] == xf
-        A(tf)[2] == yf
-
         0.1 ≤ tf ≤ 20.0 
 
-        A'(t) == [ A(t)[3] * sin(u(t)), A(t)[3] * cos(u(t)), g * cos(u(t)) ]
+        ẋ(t) == [ x₃(t) * sin(u(t)), x₃(t) * cos(u(t)), g * cos(u(t)) ]
 
         tf → min
     end
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 9d73bf87..66d74692 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -50,17 +50,17 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        A ∈ R³, state
+        x ∈ R³, state
         u ∈ R, control
 
-        A(t0) == [x0, y0, v0]
+        x(t0) == [x0, y0, v0]
         
-        A(tf)[1] == xf
-        A(tf)[2] == yf
+        x₁(tf) == xf
+        x₂(tf) == yf
 
         0.1 ≤ tf ≤ 20.0
 
-        ∂(A)(t) == [ A(t)[3] * sin(u(t)), A(t)[3] * cos(u(t)), g * cos(u(t)) ]
+        ∂(x)(t) == [ x₃(t) * sin(u(t)), x₃(t) * cos(u(t)), g * cos(u(t)) ]
 
         tf → min
     end

From 8bbfbd877f31413f9308183895af53f0495cbafd Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sun, 8 Feb 2026 11:07:05 +0100
Subject: [PATCH 11/24] bug fix 9

---
 ext/OptimalControlModels/brachistochrone.jl     | 4 ++--
 ext/OptimalControlModels_s/brachistochrone_s.jl | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index f59519b7..717d0bdf 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -36,14 +36,14 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x ∈ R³, state 
+        x ∈ R³, state
         u ∈ R, control
 
         x(t0) == [x0, y0, v0]
         
         x₁(tf) == xf
         x₂(tf) == yf
-        
+
         0.1 ≤ tf ≤ 20.0 
 
         ẋ(t) == [ x₃(t) * sin(u(t)), x₃(t) * cos(u(t)), g * cos(u(t)) ]
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 66d74692..43035c12 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -60,7 +60,7 @@ function OptimalControlProblems.brachistochrone_s(
 
         0.1 ≤ tf ≤ 20.0
 
-        ∂(x)(t) == [ x₃(t) * sin(u(t)), x₃(t) * cos(u(t)), g * cos(u(t)) ]
+        ẋ(t) == [ x₃(t) * sin(u(t)), x₃(t) * cos(u(t)), g * cos(u(t)) ]
 
         tf → min
     end

From 3db0289263382f8ff88dd35e659f8855b0d86f82 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sun, 8 Feb 2026 11:19:23 +0100
Subject: [PATCH 12/24] vsy claude !

---
 ext/OptimalControlModels/brachistochrone.jl     | 4 +++-
 ext/OptimalControlModels_s/brachistochrone_s.jl | 4 +++-
 2 files changed, 6 insertions(+), 2 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 717d0bdf..912d0161 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -46,7 +46,9 @@ function OptimalControlProblems.brachistochrone(
 
         0.1 ≤ tf ≤ 20.0 
 
-        ẋ(t) == [ x₃(t) * sin(u(t)), x₃(t) * cos(u(t)), g * cos(u(t)) ]
+        ẋ₁(t) == x₃(t) * sin(u(t))
+        ẋ₂(t) == x₃(t) * cos(u(t))
+        ẋ₃(t) == g * cos(u(t))
 
         tf → min
     end
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 43035c12..89f77018 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -60,7 +60,9 @@ function OptimalControlProblems.brachistochrone_s(
 
         0.1 ≤ tf ≤ 20.0
 
-        ẋ(t) == [ x₃(t) * sin(u(t)), x₃(t) * cos(u(t)), g * cos(u(t)) ]
+        ẋ₁(t) == x₃(t) * sin(u(t))
+        ẋ₂(t) == x₃(t) * cos(u(t))
+        ẋ₃(t) == g * cos(u(t))
 
         tf → min
     end

From 6364334b0291e020eae10974ec8e6c3a14de9418 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Sun, 8 Feb 2026 11:25:25 +0100
Subject: [PATCH 13/24] claude 2

---
 ext/OptimalControlModels/brachistochrone.jl     | 8 ++++++--
 ext/OptimalControlModels_s/brachistochrone_s.jl | 9 ++++++---
 2 files changed, 12 insertions(+), 5 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 912d0161..deb7c9de 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -36,10 +36,14 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x ∈ R³, state
+        x₁ ∈ R, state
+        x₂ ∈ R, state
+        x₃ ∈ R, state
         u ∈ R, control
 
-        x(t0) == [x0, y0, v0]
+        x₁(t0) == x0
+        x₂(t0) == y0
+        x₃(t0) == v0
         
         x₁(tf) == xf
         x₂(tf) == yf
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 89f77018..dc7e536b 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -16,7 +16,6 @@ It then performs direct transcription to generate a discrete optimal control pro
 - `docp`: The direct optimal control problem object, representing the discretised problem.
 
