On a problem in tracking

Abstract We consider the problem of bringing the controlled motion z ( t ) into a neighborhood of the random point y ( t ). The displacements of y ( t ) represent a stochastic diffusion process [1], and the motion of z ( t ) is described by linear differential equations involving the control function u . The control function u [ t , y , z ] is formed at each instant of time t on the basis of the realized values of y ( t ) and z ( t ). It is shown that the problem of bringing the point z ( t ) into an ϵ-neighborhood of y ( T )( T > 0) with a probability p u [ t , y , z ] if the motion of z ( t ) is completely controlled in a certain sense and the parameters of the process y ( t ) are held within certain bounds. When the average value M { y ( t )} is described by linear equations, we obtain an explicit form of the control function u which is a linear function of y and z . Several optimal control problems are discussed incidentally. The problem is solved by the Liapunov function method [2,3] modernized for the present problem. This modernization makes use of concepts from the theory of dynamic programming [4].