# two-point boundary value problems

Consider the problem of minimizing an integral cost $\int_0^T c(x(t), u(t))\, dt$ over measurable controls $u:[0, T]\to \mathbb U$, subject to a finite-dimensional control system $\dot x(t) = f(x(t), u(t))$ with given initial condition $x(0) = \bar x$ and given final condition $x(T) = \hat x$, assuming that $T > 0$ is given, the maps $c:\mathbb R^d\times\mathbb R^m\to\mathbb R$ and $f:\mathbb R^d\times\mathbb R^m \to \mathbb R^d$ is continuously differentiable, and that $\mathbb U\subset\mathbb R^m$ is non-empty and compact. (It is known that under milder assumptions on the function $c:\mathbb R^d\times\mathbb R^m\to\mathbb R$ there exists a solution of the above problem; see, e.g., doi:10.1007/978-1-4612-6380-7 for the relevant theory.)

Recall that the Pontryagin maximum principle (PMP) provides necessary conditions for solutions to the preceding optimal control problem; the resulting characterization of optimal controls turns out in the form of a two-point boundary value problem that looks like the following:

If $u_\star:[0, T]\to\mathbb U$ solves the above problem and $x_\star:[0, T]\to\mathbb R^d$ is the corresponding solution of the control system, then there exists a pair $(\eta, p)$ with $\eta\in\{0, 1\}$, and $p:[0, T]\to\mathbb R^d$ absolutely continuous, satisfying $$\dot x_\star(t) = f(x_\star(t), u_\star(t))\quad\text{for a.e. }t \text{ with }x_\star(0) = \bar x \text{ and } x_\star(T) = \hat x,$$ $$-\dot p(t) = \partial_x f(x_\star(t), u_\star(t)) p(t) - \eta \partial_x c(x_\star(t), u_\star(t))\quad\text{for a.e. }t,$$ $$u_\star(t)\in\text{argmax}_{v\in\mathbb U} \bigl\{ \langle p(t), f(x_\star(t), v)\rangle - \eta c(x_\star(t), v) \bigr\},$$ plus a few other conditions. Of course, the preceding description assumes that $f$ and $c$ are continuously differentiable, but there are results that relax this requirement, most notably those proposed by Francis Clarke: see, e.g., doi:10.1007/978-1-4471-4820-3, Chapter 22, for the relevant material. While it is currently not known (see the thread posted a few years ago: existence of optimal control) whether the optimal control problem above admits solutions if the instantaneous cost function $c$ is discontinuous in $u$, the PMP proposed by Clarke (see Theorem 22.26 in the reference above) nevertheless provides necessary conditions for optimality.

Notice that the characterization above requires us to solve two $d$-dimensional o.d.e.s simultaneously with $2d$ given boundary conditions; however, there are no boundary conditions for $p$. This is precisely the case that I'm interested in: two-point boundary value problems such as the one above in terms of the joint variables $(x, p)$ for $p$ has no given boundary condition. I'm seeking pointers to any relevant articles/books/lecture-notes that treats this topic.