Optimal Control For Linear Semistabilization With Singular System

The optimal control problem for linear semistabilization with continuous singular system is discussed in this paper.Semistability,a kind of stability between Lyapunov stability and asymptotic stability,is suitable for systems with continuous equilibrium state.Furthermore,the semistable equilibrium state depends not only on the system,but also on the initial state of the system.Because of the singularity of the system,the method which can be used to deal with nonsingular system and the optimal control problem of linear semistabilization with nonsingular system can not be directly applied to the optimal control problem of linear semistabilization with singular system.In this paper,the optimal linear control problem is extended to address semistability with singular system.In order to solve this problem,the singular system is decomposed into fast system and slow system by normalizability decomposition method,in which the slow system is nonsingular system.The relationship of equivalent semistability between the singular system and slow system is proved,and the judging conditions of semistability with the singular system is given.Then the optimal linear control problem for semistabilization with the singular system is equivalently transformed into a semistable optimal control problem with slow system which is obtained by decomposing the singular system,and the sufficient conditions to judge semistable optimal control is given by establishing and solving the related Riccati equatiuon.A numerical example and an Leontief model in economics are presented to demonstrate the efficacy of the proposed frameworks of optimal semistabilization control with singular system.