Active control of forced vibrations in a beam via Maximum principle

Open-loop optimal control problem is formulated and applied to damp out the forced vibrations of a simply supported beam where the control action is implemented using piezoelectric actuators. A Maximum principle is derived to obtain the optimal control law. A convex performance index is introduced as a weighted quadratic functional of the displacement and velocity which is to be minimized at a fixed terminal time with a penalty term related to the minimum expenditure of actuation energy. The Maximum principle given in this paper involves a Hamiltonian which contains an adjoint variable and the control function. The problem is reduced to solving a system of coupled partial differential equations for the state variable and the adjoint variable subject to boundary, initial and terminal conditions. The method of solution involves Fourier-sine expansion to transform the original problem into the optimal control of lumped parameter system where the optimal control is determined by the derived Maximum principle. A numerical example is given to demonstrate the applicability and the efficiency of the proposed method.