 # Example
-
 ```julia-repl
 julia> using OptimalControlProblems
 
@@ -50,10 +49,14 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x ∈ R³, state
+        x₁ ∈ R, state
+        x₂ ∈ R, state
+        x₃ ∈ R, state
         u ∈ R, control
 
-        x(t0) == [x0, y0, v0]
+        x₁(t0) == x0
+        x₂(t0) == y0
+        x₃(t0) == v0
         
         x₁(tf) == xf
         x₂(tf) == yf

From a4b81193beee495268a7c8e91beba1bddcaa0b3d Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Wed, 11 Feb 2026 09:53:29 +0100
Subject: [PATCH 14/24] test

---
 ext/OptimalControlModels/brachistochrone.jl     | 9 ++-------
 ext/OptimalControlModels_s/brachistochrone_s.jl | 8 ++------
 2 files changed, 4 insertions(+), 13 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index deb7c9de..de089445 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -8,7 +8,6 @@ The goal is to move from point A to point B in minimum time under gravity.
 - `::OptimalControlBackend`: The backend type.
 - `grid_size`: Number of time steps.
 - `parameters`: Optional parameter overrides.
-
 ```julia-repl
 julia> using OptimalControlProblems
 
@@ -36,14 +35,10 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x₁ ∈ R, state
-        x₂ ∈ R, state
-        x₃ ∈ R, state
+        x = (x₁, x₂, x₃) ∈ R³, state
         u ∈ R, control
 
-        x₁(t0) == x0
-        x₂(t0) == y0
-        x₃(t0) == v0
+        x(t0) == [x0, y0, v0]
         
         x₁(tf) == xf
         x₂(tf) == yf
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index dc7e536b..a7decf85 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -49,14 +49,10 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x₁ ∈ R, state
-        x₂ ∈ R, state
-        x₃ ∈ R, state
+        x = (x₁, x₂, x₃) ∈ R³, state
         u ∈ R, control
 
-        x₁(t0) == x0
-        x₂(t0) == y0
-        x₃(t0) == v0
+        x(t0) == [x0, y0, v0]
         
         x₁(tf) == xf
         x₂(tf) == yf

From 88113301d99a2f885d34ed3849dbe8f24402d1df Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Wed, 11 Feb 2026 10:16:46 +0100
Subject: [PATCH 15/24] test 2

---
 ext/OptimalControlModels/brachistochrone.jl     | 2 +-
 ext/OptimalControlModels_s/brachistochrone_s.jl | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index de089445..43437767 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -35,7 +35,7 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x = (x₁, x₂, x₃) ∈ R³, state
+        x ∈ R³, state
         u ∈ R, control
 
         x(t0) == [x0, y0, v0]
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index a7decf85..e078fb95 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -49,7 +49,7 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x = (x₁, x₂, x₃) ∈ R³, state
+        x ∈ R³, state
         u ∈ R, control
 
         x(t0) == [x0, y0, v0]

From 8fc7ca56a7416a5e5da77bc19edffe96cc996f24 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Wed, 11 Feb 2026 10:26:23 +0100
Subject: [PATCH 16/24] test 3

---
 ext/OptimalControlModels/brachistochrone.jl     | 12 +++++++-----
 ext/OptimalControlModels_s/brachistochrone_s.jl | 12 +++++++-----
 2 files changed, 14 insertions(+), 10 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 43437767..850c43cf 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -35,23 +35,25 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x ∈ R³, state
+        z = (x₁, x₂, x₃) ∈ R³, state
         u ∈ R, control
 
