On the “bang-bang” control problem

Abstract

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