Control theory is used nowadays everywhere e.g., in autopilot, autonomous vehicles, airplanes, and battery management systems to name a few. In simple scenarios, open-loop control is used. But for complicated real-life cases, closed-loop control is crucial. When exogenous disturbance appears, feedback control is necessary for robotics, flight control, iron -dome, the defense industry associated with gun recoil, and optimal design systems. In various flight problems, we know that turbulence can occur at any time. So, minimizing the passenger’s discomfort along with fuel consumption is necessary. In mathematical terms, we need to minimize some energy in terms of cost functional or performance index. In nature, various phenomena such as turbulence, shock wave propagation, traffic flow, cosmology, and gas dynamics are modeled by PDEs. Along with PDEs when we consider different boundary conditions, we further solve a different kind of control problems in applications. For example, in the thermal process, one could not actuate the temperature on the boundary, but how much heat is flowing through the boundary can be measured. In that case, instead of the Dirichlet boundary, the Neumann boundary condition will be helpful. In most of the control problems, we solve the optimal feedback control problem, where we have applied so-called the Riccati-based approach for solving linear control problems and Hamilton-Jacobi-Bellman type equations for solving nonlinear control problems. For numerical treatment of the above-mentioned problem, we solve them using the finite element method even for complex geometry. Nowadays, the finite element method is used in solving various civil and mechanical engineering problems.