Finite-Dimensional Boundary Control of a Wave Equation With Viscous Friction and Boundary Measurements

Anton Selivanov, Emilia Fridman

Research output: Contribution to journalArticlepeer-review


Recently, a constructive approach to the design of finite-dimensional observer-based controller has been proposed for parabolic partial differential equations&#x00A0;(PDEs). This paper extends it to hyperbolic PDEs. Namely, we design a finite-dimensional, output- feedback, boundary controller for a wave equation&#x00A0;with in-domain viscous friction. The control-free system is unstable for any friction coefficient due to an external force. Our approach is based on modal decomposition: an observer-based controller is designed for a finite-dimensional projection of the wave equation&#x00A0;on <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> eigenfunctions (modes) of the Sturm-Liouville operator. The danger of this approach is the &#x201C;spillover&#x201D; effect: such a controller may have a deteriorating effect on the stability of the unconsidered modes and cause instability of the full system. Our main contribution is an appropriate Lyapunov-based analysis leading to linear matrix inequalities (LMIs) that allow one to find a controller gain and number of modes, <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>, guaranteeing that the &#x201C;spillover&#x201D; effect does not occur. An important merit of the derived LMIs is that their complexity does not change when <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> grows. Moreover, we show that appropriate <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> always exists and, if the LMIs are feasible for some <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>, they remain so for <inline-formula><tex-math notation="LaTeX">$N+1$</tex-math></inline-formula>.

Original languageEnglish
Pages (from-to)1-8
Number of pages8
JournalIEEE Transactions on Automatic Control
StateAccepted/In press - 2023


  • Aerospace electronics
  • Damping
  • Friction
  • Linear matrix inequalities
  • Observers
  • Propagation
  • Reduced order systems


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