The relationship between the stability and the optimality of linear systems

Abstract

Different from the inverse problem put forward by R. E. Kalman, another kind of inverse problem of linear optimal control is proposed and dicussed in this paper, which is as follows. If an asymptotically stable system$\dot x = \tilde Ax$ and a performance index$J = \int_0^\infty {(x^\tau Qx + u^\tau u)} dt$ are given, when can à be decomposed into Ã=A+BK^T so that the control lawu=K^Tx is optimal for the system$\dot x = Ax + Bu$ and the index J? This paper gives the solution for the problem and presents certain correspondence between the asymptotically stable system and the optimal system.