-        x(t0) == [x0, y0, v0]
+        z(t0) == [x0, y0, v0]
         
         x₁(tf) == xf
         x₂(tf) == yf
 
         0.1 ≤ tf ≤ 20.0 
 
-        ẋ₁(t) == x₃(t) * sin(u(t))
-        ẋ₂(t) == x₃(t) * cos(u(t))
-        ẋ₃(t) == g * cos(u(t))
+        ż(t) == dynamics(x₃(t), u(t), g)
 
         tf → min
     end
 
+    function dynamics(x₃, u, g)
+        return [x₃ * sin(u), x₃ * cos(u), g * cos(u)]
+    end
+
     init = (
         state = [5.0, 7.5, 5.0], 
         control = 1.5, 
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index e078fb95..e9888425 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -49,23 +49,25 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x ∈ R³, state
+        z = (x₁, x₂, x₃) ∈ R³, state
         u ∈ R, control
 
-        x(t0) == [x0, y0, v0]
+        z(t0) == [x0, y0, v0]
         
         x₁(tf) == xf
         x₂(tf) == yf
 
         0.1 ≤ tf ≤ 20.0
 
-        ẋ₁(t) == x₃(t) * sin(u(t))
-        ẋ₂(t) == x₃(t) * cos(u(t))
-        ẋ₃(t) == g * cos(u(t))
+        ż(t) == dynamics(x₃(t), u(t), g)
 
         tf → min
     end
 
+    function dynamics(x₃, u, g)
+        return [x₃ * sin(u), x₃ * cos(u), g * cos(u)]
+    end
+
     # initial guess
     init = (
         state = [5.0, 7.5, 5.0], 

From 2f13b0d75043b21b4d6701b02c30c94fc04496bd Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Wed, 11 Feb 2026 10:33:22 +0100
Subject: [PATCH 17/24] test 4

---
 ext/OptimalControlModels/brachistochrone.jl     | 4 +++-
 ext/OptimalControlModels_s/brachistochrone_s.jl | 7 ++++---
 2 files changed, 7 insertions(+), 4 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 850c43cf..4204cf15 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -45,7 +45,9 @@ function OptimalControlProblems.brachistochrone(
 
         0.1 ≤ tf ≤ 20.0 
 
-        ż(t) == dynamics(x₃(t), u(t), g)
+        ẋ₁(t) == x₃(t) * sin(u(t))
+        ẋ₂(t) == x₃(t) * cos(u(t))
+        ẋ₃(t) == g * cos(u(t))
 
         tf → min
     end
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index e9888425..891aa4fd 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -45,7 +45,6 @@ function OptimalControlProblems.brachistochrone_s(
 
     # model
     ocp = @def begin
-        
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
@@ -57,9 +56,11 @@ function OptimalControlProblems.brachistochrone_s(
         x₁(tf) == xf
         x₂(tf) == yf
 
-        0.1 ≤ tf ≤ 20.0
+        0.1 ≤ tf ≤ 20.0 
 
-        ż(t) == dynamics(x₃(t), u(t), g)
+        ẋ₁(t) == x₃(t) * sin(u(t))
+        ẋ₂(t) == x₃(t) * cos(u(t))
+        ẋ₃(t) == g * cos(u(t))
 
         tf → min
     end

From f1c2edf601988b9e54d65b5fbeec40c4e49e7fff Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Wed, 11 Feb 2026 10:37:25 +0100
Subject: [PATCH 18/24] test 5

---
 ext/OptimalControlModels/brachistochrone.jl     |  8 ++------
 ext/OptimalControlModels_s/brachistochrone_s.jl | 11 ++++-------
 2 files changed, 6 insertions(+), 13 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 4204cf15..de089445 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -35,10 +35,10 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        z = (x₁, x₂, x₃) ∈ R³, state
+        x = (x₁, x₂, x₃) ∈ R³, state
         u ∈ R, control
 
-        z(t0) == [x0, y0, v0]
+        x(t0) == [x0, y0, v0]
         
         x₁(tf) == xf
         x₂(tf) == yf
@@ -52,10 +52,6 @@ function OptimalControlProblems.brachistochrone(
         tf → min
     end
 
-    function dynamics(x₃, u, g)
-        return [x₃ * sin(u), x₃ * cos(u), g * cos(u)]
-    end
-
     init = (
         state = [5.0, 7.5, 5.0], 
         control = 1.5, 
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 891aa4fd..a7decf85 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -45,18 +45,19 @@ function OptimalControlProblems.brachistochrone_s(
 
