Optimal Distributed Control Of Related Models For A Class Of Coupled Phase Field System

As an important part of modern control theory,optimal control theory is formed and developed under the promotion of space technology.The main problem studied is to select an admissible control law for a controlled dynamic system or motion process according to the established mathematical model of the controlled object,such that the controlled object reaches the predetermined goal and optimizes a given performance index.From a mathematical point of view,the optimal control problem is to solve a class of functional extremum problems with constraints.In recent years,the optimal control theory has developed greatly in depth and breadth,especially the optimal control theory of partial differential equations has become one of the research hot spots in the field of partial differential equations and has been widely used in many disciplines.This paper focuses on the optimal distributed control problem of related models for a class of coupled phase field system.First of all,we considered a coupled Allen-Cahn/Cahn-Hilliard system in the sec-ond chapter,it is used to describe simultaneous order-disorder and phase separation in binary alloys on a BBC lattice in the neighborhood of the triple point.This system model was proposed in[15]by Cahn and Novick-Cohen in 1994,some related research works can be found in[13,73,74].Here we mainly studied the case of the lattice spac-ing h=1 and the nonlinear term f(s)=s3-?s(s?R and ? is a given real constant),it can be summarized as the following coupled equations(?)=-2v3-6u2v+(2?-?)v+?v,in(x,t)??×(0,T),(?)=?(2v3+6uv2-2?u-?u),in(x,t)??×(0,T),where C R3 is a nonempty bounded connected open set which has a smooth boundary(?)? and T>0.Here conserved quantity u is the concentration of one of the components,v denotes an order parameter.In addition,? represents the location of the system within the phase diagram.In the past few years,some experts and scholars have also paid attention to the above-mentioned coupled equations,see literatures[59,111].But as far as we know,the optimal distributed control problem of this coupled system has not been studied.As mentioned before,the study of optimal distribution control lies in the minimization of the cost functional,hence we set Q(?)?×(0,T)and consider the following cost functional where uQ ?L2(Q)and u??L2(?)are given target functions and(?)1,(?)2,(?)3 are given nonnegative constants but not all zero.Moreover,???ad is constraint control term,the set of admissible controls ?ad denotes ??L?(Q)and ?min ????max in Q almost everywhere for all ???ad where the given functions ?min,?max?L?(Q)and?min??max in Q almost everywhere.In addition,the cost functional J(u,?)satisfies the state system(?)=-2v3-6u2v+(2?-?)v+?v+?,in Q,(?)=?(2u3+6uv2-2?u-?u),on Q,v=u=?u=0,on ?,v(x,0)=v0(x),u(x,0)=u0(x),in ?,here ?(?)?×(0,T),Our work is to obtain the weln-posedness of the state system and the existence and uniqueness of the strong solution with the help of the framework of Galerkin approximation,then we get the continuous dependence of the strong solution of the system on the constraint control term ?.Next we derived the existence of optimal control and discussed differentiable properties of the control-to-state operator and first-order necessary optimal conditions satisfied by the optimal distributed control problem of the state system.In the third and fourth chapters,we mainly consider the models for a class of phase field system with temperature,this system is coupled by an Allen-Cahn equation,a Cahn-Hilliard equation and equation(?)=div q.In the theory of thermodynamics,the vector q represents thermal flux and H denotes enthalpy which is equivalent to the total heat content of the system.On the basis of the classical Fourier heat conduction law,the coupled system with temperature can be further expressed as the following equivalent form(?)=?-f(u+v)+f(u-v)-?v+h2?v,in Q,(?)=h2?(f(u+v)+f(u-v)-h2?u),in Q,(?)=??-(?),in Q,v=u=?u=?=0,on ?,v(x,0)=v0(x),u(x,0)=u0(x),?(x,0)=?0(x),in?,where u,v,h and a have the same meaning as mentioned above,? means relative temperature.For the case of h=1,?=0 and the nonlinear term satisfies certain conditions,optimal distributed control problem of the above system with temperature is to study the cost functional J1(u,v,?)of satisfying the control constraints ???ad and state system(?)=?-f(u+v)+f(u-v)+?v,in Q,(?)=?(f(u+v)+f(u-v)-?u),in Q,(?)=??-(?)+?,in Q v=u=?u=?=0,on ?,v(x,0)=v0(x),u(x,0)=u0(x),?(x,0)=?0(x),in ?,the cost functional J1(u,v,?)is defined by where(?)i(i=4,5,6,7,8)are given non-negative constants but not all zero,uQ,vQ?L2(Q),u?,v? ?H1(?)are the given objective functions.In the third chapter,we firstly use the Galerkin method to obtain the well-posedness of the phase field system with temperature and the existence and uniqueness of the strong solution of this system.In addition,the continuous dependence of the strong solution on the constrained control item ?1 is also obtained.Based on these results,we derive the existence of optimal control,differentiability of the control-to-state operator and the first-order necessary optimality conditions.When we consider the phase field system with temperature on the basis of the type ? law in the thermodynamic theory,it can be written in the following form(?)=(?)-f(u+v)+f(u-v)-?v+h2?v,in Q,(?)=h2?(f(u+v)+f(u-v)-h2?u),in Q,(?)=k1?(?)+k??—(?),in Q,v=u=?u=?=0,on ?,v(x,0)=v0(x),u(x,0)=u0(x),?(x,0)=?0(x),(?)(x,0)=?1(x),in Q,where ? represents the thermal displacement variable,which satisfies 0=(?)for relative temperature 0.In addition,k and k1 are two positive numbers.Therefore,in the fourth chapter,we aim at the situation where h=k=k1=1,?=0 and the nonlinear term meets certain conditions and also consider the optimal distributed control problem of the coupled system.Specifically,we studied the cost functional J1(u,V,?)which subject to control constraints ???ad and the following state system(?)=(?)-f(u+v)+f(u-v)+?v,in Q,(?)=?(f(u+v)+f(u-v)-?u),in Q,(?)=?(?)+??-(?)+?,in Q,v=u=?u=?=0,on ?,v(x,0)=v0(x),u(x,0)=u0(x),?(x,0)=?0(x),(?)(x,0)=?1(x),in Q.Similar to the previous discussion framework,Firstly we apply Galerkin method to deduce the existence and uniqueness of strong solution of the state system and the continuous dependence of strong solution on constraint control term ?.Afterwards,we have also made relevant discussions on the existence of optimal control,differentiablity of the control-to-state operator and first-order necessary optimality conditions satisfied by optimal distributed control problem of the system.