Singular control

Remarks

Where $\Phi(t) > 0$ the maximum condition forces $u(t) = b$, and where $\Phi(t) < 0$ it forces $u(t) = a$. Where $\Phi$ vanishes only at isolated times, the control is bang–bang. Example: minimize $\int_0^2 x^2 \, dt$ for $x' = u$, $x(0) = 1$, $u \in [-1, 1]$, with free final state. The optimal control is $u = -1$ on $[0, 1]$ and $u = 0$ on $[1, 2]$, with $x(t) = 1 - t$ on $[0, 1]$ and $x = 0$ on $[1, 2]$. Here $H = p u - x^2$, $\Phi = p$, and the costate equation is $p' = 2x$ with $p(2) = 0$. So $p = 0$ on $[1, 2]$, and the control is singular there, with the interior value $u = 0$. Non-example: in the same problem, the control is not singular on $[0, 1)$, since $p(t) = -(1 - t)^2 < 0$ there. To find a singular control, one differentiates $\Phi$ in time until $u$ appears and solves for $u$.