Simple exponential: energy minimisation
The energy minimisation simple exponential problem consists in minimising
\[ \frac{1}{2}\int_{0}^{1} u^2(t) \, \mathrm{d}t\]
subject to the constraints
\[ \dot x(t) = - x(t) + u(t), u(t) \in \mathbb{R}\]
and the limit conditions
\[ x(0) = -1, \quad x(1) = 0.\]
You can access the problem in the CTProblems package:
using CTProblems
prob = Problem(:exponential, :energy)
title = simple exponential - energy min
model (Type) = OptimalControlModel{Autonomous, Fixed}
solution (Type) = OptimalControlSolution
Then, the model is given by
prob.model
The (autonomous) optimal control problem is given by:
t ∈ [t0, tf], time
x ∈ R, state
u ∈ R, control
x(t0) == x0, initial_con
x(tf) == xf, final_con
ẋ(t) == -(x(t)) + u(t)
∫(0.5 * u(t) ^ 2) → min
The (autonomous) optimal control problem is of the form:
minimize J(x, u) = ∫ f⁰(x(t), u(t)) dt, over [0, 1]
subject to
ẋ(t) = f(x(t), u(t)), t in [0, 1] a.e.,
ϕl ≤ ϕ(x(0), x(1)) ≤ ϕu,
where x(t) ∈ R and u(t) ∈ R.
Declarations (* required):
╭────────┬────────┬──────────┬──────────┬───────────┬────────────┬─────────────╮
│ times* │ state* │ control* │ variable │ dynamics* │ objective* │ constraints │
├────────┼────────┼──────────┼──────────┼───────────┼────────────┼─────────────┤
│ V │ V │ V │ X │ V │ V │ V │
╰────────┴────────┴──────────┴──────────┴───────────┴────────────┴─────────────╯
You can plot the solution.
using Plots
plot(prob.solution)