On the “bang-bang” control problem

Let $S$ be a physical system whose state at any time is described by an $n$-dimensional vector $x\left ( t \right )$, where $x\left ( t \right )$ is determined by a linear differential equation $dz/dt = Az$, with $A$ a constant matrix. Application of external influences will yield an inhomogeneous equation, $dz/dt = Az + f$, where $f$, the “forcing term", represents the control. A problem of some importance in the theory of control circuits is that of choosing $f$ so as to reduce $z$ to 0 in minimum time. If $f$ is restricted to belong to the class of vectors whose $i$th components can assume only the values $\pm {b_i}$, the control is said to be of the “bang-bang” type.