     # model
     ocp = @def begin
+        
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        z = (x₁, x₂, x₃) ∈ R³, state
+        x = (x₁, x₂, x₃) ∈ R³, state
         u ∈ R, control
 
-        z(t0) == [x0, y0, v0]
+        x(t0) == [x0, y0, v0]
         
         x₁(tf) == xf
         x₂(tf) == yf
 
-        0.1 ≤ tf ≤ 20.0 
+        0.1 ≤ tf ≤ 20.0
 
         ẋ₁(t) == x₃(t) * sin(u(t))
         ẋ₂(t) == x₃(t) * cos(u(t))
@@ -65,10 +66,6 @@ function OptimalControlProblems.brachistochrone_s(
         tf → min
     end
 
-    function dynamics(x₃, u, g)
-        return [x₃ * sin(u), x₃ * cos(u), g * cos(u)]
-    end
-
     # initial guess
     init = (
         state = [5.0, 7.5, 5.0], 

From c3d34108bc9dc764bb2d7dc7be4cc127fcd61afb Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Wed, 11 Feb 2026 10:42:04 +0100
Subject: [PATCH 19/24] test 6

---
 ext/OptimalControlModels/brachistochrone.jl     | 14 +++++++-------
 ext/OptimalControlModels_s/brachistochrone_s.jl | 14 +++++++-------
 2 files changed, 14 insertions(+), 14 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index de089445..ebef9787 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -35,19 +35,19 @@ function OptimalControlProblems.brachistochrone(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x = (x₁, x₂, x₃) ∈ R³, state
+        z = (px, py, v) ∈ R³, state
         u ∈ R, control
 
-        x(t0) == [x0, y0, v0]
+        z(t0) == [x0, y0, v0]
         
-        x₁(tf) == xf
-        x₂(tf) == yf
+        px(tf) == xf
+        py(tf) == yf
 
         0.1 ≤ tf ≤ 20.0 
 
-        ẋ₁(t) == x₃(t) * sin(u(t))
-        ẋ₂(t) == x₃(t) * cos(u(t))
-        ẋ₃(t) == g * cos(u(t))
+        ṗx(t) == v(t) * sin(u(t))
+        ṗy(t) == v(t) * cos(u(t))
+        v̇(t) == g * cos(u(t))
 
         tf → min
     end
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index a7decf85..fdaafff4 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -49,19 +49,19 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
-        x = (x₁, x₂, x₃) ∈ R³, state
+        z = (px, py, v) ∈ R³, state
         u ∈ R, control
 
-        x(t0) == [x0, y0, v0]
+        z(t0) == [x0, y0, v0]
         
-        x₁(tf) == xf
-        x₂(tf) == yf
+        px(tf) == xf
+        py(tf) == yf
 
         0.1 ≤ tf ≤ 20.0
 
-        ẋ₁(t) == x₃(t) * sin(u(t))
-        ẋ₂(t) == x₃(t) * cos(u(t))
-        ẋ₃(t) == g * cos(u(t))
+        ṗx(t) == v(t) * sin(u(t))
+        ṗy(t) == v(t) * cos(u(t))
+        v̇(t) == g * cos(u(t))
 
         tf → min
     end

From 31f0be7ea03283edb2c20718b31c67c6ac307464 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Thu, 12 Feb 2026 09:23:25 +0100
Subject: [PATCH 20/24] normalement c'est bon !

---
 ext/OptimalControlModels/brachistochrone.jl   | 43 +++++++++++++------
 .../brachistochrone_s.jl                      | 11 +++--
 2 files changed, 36 insertions(+), 18 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index ebef9787..4be0533f 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -1,18 +1,30 @@
 """
 $(TYPEDSIGNATURES)
 
-Constructs the **Brachistochrone** optimal control problem.
-The goal is to move from point A to point B in minimum time under gravity.
+Constructs an **OptimalControl problem** representing the Brachistochrone problem using the OptimalControl backend.
+The function sets up the state and control variables, boundary conditions, dynamics, and the objective functional.
+It then performs direct transcription to generate a discrete optimal control problem (DOCP).
 
