Optimal Control For A Class Of Kirchhoff Equations

In this paper,we consider the optimal control problem governed by the following Kirchhoff equation where a and b are given positive functions or positive constants.u is the control function,taken as the allowable control set U U={u∈L2（Ω）|m≤u（x）≤M;a.e.x∈Ω｝.The optimal control problem is formulated as:For a given ideal state yd ∈ L2（Ω）,find the optimal control pair（y*,u*）∈ H01（Ω）×U such that,subject to the above constraints on the controlled system,the performance index Reach Minimum.Firstly,the existence and the uniqueness solutions of the controlled system are proved by using the variational method and LP theory for the positive coefficients a and b and positive functions a and b,respectively.Secondly,the existence of optimal control is obtained by using the method of minimization sequence.Finally,we derive the first-order necessary condition by DubovitskiiMilyutin Theorem.