# Geometric bang-bang theorem for nonlinear optimal control

The classical bang-bang theorem is usually stated for linear systems (e.g. Control Theory from the Geometric Viewpoint by Agrachev-Sachkov, p. 209). Sussman proved a nice generalization for systems affine in control $$\dot{x}=f_0(x)+f_1(x)u$$, which gives geometric conditions in terms of Lie brackets of $$f_i$$ for the control to be bang-bang (or not), bounds on the number of switches, etc. They depend only on the Lie structure (which is the analog of the Riemannian curvature here), and hence are invariant under coordinate transformations, linear or nonlinear. Unfortunately, the versions in his papers (e.g. An introduction to the coordinate-free Maximum Principle, p.65) are always stated for the time minimization problem only. Wikipedia mentions, without a reference, that bang-bang solutions also arise when the Hamiltonian is linear in the control variable". Examples readily confirm that for functionals of the form $$\int_0^T L_0(x)+L_1(x)u\,dt+l(x(T))$$ (the so-called Bolza problem).

But I could not find the theory spelled out for such problems, e.g. general conditions that rule out singular controls, bounds on the number of switches, etc., in terms of $$f_i,L_j$$. I am aware of the formal trick that eliminates the integral by introducing new "time", but it involves division of $$f_i$$ by $$L$$, and even if $$L_1=0$$ tracking formulas, and technical assumptions, through the transformation is messy.

Is there some principled difficulty with extending the theory to the Bolza functionals? Is it written up somewhere explicitly?