Optimal feedback control law for viscoelastic materials with memory effects

Viscoelastic materials have excellent properties of absorbing vibrational energy which makes their use very attractive in structural, aerospace and biomechanics engineering applications. The macroscopic dynamical behaviour of such materials depends on the time history, or memory, of the strain. The stress-strain viscoelastic relation can be described by a convolution integral with a memory kernel, according to Boltzmann's formulation of hereditary elasticity, or by using Caputo or Riemann-Liouville fractional derivatives. In order to emphasize the vibrations damping attitude of these materials, by actively controlling their stress-strain behaviour, novel optimal control logics are required which involve memory effects. This paper deals with a feedback control strategy applied to a structural-dynamic problem described by integral-differential equations. It is shown how to obtain a feedback control, called PD, i.e. Proportional-Nth-order-Derivatives control, by using a variational approach. Numerical simulations show how the PD controller is an effective tool to improve the viscoelastic materials performance.