On Discontinuous Optimal Control

Abstract

Consider time-optimal control problem

$$ \dot x(t) = f(x,(t),u(t)); $$

(1)

$$ x(0) = {{x}_{0}},x(T){{x}_{1}}; $$

(2)

$$ u(t) \in U; $$

(3)

$$ T \to \min $$

(4)

where phase vector x belongs to the state space R ⁿ,U is compact convex set. Due to Filippov’s existence theorem [1] it is natural to take Labesgue’s measurable functions satisfying constraints (3) as the class of admissible controls. But the following questions arise: if the optimal control does exist which regular properties might it have? Is the measurability hypothesis vitally important? Or is it possible to ensure that the optimal control actually belongs to some more regular functional class: the class of piecewise continuous functions, the class of Riemann integrable functions, etc.?