| Abstract: |
| Piezoelectric materials exhibit electric responses to mechanical stress, and mechanical responses to electric stress. The electrostatic and magnetizable PDE models, describing the longitudinal oscillations on the beam, with boundary feedback sensors/actuators are known to have exponentially stable solutions.
Firstly, a thorough analysis for the maximal decay rate via the optimal choice of feedback sensor amplifiers is discussed. Next, standard Finite Differences and novel order-reduction-based Finite Differences model reductions for these PDEs are proposed. In certain cases, numerical filtering for the spurious high-frequency modes may be unavoidable. These modes simply cause the loss of uniform gap among the eigenvalues as the discretization parameter tends to zero. The exponential decay of the solutions, mimicking the PDE counterparts, can be retained uniformly. A thorough analysis for the maximal decay rate via the optimal choice of feedback sensor amplifiers and the discretization parameter is discussed.
Finally, several interactive numerical tests by Wolfram Demonstrations Projects (WDP) are shared to support our results. These are simply interactive visualizations of the controlled dynamics preserving control-theoretic properties of the PDEs such as observability, controllability, stabilizability. As you move a Demonstration`s controls, you see a change in its output that helps you understand the controlled dynamics with optimal feedback controllers. |
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