On the necessary and sufficient optimality conditions for a beam system

In this thesis, optimality conditions of a beam model based on Mindlin's gradient elasticity theory is studied. The beam system depends on the external excitation function, non-homogeneous boundary conditions and some mixed integral constraints including ineqality/equality on the control function and state variable. Before obtaining the optimality conditions of the system, energy integral method is employed for proving the uniqueness of the solition of the beam system. Controllability properties of the system is also discussed.Adjoint system corrosponding to beam system is derived with suitable terminal conditions for achieving the maximum principle. In order to show the confirmation of the obtained theoretical results. A real mechanical problem is illustrated and results are presented in the table and graphical froms.