Method of Tikhonov regularization in a problem of optimal control of a nonlinear parabolic system (Q910704)

For a system described by the equation \(y'(v)-\Delta y(v)+| y(v)|^ qy(v)=v\) and some boundary conditions the problem of minimizing the cost function \(I(v)=\| y(v,T)-z\|^ 2\), where T is a final time, is considered. The existence of an optimal control \(v_ 0\) is proved. With the method of the quasiconjugate system the maximum principle \(v_ 0p=\max_{v\in U}vp\), where p is a conjugate state, is derived. The necessary optimality conditions admit a singularity. Applying Tikhonov's regularization method, the cost functions \(I_ k(v)=I(v)+\alpha_ k\| v\|^ 2\), where \(\alpha_ k>0\), \(\alpha_ k\to 0\) for \(k\to \infty\), are considered. The solutions of the regularized problems are \(v_ k=P(P_ k/2\alpha_ k)\), where p is a projector to a set U. The convergence of the regularization method is proved. The necessary optimality conditions for the regularized problems are obtained by an iterative method.