The Bang Bang Servo Problem Treated by Variational Techniques

In this paper variational techniques are used to establish the following result: suppose a dynamical system is governed by the differential equation y ˙ = A y + f ( t ) ( 1 ) where A is a real constant matrix with distinct eigenvalues. Suppose that these eigenvalues are further restricted to have nonpositive real parts but are not required to be purely real. Finally let each component φ(t) of the vector forcing functing f(t) satisfy, for all t , the conditions | ϕ i ( t ) | ≦ γ i ( i = 1 , 2 , ⋅ ⋅ ⋅ , n ) ( 2 ) where the γi's are preassingned constants. It is shown that, given an arbitrary initial condition y (0), the forcing function that will bring the system to its equilibrium position in the shortest possible time is such that φi(t)=±γi , and the instants of time at which φi(t) changes from ± γi to ± γi are obtained by considering the output of the adjoint system. Further relationship between the given system and the adjoint system is discussed in the paper. It is also shown that this solution, obtained by variational techniques, implies the concept of switching surfaces.