Optimal control of a nonlinear parabolic–elliptic system

In this paper, we shall study the problem of optimal control of the parabolic–elliptic system ut+(f(t,x,u))x+g(t,x,u)+Px−(a(t,x)ux)x=f0+B∗νut+(f(t,x,u))x+g(t,x,u)+Px−(a(t,x)ux)x=f0+B∗ν and −Pxx+P=h(t,x,u,ux)+k(t,x,u)−Pxx+P=h(t,x,u,ux)+k(t,x,u) with initial data u|t=0=u0.u|t=0=u0. The existence and uniqueness of weak solution to the parabolic–elliptic system are given in a short interval. According to the variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of the solution is related to the control item and initial value in the special Hilbert space. The optimal control of the parabolic–elliptic system with the initial data is given and the existence of an optimal solution to the parabolic–elliptic system is proved.