Double integrator: energy minimisation

The energy minimisation problem consists in minimising the cost functional in Lagrange form

\[ \frac{1}{2}\int_{t_0}^{t_f} u^2(t) \, \mathrm{d}t\]

subject to the constraints for $t \in [t_0, t_f]$ a.e.

\[ \dot x_1(t) = x_2(t), \quad \dot x_2(t) = u(t) \in \mathbb{R},\]

and the limit conditions

\[ x(t_0) = (a, b), \quad x(t_f) = x_f,\]

with $a$, $b \in \mathbb{R}$, $x_f \in \mathbb{R}^2$ and $t_f \ge t_0 = 0$ fixed. Denoting $H(x, p, u) \coloneqq p_1 x_2 + p_2 u - u^2/2$ the pseudo-Hamiltonian (in the normal case), then, by the Pontryagin Maximum Principle, the maximising control is given in feedback form by

\[ u(x, p) \coloneqq p_2.\]

Thus, solving this optimal control problem leads to solve the following boundary value problem:

\[ \left\{ \begin{aligned} & \dot{x}_1(t) = x_2(t), \quad \dot{x}_2(t) = u(x(t), p(t)), \quad \dot{p}_1(t) = 0, \quad \dot{p}_2(t) = -p_1(t), \\ & x(0) = (a, b), \quad x(t_f) = x_f. \end{aligned} \right.\]

where $p(t) = (p_1(t), p_2(t))$ is the costate vector. Integrating first the differential system, we get

\[ \left\{ \begin{aligned} p_1(t) &= \alpha, \\[0.3em] p_2(t) &= -\alpha t + \beta, \\[0.3em] x_2(t) &= b + \int_0^t (-\alpha s + \beta) \mathrm{d} s = b + \beta t - \frac{\alpha}{2} t^2, \\[0.3em] x_1(t) &= a + b t + \frac{\beta}{2} t^2 - \frac{\alpha}{6} t^3, \end{aligned} \right. \]

with $p(0) = (\alpha, \beta)$ the parameters to identify. Solving the boundary value problem, that is finding $(\alpha, \beta)$, is then equivalent to solve the shooting equation

\[ S(\alpha, \beta) \coloneqq \begin{pmatrix} \displaystyle a + b\, t_f + \frac{\beta}{2} t_f^2 - \frac{\alpha}{6} t_f^3 \\[0.5em] \displaystyle b + \beta\, t_f - \frac{\alpha}{2} t_f^2 \end{pmatrix} - x_f = A\, p_0 - B = 0,\]

where we set

\[ A \coloneqq \begin{pmatrix} -\frac{t_f^3}{6} & \frac{t_f^2}{2} \\[0.5em] -\frac{t_f^2}{2} & t_f \end{pmatrix}, \quad B \coloneqq ( -a-b t_f, -b) + x_f \quad \text{and} \quad p_0 \coloneqq (\alpha, \beta).\]

Note that the shooting function$S$ is linear. If $t_f \ne 0$, then, $A$ is invertible since $\det A = t_f^4/12$ and the solution is then given by

\[ p^*_0 = A^{-1} B.\]

For an illusration we set $t_f=1$, $x_0 = (-1, 0)$ and $x_f = (0, 0)$, see the following figure. The red sphere represents the solution.

You can access the problem from the CTProblems.jl package:

using CTProblems
prob = Problem(:integrator, :energy, :x_dim_2, :u_dim_1, :lagrange, :noconstraints)

title           = Double integrator energy - minimise ∫ u²
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 ∈ R², state
    u ∈ R, control
    x(t0) == [-1, 0], initial_con
    x(tf) == [0, 0], final_con
    ẋ(t) == A * x(t) + B * 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) ∈ R² 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, size=(700, 700))