Optimal pointwise control for a parallel system of Euler–Bernoulli beams

The optimal control of a distributed system consisting of two Euler–Bernoulli beams coupled in parallel with pointwise controllers is considered. An index of performance is formulated which consists of a modified energy functional of two coupled structures at a specified time and penalty functions involving the point control forces. The minimization of the performance index over these forces is subject to the equation of motion governing the structural vibrations, the imposed initial condition as well as the boundary conditions. A maximum principle is derived for optimal point controls of one-dimensional coupled structures undergoing transverse vibrations. The optimal control law is obtained using a maximum principle and the applicability of the results is demonstrated. A method of solution for such a type of structure is suggested by using the eigenfunction expansion and the maximum principle. The solution involves reducing the original problem to a system of ordinary differential equations. The effectiveness of this approach is illustrated numerically by comparing the behavior of the controlled and uncontrolled problem.