 # Arguments
-- `::OptimalControlBackend`: The backend type.
-- `grid_size`: Number of time steps.
-- `parameters`: Optional parameter overrides.
+
+- `::OptimalControlBackend`: Placeholder type to specify the OptimalControl backend or solver interface.
+- `grid_size::Int=grid_size_data(:brachistochrone)`: (Keyword) Number of discretisation points for the direct transcription grid.
+- `parameters::Union{Nothing,NamedTuple}=nothing`: (Keyword) Custom parameters to override defaults.
+
+# Returns
+
+- `docp`: The direct optimal control problem object, representing the discretised problem.
+
+# Example
 ```julia-repl
 julia> using OptimalControlProblems
 
 julia> docp = OptimalControlProblems.brachistochrone(OptimalControlBackend(); N=100);
 ```
+
+# References
+
+- Dymos Brachistochrone: https://openmdao.github.io/dymos/examples/brachistochrone/brachistochrone.html
 """
 function OptimalControlProblems.brachistochrone(
     ::OptimalControlBackend,
@@ -34,27 +46,30 @@ function OptimalControlProblems.brachistochrone(
     ocp = @def begin
         tf ∈ R, variable
         t ∈ [t0, tf], time
-
+        
         z = (px, py, v) ∈ R³, state
         u ∈ R, control
 
-        z(t0) == [x0, y0, v0]
+        0.1 ≤ tf ≤ 20.0 
+
+        px(t0) == x0
+        py(t0) == y0
+        v(t0)  == v0
         
         px(tf) == xf
         py(tf) == yf
 
-        0.1 ≤ tf ≤ 20.0 
-
-        ṗx(t) == v(t) * sin(u(t))
-        ṗy(t) == v(t) * cos(u(t))
-        v̇(t) == g * cos(u(t))
+        ∂(px)(t) == v(t) * sin(u(t))
+        ∂(py)(t) == v(t) * cos(u(t))
+        ∂(v)(t)  == g * cos(u(t))
 
+        # Objectif
         tf → min
     end
 
     init = (
-        state = [5.0, 7.5, 5.0], 
-        control = 1.5, 
+        state = [x0, y0, v0], 
+        control = 0.0, 
         variable = 1.0
     )
 
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index fdaafff4..73b1d900 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -49,26 +49,29 @@ function OptimalControlProblems.brachistochrone_s(
         tf ∈ R, variable
         t ∈ [t0, tf], time
 
+    
         z = (px, py, v) ∈ R³, state
         u ∈ R, control
 
+       
         z(t0) == [x0, y0, v0]
         
+       
         px(tf) == xf
         py(tf) == yf
 
         0.1 ≤ tf ≤ 20.0
 
-        ṗx(t) == v(t) * sin(u(t))
-        ṗy(t) == v(t) * cos(u(t))
-        v̇(t) == g * cos(u(t))
+        ∂(px)(t) == v(t) * sin(u(t))
+        ∂(py)(t) == v(t) * cos(u(t))
+        ∂(v)(t)  == g * cos(u(t))
 
         tf → min
     end
 
     # initial guess
     init = (
-        state = [5.0, 7.5, 5.0], 
+        state = [x0, y0, v0], 
         control = 1.57, 
         variable = 2.0
     )

From 82493ebc4ca00083b1d289c5595f6d6c61707263 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Thu, 12 Feb 2026 11:57:02 +0100
Subject: [PATCH 21/24] gemini cli

---
 ext/JuMPModels/brachistochrone.jl               | 4 ++--
 ext/OptimalControlModels/brachistochrone.jl     | 2 +-
 ext/OptimalControlModels_s/brachistochrone_s.jl | 2 +-
 3 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/ext/JuMPModels/brachistochrone.jl b/ext/JuMPModels/brachistochrone.jl
index 7d56ff07..acfc36ca 100644
--- a/ext/JuMPModels/brachistochrone.jl
+++ b/ext/JuMPModels/brachistochrone.jl
@@ -106,8 +106,8 @@ function OptimalControlProblems.brachistochrone(
             # dx/dt = v * sin(u)
             dx[i = 0:N], v[i] * sin(u[i])
             
-            # dy/dt = v * cos(u)
-            dy[i = 0:N], v[i] * cos(u[i])
+            # dy/dt = -v * cos(u)
+            dy[i = 0:N], -v[i] * cos(u[i])
             
             # dv/dt = g * cos(u)
             dv[i = 0:N], g * cos(u[i])
diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 4be0533f..42cc77e9 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -60,7 +60,7 @@ function OptimalControlProblems.brachistochrone(
         py(tf) == yf
 
