OPTIMAL CONTROL FOR VIBRATING SYSTEMS

A linear regulator problem for mechanical vibrating systems is studied in the second-order formulation. We exploit the secondorder form of the differential equations involved, and solve the problem without the traditional use of a Riccati equation. In its natural representation, the optimal control problem gives rise to minimisation of a functional depending on second derivatives. The Euler–Lagrange equations and the transversality conditions developed lead to a linear fourth-order differential equation that determines the optimal control. The results are demonstrated by examples.