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(t), where x(t) is determined by a linear differential equation Z = Az, with A a constant matrix. Application of external influences will yield an inhomogeneous equation, Z = 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 =b sub i, the control is said to be of the 'bang-bang' type. The case where all the solutions of Z = Az approach zero as t approaches infinity is considered.