Open-loop optimal control theory is formulated and applied to damp out the
vibrations of a beam where the control action is implemented using piezocer
amic actuators. The optimal control law is derived by using a maximum princ
iple developed for one-dimensional structures where the control function ap
pears in the boundary conditions in the form of a moment. The objective fun
ction is specified as a weighted quadratic functional of the displacement a
nd velocity which is to be minimized at a specified terminal time using con
tinuous piezoelectric actuators. The expenditure of control force is includ
ed in the objective functional as a penalty term. The explicit solution of
the problem is developed for cantilever beams using eigenfunction expansion
s of the state and adjoint variables. The effectiveness of the proposed con
trol mechanism is assessed by plotting the displacement and velocity agains
t time. It is shown that both quantities are damped out substantially as co
mpared to an uncontrolled beam and this reduction depends on the magnitude
of the control moment. The capabilities of piezo actuation are also investi
gated by means of control moment versus piezo and beam thickness graphs whi
ch indicate the required minimum level of voltage to be applied on piezo ma
terials in relation to geometric dimensions of the combined active/passive
structure. The graphs show the magnitude of the control moment which can be
achieved using piezoceramics in terms of problem inputs such as voltage, p
iezo and beam thicknesses. (C) 2000 Elsevier Science Ltd. All rights reserv
ed.