Global Controllability by Nice Controls

Let ẋ = f(x, u(t)) be a control system, where the state x resides in a finite-dimensional, differentiable manifold M, and the control u takes values in a set Ω, called the control space, which for our purposes is assumed to be a metric space. Such a system is called globally controllable if, given any pair of points x ₀, x ₁ in M, there exists an “admissible” control u : ℝ → Ω that steers x ₀ to x ₁ along a trajectory of the system corresponding to u. In optimal control problems, one must usually allow controls to be only Lebesgue measurable functions of t if one is to have any hope of proving a priori existence of optimal controls. Of course, once existence has been established within the class of measurable controls, it is a natural problem to examine what additional regularity properties, if any, an optimal control might have.