Optimal control problem for deflection plate with crack
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We consider a control problem where the state variable is defined as the solution of a variational inequality. This system describes the vertical displacement of points of a thin plate with the presence of crack inside . As the control we define the force that originates the deection of the plate. In order to get the system of optimality for the control problem we use a penalized problem  and its reformation as a Lagrangian problem. We prove the existence of a Lagrange multiplier to obtain a system of optimality to the exact problem via Lagrangian. Applying the method of bounded increments  we get the final result that characterizes the optimal state and control.