A Spectral Method For Phase-field Model Of Solidification Phenomena With Control Constrained Optimal Control Problem

The paper is dedicated to the study of the optimal control problem with the control constrained,and its state equation is coupled nonlinear parabolic equation-s,which can apply to the phase-field model of solidification phenomena.At first,by the condition of controlling restricted target functionals,we can get the optimality conditions of the optimal control problem,namely,we obtain a coupled system of s-tate equation,co-state equation and a variational inequality,which can replace the original minimization problem.Furthermore,we establish the spectral discretization scheme of optimal control problem by linear combination of Legendre polynomials in time and space.Finally,by using Lipschitz condition,the nonlinear item of error equation is turned into the linear issue,and we can derive the prior error estimate of the optimal control problem by leading into some intermediate variables and cor-responding error equations,which are combined the properties of L2 projector and other projection operators.