Direct collocation

Remarks

Each choice of collocation points corresponds to an implicit Runge–Kutta method; the state on each interval is a polynomial interpolant of its values. With $s = 2$ and the endpoints $\tau_k, \tau_{k+1}$ as collocation points, clause 2 becomes the trapezoidal rule $x_{k+1} - x_k = \frac{h_k}{2} \left( f(x_k, u_k, \tau_k) + f(x_{k+1}, u_{k+1}, \tau_{k+1}) \right)$ with $h_k = \tau_{k+1} - \tau_k$. Hermite–Simpson collocation uses the endpoints and the midpoint with a cubic state. The nonlinear program is large but sparse, and it is usually solved by sequential quadratic programming or by interior-point methods.

Example: minimise $\int_0^1 \frac{1}{2} u^2 \, dt$ with $x' = u$, $x(0) = 0$, $x(1) = 1$, using trapezoidal collocation on a uniform mesh of step $h = 1/N$. The program is: minimise $\sum_{k=0}^{N-1} \frac{h}{4} (u_k^2 + u_{k+1}^2)$ subject to $x_{k+1} - x_k = \frac{h}{2} (u_k + u_{k+1})$, $x_0 = 0$, $x_N = 1$. The choice $u_k = 1$, $x_k = k h$ satisfies all constraints. It also satisfies the optimality conditions, with all multipliers of the dynamics constraints equal to one constant ($1$ or $-1$, by sign convention). It reproduces the exact solution $u = 1$, $x(t) = t$ at the grid points.

Non-example: direct single shooting integrates the dynamics forward from a guessed control and optimises only over the control values. It is not collocation, because the states are not unknowns of the program.