Trajectory optimization

Definition

Let $U \subseteq \R^m$ be nonempty, and let $f \colon \R^n \times U \times \R \to \R^n$, $L \colon \R^n \times U \times \R \to \R$, $\phi \colon \R \times \R^n \times \R \times \R^n \to \R$ be continuous. Let $c \colon \R^n \times U \times \R \to \R^q$ give path constraints and $b \colon \R \times \R^n \times \R \times \R^n \to \R^r$ give boundary conditions. A trajectory optimization problem asks for times $t_0 < t_f$, a measurable control $u \colon [t_0, t_f] \to U$ and an absolutely continuous state $x \colon [t_0, t_f] \to \R^n$ that minimise $$\phi(t_0, x(t_0), t_f, x(t_f)) + \int_{t_0}^{t_f} L(x(t), u(t), t) \, dt$$ subject to:

1. the control system $x'(t) = f(x(t), u(t), t)$ for almost every $t$;
2. the path constraints $c(x(t), u(t), t) \le 0$ componentwise for almost every $t$;
3. the boundary conditions $b(t_0, x(t_0), t_f, x(t_f)) = 0$.

Trajectory optimization is the numerical solution of such problems, which yields an open-loop control and its trajectory.