Analysis of an optimization problem for a piezoelectric energy harvester

  • Barbara KaltenbacherEmail author
  • Pavel Krejčí


The problem of optimal energy harvesting for a piezoelectric element driven by mechanical vibrations is stated in terms of an ODE system with hysteresis under the time derivative coupling a mechanical oscillator with an electric circuit with or without inductance. In the piezoelectric constitutive law, both the self-similar piezoelectric butterfly character of the hysteresis curves and feedback effects are taken into account in a thermodynamically consistent way. The physical parameters of the harvester are chosen to be the control variable, and the goal is to maximize the harvested energy for a given mechanical load and a given time interval. If hysteresis is modeled by the Preisach operator, the system is shown to be well-posed with continuous data dependence. For the special case of the play operator, we derive first-order necessary optimality conditions and an explicit form of the gradient of the total harvested energy functional in terms of solutions to the adjoint system.


Hysteresis Piezoelectricity Differential equations Optimal control 



The authors thank both reviewers for their valuable comments. Moreover, financial support by the GAČR Grant GA15-12227S, RVO: 67985840, and FWF Grant P30054, is gratefully acknowledged.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for MathematicsAlpen-Adria-UniversitätKlagenfurtAustria
  2. 2.Institute of MathematicsCzech Academy of SciencesPrague 1Czech Republic

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