Active control procedure for minimizing the dynamic response of carbon nanotube reinforced plates

The current paper aims to present an active control procedure for optimally minimizing the dynamic response of functionally graded plates in various edge conditions. The plates are reinforced by four distribution types of uniaxially aligned straight carbon nanotubes in the through-thickness plate direction. The plates are subjected to an active force distributed on the upper face; this force is taken as a control tool for the minimization process. The plate governing equations are derived based on a higher-order plate theory, and the target function is formulated including the plate total energy as a measure for the dynamic response and the control energy according to the Pareto principle. The control problem is solved using the Lyapunov-Bellman theory. Numerical analyses are presented to examine the given control procedure's efficiency and illustrate the influence of reinforcement type, the order of the plate theory, boundary conditions, and volume fractions on the control process. To the best of our knowledge, this study represents one of the first implementations of a Lyapunov–Bellman theory for carbon nanotube reinforced plates.