Minimal time control with multiple saturation limits

General rules are proved for minimal time control of a linear system with the constraints that both the manipulated variable
m(t)
and its derivative
\dot{m}(t)
are amplitude limited: 1)
\dot{m}(t)
is always at its extreme value unless
m(t)
is at its extreme value, 2) for normal systems the minimal time path is unique and consequently optimum switching boundaries can be defined, and 3) the choice of
\dot{m}(t)
maximizes a Hamiltonian with a modified adjoint function. The above rules are applied to third-order control systems with decidedly favorable results.