         ∂(px)(t) == v(t) * sin(u(t))
-        ∂(py)(t) == v(t) * cos(u(t))
+        ∂(py)(t) == -v(t) * cos(u(t))
         ∂(v)(t)  == g * cos(u(t))
 
         # Objectif
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 73b1d900..27374799 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -63,7 +63,7 @@ function OptimalControlProblems.brachistochrone_s(
         0.1 ≤ tf ≤ 20.0
 
         ∂(px)(t) == v(t) * sin(u(t))
-        ∂(py)(t) == v(t) * cos(u(t))
+        ∂(py)(t) == -v(t) * cos(u(t))
         ∂(v)(t)  == g * cos(u(t))
 
         tf → min

From ad0f6d43ccdef449ef9099ba49d4dce4d29528f6 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Thu, 12 Feb 2026 18:20:00 +0100
Subject: [PATCH 22/24] bug jump

---
 ext/JuMPModels/brachistochrone.jl | 5 +----
 1 file changed, 1 insertion(+), 4 deletions(-)

diff --git a/ext/JuMPModels/brachistochrone.jl b/ext/JuMPModels/brachistochrone.jl
index acfc36ca..32f54b76 100644
--- a/ext/JuMPModels/brachistochrone.jl
+++ b/ext/JuMPModels/brachistochrone.jl
@@ -64,10 +64,7 @@ function OptimalControlProblems.brachistochrone(
 
     model[:time_grid] = () -> range(t0, value(tf), N+1)
     model[:state_components] = ["x", "y", "v"]
-    
-    model[:costate_components] = String[]
-    # ----------------------
-    
+    model[:costate_components] = ["∂x", "∂y", "∂v"]
     model[:control_components] = ["u"]
     model[:variable_components] = ["tf"]
 

From a379ed5f90bba2ca122f9a12f1d01330be2a7178 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Thu, 12 Feb 2026 18:39:01 +0100
Subject: [PATCH 23/24] c'est bon !

---
 ext/JuMPModels/brachistochrone.jl           | 32 ++++++++++-----------
 ext/OptimalControlModels/brachistochrone.jl |  6 ++--
 2 files changed, 20 insertions(+), 18 deletions(-)

diff --git a/ext/JuMPModels/brachistochrone.jl b/ext/JuMPModels/brachistochrone.jl
index 32f54b76..71319d34 100644
--- a/ext/JuMPModels/brachistochrone.jl
+++ b/ext/JuMPModels/brachistochrone.jl
@@ -54,8 +54,8 @@ function OptimalControlProblems.brachistochrone(
         model,
         begin
             0.1 <= tf <= 20.0, (start = 2.0)
-            x[0:N]
-            y[0:N]
+            px[0:N]
+            py[0:N]
             v[0:N]
             u[0:N]
         end
@@ -63,15 +63,15 @@ function OptimalControlProblems.brachistochrone(
 
 
     model[:time_grid] = () -> range(t0, value(tf), N+1)
-    model[:state_components] = ["x", "y", "v"]
-    model[:costate_components] = ["∂x", "∂y", "∂v"]
+    model[:state_components] = ["px", "py", "v"]
+    model[:costate_components] = ["∂px", "∂py", "∂v"]
     model[:control_components] = ["u"]
     model[:variable_components] = ["tf"]
 
     for i in 0:N
         alpha = i / N
-        set_start_value(x[i], x0 + alpha * (xf - x0))
-        set_start_value(y[i], y0 + alpha * (yf - y0))
+        set_start_value(px[i], x0 + alpha * (xf - x0))
+        set_start_value(py[i], y0 + alpha * (yf - y0))
         set_start_value(v[i], v0 + alpha * 10.0) # Estimate speed
         set_start_value(u[i], 1.57) # ~90 degrees
     end
@@ -81,13 +81,13 @@ function OptimalControlProblems.brachistochrone(
         model,
         begin
             # Start
-            x[0] == x0
-            y[0] == y0
+            px[0] == x0
+            py[0] == y0
             v[0] == v0
             
             # End
-            x[N] == xf
-            y[N] == yf
+            px[N] == xf
+            py[N] == yf
             # v[N] is free
         end
     )
@@ -100,11 +100,11 @@ function OptimalControlProblems.brachistochrone(
             dt, (tf - t0) / N
 
