For loop and control of ensemble

[ocots]

Consider a parameterized dynamics $f(x, u, \theta)$, with $x\in\mathbb{R}$ and an optimal control problem where we have the dynamics:

$$ \dot{x}_i(t) = f(x_i(t), u(t), \theta_i), \quad i = 1,\ldots,N. $$

and some constraints of the form:

$$ c(x_i(t_0), x_i(t_f), \theta_i) = 0, \quad i = 1,\ldots,N. $$

It would be nice to have the possibility to write something like:

N = 100
ocp = @def begin
    t ∈ [0, tf], time
    x ∈ R^(2N), state
    u ∈ R, control 

    x[i](0) == [1, 0], for i ∈ 1:N end
    x[i](tf) == [0, 0], for i ∈ 1:N end

    ẋ[i](t) == f(xi(t), u(t), θ[i]), for i ∈ 1:N end

    ∫( u(t)^2 ) → min
end

[ocots]

For the moment, we can write something like that:

using OptimalControl
using NLPModelsIpopt
using Plots

c0(N) = repeat([-1, 0], N)
cf(N) = repeat([ 0, 0], N)
f(x, u) = [x[2], u]
f(x, u, N) = reduce(vcat, [f(x[2i-1:2i], u) for i ∈ 1:N])

N = 3
ocp = @def begin
    t ∈ [0, 1], time
    x ∈ R^(2N), state
    u ∈ R, control
    x(0) == c0(N)
    x(1) == cf(N)
    ẋ(t) == f(x(t), u(t), N)
    ∫( 0.5u(t)^2 ) → min
end

sol = solve(ocp)

[ocots]

@ox217 Welcome.

[ocots]

@jbcaillau @PierreMartinon Comments are welcomed.

[ox217]

Thank you @ocots

I would like a slightly more general dynamic:
$\dot x_i=f(x,u,\theta_i)$,
and the same for the final constraints
$c_i(x(t_0),x(t_f),\theta_i)\le 0$,

In other words, the dynamics of a single state can be affected by other states.

[ocots]

Can you try to write an example in Julia with the @def macro?

[ox217]

Now, I think the example that you wrote is ok, I didn't note that $x_i\in R^{N}$ with $N\ge 2$.

I add the final constraint:

N = 100
ocp = @def begin
    t ∈ [0, tf], time
    x ∈ R^(2N), state
    u ∈ R, control 

    x[i](0) == [1, 0], for i ∈ 1:N end
    x[i](tf) == [0, 0], for i ∈ 1:N end

    ẋ[i](t) == f(xi(t), u(t), θ[i]), for i ∈ 1:N end
    c(x[i](tf),θ[i]) <=0  for i ∈ 1:N end

    ∫( u(t)^2 ) → min
end