Stabilization of an Unstable Wave Equation Using an Infinite Dimensional Dynamic Controller

This paper deals with the stabilization of an anti-stable string equation with Dirichlet actuation where the instability appears because of the uncontrolled boundary condition. Then, infinitely many unstable poles are generated and an infinite dimensional control law is therefore proposed to exponentially stabilize the system. The idea behind the choice of the controller is to extend the domain of the PDE so that the anti-damping term is compensated by a damping at the other boundary condition. Additionally, notice that the system can then be exponentially stabilized with a chosen decay-rate and is robust to uncertainties on the wave speed and the anti-damped coefficient of the wave equation, with the only use of a point-wise boundary measurement. The efficiency of this new control strategy is then compared to the backstepping approach.

[1]  Miroslav Krstic,et al.  Adaptive control of an anti-stable wave PDE , 2009, 2009 American Control Conference.

[2]  M. Krstić.  Boundary Control of PDEs: A Course on Backstepping Designs , 2008 .

[3]  Omer Morgolt,et al.  A Dynamic Control Law for the Wave Equation * , 2002 .

[4]  R. Datko.  Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks , 1988 .

[5]  Shuzhi Sam Ge,et al.  Adaptive Control of a Flexible Crane System With the Boundary Output Constraint , 2014, IEEE Transactions on Industrial Electronics.

[6]  J. Lagnese.  Decay of solutions of wave equations in a bounded region with boundary dissipation , 1983 .

[7]  Martin Gugat,et al.  Exponential Stabilization of the Wave Equation by Dirichlet Integral Feedback , 2015, SIAM J. Control. Optim..

[8]  Jack K. Hale,et al.  Effects of Small Delays on Stability and Control , 2001 .

[9]  Miroslav Krstic,et al.  Output-feedback adaptive control of a wave PDE with boundary anti-damping , 2014, Autom..

[10]  J. Lagnese,et al.  An example of the effect of time delays in boundary feedback stabilization of wave equations , 1985, 1985 24th IEEE Conference on Decision and Control.

[11]  Enrique Zuazua,et al.  Exact Controllability and Stabilization of a Vibrating String with an Interior Point Mass , 1995 .

[12]  Miroslav Krstic,et al.  Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary , 2009, 2009 American Control Conference.

[13]  Matthieu Barreau,et al.  Lyapunov Stability Analysis of a String Equation Coupled With an Ordinary Differential System , 2017, IEEE Trans. Autom. Control..

[14]  Scott Whitney,et al.  Forced wave motion with internal and boundary damping. , 2012, Journal of applied physics.

[15]  Fathi Hassine.  Rapid exponential stabilization of a 1-D transmission wave equation with in-domain anti-damping , 2018, ArXiv.

[16]  Sophie Tarbouriech,et al.  Wave Equation With Cone-Bounded Control Laws , 2016, IEEE Transactions on Automatic Control.

[17]  G. Bastin,et al.  Stability and Boundary Stabilization of 1-D Hyperbolic Systems , 2016 .

[18]  F. Gouaisbaut,et al.  Input/output stability of a damped string equation coupled with ordinary differential system , 2018, International Journal of Robust and Nonlinear Control.

[19]  R. Triggiani,et al.  Uniform stabilization of the wave equation with dirichlet-feedback control without geometrical conditions , 1992 .

[20]  M. Krstić.  Delay Compensation for Nonlinear, Adaptive, and PDE Systems , 2009 .

[21]  Gilberto Flores,et al.  Dynamics of a Damped Wave Equation Arising from MEMS , 2014, SIAM J. Appl. Math..

[22]  Georges Bastin,et al.  A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws , 2007, IEEE Transactions on Automatic Control.

[23]  Alexandre Seuret,et al.  Tractable sufficient stability conditions for a system coupling linear transport and differential equations , 2017, Syst. Control. Lett..

[24]  Miroslav Krstic,et al.  Output-feedback stabilization of an unstable wave equation , 2008, Autom..

[25]  E. Zuazua,et al.  Stability Results for the Wave Equation with Indefinite Damping , 1996 .

[26]  Nicolas Marchand,et al.  Event-based control of linear hyperbolic systems of conservation laws , 2016, Autom..

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