             # Dynamics expressions (Dymos formulation)
-            # dx/dt = v * sin(u)
-            dx[i = 0:N], v[i] * sin(u[i])
+            # dpx/dt = v * sin(u)
+            dpx[i = 0:N], v[i] * sin(u[i])
             
-            # dy/dt = -v * cos(u)
-            dy[i = 0:N], -v[i] * cos(u[i])
+            # dpy/dt = -v * cos(u)
+            dpy[i = 0:N], -v[i] * cos(u[i])
             
             # dv/dt = g * cos(u)
             dv[i = 0:N], g * cos(u[i])
@@ -115,8 +115,8 @@ function OptimalControlProblems.brachistochrone(
     @constraints(
         model,
         begin
-            ∂x[i = 1:N], x[i] == x[i - 1] + 0.5 * dt * (dx[i] + dx[i - 1])
-            ∂y[i = 1:N], y[i] == y[i - 1] + 0.5 * dt * (dy[i] + dy[i - 1])
+            ∂px[i = 1:N], px[i] == px[i - 1] + 0.5 * dt * (dpx[i] + dpx[i - 1])
+            ∂py[i = 1:N], py[i] == py[i - 1] + 0.5 * dt * (dpy[i] + dpy[i - 1])
             ∂v[i = 1:N], v[i] == v[i - 1] + 0.5 * dt * (dv[i] + dv[i - 1])
         end
     )
diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 42cc77e9..4994fae7 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -69,15 +69,17 @@ function OptimalControlProblems.brachistochrone(
 
     init = (
         state = [x0, y0, v0], 
-        control = 0.0, 
-        variable = 1.0
+        control = 1.57, 
+        variable = 2.0
     )
 
     docp = direct_transcription(
         ocp,
         description...;
+        lagrange_to_mayer=false,
         init = init,
         grid_size = grid_size,
+        disc_method=:trapeze,
         kwargs...
     )
 

From e47616a32e2d543fdb19f9d449cc31a3017933e5 Mon Sep 17 00:00:00 2001
From: Amiel <amielmetier@gmail.com>
Date: Thu, 12 Feb 2026 18:47:43 +0100
Subject: [PATCH 24/24] ok

---
 ext/OptimalControlModels/brachistochrone.jl     | 12 +++++++++---
 ext/OptimalControlModels_s/brachistochrone_s.jl | 11 ++++++++---
 2 files changed, 17 insertions(+), 6 deletions(-)

diff --git a/ext/OptimalControlModels/brachistochrone.jl b/ext/OptimalControlModels/brachistochrone.jl
index 4994fae7..860ea7eb 100644
--- a/ext/OptimalControlModels/brachistochrone.jl
+++ b/ext/OptimalControlModels/brachistochrone.jl
@@ -67,10 +67,16 @@ function OptimalControlProblems.brachistochrone(
         tf → min
     end
 
+    # initial guess: linear interpolation to match JuMP
+    tf_guess = 2.0
     init = (
-        state = [x0, y0, v0], 
-        control = 1.57, 
-        variable = 2.0
+        state = t -> [
+            x0 + (t - t0) / (tf_guess - t0) * (xf - x0),
+            y0 + (t - t0) / (tf_guess - t0) * (yf - y0),
+            v0 + (t - t0) / (tf_guess - t0) * 10.0
+        ],
+        control = 1.57,
+        variable = tf_guess
     )
 
     docp = direct_transcription(
diff --git a/ext/OptimalControlModels_s/brachistochrone_s.jl b/ext/OptimalControlModels_s/brachistochrone_s.jl
index 27374799..8d46e45f 100644
--- a/ext/OptimalControlModels_s/brachistochrone_s.jl
+++ b/ext/OptimalControlModels_s/brachistochrone_s.jl
@@ -69,11 +69,16 @@ function OptimalControlProblems.brachistochrone_s(
         tf → min
     end
 
-    # initial guess
+    # initial guess: linear interpolation to match JuMP
+    tf_guess = 2.0
     init = (
-        state = [x0, y0, v0], 
+        state = t -> [
+            x0 + (t - t0) / (tf_guess - t0) * (xf - x0),
+            y0 + (t - t0) / (tf_guess - t0) * (yf - y0),
+            v0 + (t - t0) / (tf_guess - t0) * 10.0
+        ],
         control = 1.57, 
-        variable = 2.0
+        variable = tf_guess
     )
 
     # discretise the optimal control problem