Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects

It is well known that magnetic energy of the piezoelectric beam is relatively small, and it does not change the overall dynamics. Therefore, the models, relying on electrostatic or quasi-static approaches, completely ignore the magnetic energy stored/produced in the beam. A single piezoelectric beam model without the magnetic effects is known to be exactly observable and exponentially stabilizable in the energy space. However, the model with the magnetic effects is proved to be not exactly observable/exponentially stabilizable in the energy space for almost all choices of material parameters. Moreover, even strong stability is not achievable for many values of the material parameters. In this paper, it is shown that the uncontrolled system is exactly observable in a space larger than the energy space. Then, by using a $$B^*$$B∗-type feedback controller, explicit polynomial decay estimates are obtained for more regular initial data. Unlike the classical counterparts, this choice of feedback corresponds to the current flowing through the electrodes, and it matches better with the physics of the model. The results obtained in this manuscript have direct implications on the controllability/stabilizability of smart structures such as elastic beams/plates with piezoelectric patches and the active constrained layer (ACL) damped beams/plates.

[1]  Marius Tucsnak,et al.  Control of plate vibrations by means of piezoelectric actuators , 1996 .

[2]  Ralph C. Smith,et al.  Smart material systems - model development , 2005, Frontiers in applied mathematics.

[3]  M. M. Dodson,et al.  Metric Diophantine Approximation on Manifolds: Notation , 1999 .

[4]  Jiashi Yang,et al.  An Introduction to the Theory of Piezoelectricity , 2004 .

[5]  Ahmet Özkan Özer,et al.  Exact Boundary Controllability Results for a Multilayer Rao-Nakra Sandwich Beam , 2014, SIAM J. Control. Optim..

[6]  Kirsten Morris,et al.  Comparison of stabilization of current-actuated and voltage-actuated piezoelectric beams , 2014, 53rd IEEE Conference on Decision and Control.

[7]  Scott W. Hansen,et al.  Several related models for multilayer sandwich plates , 2004 .

[8]  Harvey Thomas Banks,et al.  Smart material structures: Modeling, estimation, and control , 1996 .

[9]  Marius Tucsnak,et al.  How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance , 2003 .

[10]  Farid Ammar Khodja,et al.  A new relation between the condensation index of complex sequences and the null controllability of parabolic systems , 2013 .

[11]  Kaïs Ammari,et al.  Stabilization of second order evolution equations by a class of unbounded feedbacks , 2001 .

[12]  Marius Tucsnak,et al.  How to Get a Conservative Well-Posed Linear System Out of Thin Air. Part II. Controllability and Stability , 2003, SIAM J. Control. Optim..

[13]  Ruth F. Curtain,et al.  Well posedness of triples of operators (in the sense of linear systems theory) , 1989 .

[14]  Ahmet Özkan Özer,et al.  Exact boundary controllability of an abstract Mead-Marcus sandwich beam model , 2010, 49th IEEE Conference on Decision and Control (CDC).

[15]  J. Cassels,et al.  An Introduction to Diophantine Approximation , 1957 .

[16]  Florian Luca,et al.  Control of coupled parabolic systems and Diophantine approximations , 2013 .

[17]  Kirsten Morris,et al.  Strong stabilization of piezoelectric beams with magnetic effects , 2013, 52nd IEEE Conference on Decision and Control.

[18]  C. Baiocchi,et al.  Ingham-Beurling type theorems with weakened gap conditions , 2002 .

[19]  J. Lions.  Exact controllability, stabilization and perturbations for distributed systems , 1988 .

[20]  Pascal B'egout,et al.  A Generalized Interpolation Inequality and its Application to the Stabilization of Damped Equations , 2007, 1207.2030.

[21]  N. Rogacheva.  The Theory of Piezoelectric Shells and Plates , 1994 .

[22]  Marius Tucsnak,et al.  Regularity of plate equations with control concentrated in interior curves , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[23]  David Ullrich,et al.  Divided differences and systems of nonharmonic Fourier series , 1980 .

[24]  Fatiha Alabau-Boussouira,et al.  A Two-Level Energy Method for Indirect Boundary Observability and Controllability of Weakly Coupled Hyperbolic Systems , 2003, SIAM J. Control. Optim..

[25]  Steven J. Miller,et al.  An Invitation to Modern Number Theory , 2020 .

[26]  G. Weiss,et al.  Observation and Control for Operator Semigroups , 2009 .

[27]  Marius Tucsnak,et al.  SIMULTANEOUS CONTROLLABILITY IN SHARP TIME FOR TWO ELASTIC STRINGS , 2001 .

[28]  Fatiha Alabau-Boussouira,et al.  INDIRECT STABILIZATION OF LOCALLY COUPLED WAVE-TYPE SYSTEMS , 2012 .

[29]  W. T. Scott.  Approximation to real irrationals by certain classes of rational fractions , 1940 .

[30]  Bernadette Miara,et al.  Boundary Observation and Exact Control of a Quasi-electrostatic Piezoelectric System in Multilayered Media , 2007, SIAM J. Control. Optim..

[31]  Kirsten Morris,et al.  Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects , 2013, SIAM J. Control. Optim..

[32]  Marius Tucsnak.  Regularity and Exact Controllability for a Beam With Piezoelectric Actuator , 1996 .

[33]  Assia Benabdallah,et al.  RECENT RESULTS ON THE CONTROLLABILITY OF LINEAR COUPLED PARABOLIC PROBLEMS: A SURVEY , 2011 .

[34]  Enrique Zuazua,et al.  SINGULAR INTERNAL STABILIZATION OF THE WAVE EQUATION , 1998 .

[35]  Sergei Avdonin,et al.  Ingham-type inequalities and Riesz bases of divided differences , 2001 .

[36]  Kirsten Morris,et al.  Modeling an elastic beam with piezoelectric patches by including magnetic effects , 2014, 2014 American Control Conference.

[37]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[38]  Jiashi Yang,et al.  A Review of a Few Topics in Piezoelectricity , 2006 .

[39]  H. F. Tiersten,et al.  Linear Piezoelectric Plate Vibrations , 1969 .

[40]  Scott W. Hansen,et al.  Uniform stabilization of a multilayer Rao-Nakra sandwich beam , 2013 .

[41]  I. Legrain,et al.  Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure interaction , 1992 .

[42]  P Kallio,et al.  Displacement Control of Piezoelectric Actuators Using Current and Voltage , 2011, IEEE/ASME Transactions on Mechatronics.

[43]  Lena Jaeger,et al.  Fourier Series In Control Theory , 2016 .

[44]  Vilmos Komornik,et al.  Indirect internal stabilization of weakly coupled evolution equations , 2002 .

[45]  Enrique Zuazua,et al.  Boundary Controllability of a Hybrid System Consisting in Two Flexible Beams Connected by a Point Mass , 1998 .

[46]  D. Russell,et al.  The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region , 1986 .