Indirect method by shooting

Definition

Let $T > 0$, $x_0, x_1 \in \R^n$, let $U \subseteq \R^m$ be nonempty, and let $f \colon \R^n \times U \to \R^n$ and $L \colon \R^n \times U \to \R$ be continuously differentiable in $x$. Consider the problem of minimising $\int_0^T L(x, u) \, dt$ subject to $x' = f(x, u)$, $x(0) = x_0$, $x(T) = x_1$. Let $H(x, u, p) = p \cdot f(x, u) - L(x, u)$ be the Hamiltonian in the normal case of Pontryagin's maximum principle. Assume that for all $(x, p)$ the function $u \mapsto H(x, u, p)$ has a unique maximiser $u^*(x, p)$ in $U$, and that $u^*$ is continuously differentiable. For $p_0 \in \R^n$, let $(x(\cdot\,; p_0), p(\cdot\,; p_0))$ be the solution on $[0, T]$, if it exists, of $$x' = f(x, u^*(x, p)), \qquad p' = -\frac{\partial H}{\partial x}(x, u^*(x, p), p), \qquad x(0) = x_0, \qquad p(0) = p_0.$$ The shooting function is $S(p_0) = x(T; p_0) - x_1$. The indirect method by shooting computes a zero $p_0^*$ of $S$, for example by Newton's method, and returns the control $u(t) = u^*(x(t; p_0^*), p(t; p_0^*))$.