Optimal control in the regularized problem

As we know the vertical displacements of the plate defined from (2.7), (2.8) can be found as a limit of solutions to the problem (2.9)-(2.11). Two questions arise in this case. The first one is the following. Is it possible to solve an optimal control problem like (2.19) when w = w/ is defined from (2.9)-(2.11) The second question concerns relationships between solutions of (2.19) and those of the regularized optimal control problem. Our goal in this subsection is to answer these questions. 

First of all let us formulate the regularized optimal control problem. If the set F is introduced in similar way and w/ = w is found from the equation 

Theorem 2.3. Let the above hypotheses be fulfilled. Then a solution of the problem (2.23) exists. 

We briefly indicate the line of the proof in so far as it follows the arguments of Theorem 2.2. It can be assumed for a minimizing sequence / that as n — 00 

This convergence allows one to pass to the limit in the identity for w , as n — 00, and to obtain 

---