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The problems of optimal control (OCPs) related to PDEs are a very active area of research. These problems deal with the processes of mechanical engineering, heat aeronautics, physics, hydro and gas dynamics, the physics of plasma and other real life problems. In this paper, we deal with a class of the constrained OCP for parabolic systems. It is converted to new unconstrained OCP by adding a penalty function to the cost functional. The existence solution of the considering system of parabolic optimal control problem (POCP) is introduced. In this way, the uniqueness theorem for the solving POCP is introduced. Therefore, a theorem for the sufficient differentiability conditions has been proved.

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In this paper, we have presented the numerical investigation of the geometric phase and field entropy squeezing for a two-level system interacting with coherent field under decoherence effect during the time evolution. The effects of the initial state setting and atomic dissipation damping parameter on the evolution of the geometric phase and entropy squeezing have been examined. We have reported some new results related to the periodicity and regularity of geometric phase and entropy squeezing.

A Boundary Element Method (BEM) for solving two (2D) dimensional static problem in materials under magnetic field as an external force is illustrated. A non-dimensional fundamental solution for stresses and displacements are obtained using Galerkin vectors. A reciprocity theorem is obtained. Integral representation of the displacement components is obtained. The formulation was applied to a specific problem for a thick plate. The results are compared with the exact analytical solutions obtained by using the Fourier exponential to transform for validation. The effect of magnetic field is calculated numerically and displayed graphically.