Does there exist a (weak) version of Pontryagin's minimum principle in which the control is allowed to be just Lebesgue integrable? I am mostly familiar with the 1975 text of Fleming & Rishel, wherein the existence results for optimal control problems (in Chapter 3) assert the existence of a Lebesgue-integrable control, whereas Pontryagin's principle itself (in Chapter 2) is stated only for piecewise continuous controls. The authors discuss this gap in Chapter 3, Section 6.1, but don't provide any stronger existence results except for linear problems.

Since that book is almost 50 years old, I wonder whether better results are available; I'm hoping for either:

- Existence of a piecewise continuous optimal control; or
- A version of Pontryagin that holds for Lebesgue-integrable controls.

My problem of interest is linear in the control but nonlinear and nonconvex in the state variables.