Numerical optimization of control strategies for coupled vibrational systems

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Abstract
The study examines the numerical solution of vibration control problems in a coupled system consisting of two interacting objects. The problem is solved under the assumption that the left boundary of the distributed system is fixed, while an object with lumped parameters is attached to the right boundary, where a boundary control action is also applied to the distributed system. Special attention is given to obtaining a numerical solution to the problem. The solution is approached using two methods: the gradient projection method and, due to the linearity of the boundary problem concerning phase coordinates and control inputs, the method of successive approximations. By introducing an additional variable, the one-dimensional wave equation is approximated using the method of lines, transforming it into a system of ordinary differential equations of the 2n-th order. The resulting variational problem for the system with lumped parameters is then numerically solved based on Pontryagin’s maximum principle. The approximately optimal controls, obtained using the gradient projection method with a specially chosen step size, form a minimizing sequence of controls. Based on the numerical results, functional convergence is established. The method of successive approximations provides an optimal control solution as early as the second iteration, regardless of the initial control. This demonstrates the method’s efficiency and reliability for solving linear optimal control problems. The developed numerical techniques can be applied to optimize the dynamic behavior of complex mechanical structures, enhance system stability, and improve operational efficiency in various engineering applications

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