Boundary stabilization of first-order hyperbolic equations with input delay

  • Hideki SanoEmail author
  • Masashi Wakaiki
Original Paper


This paper is concerned with the boundary stabilization problem of first-order hyperbolic equations with input delay. In our previous work, the same problem for parabolic equations with input delay was addressed. For the hyperbolic equation, the element of time lag is similarly replaced by a transport equation and a backstepping method is employed. However, unlike in the case of the parabolic equation, it is difficult to obtain the eigenvalues and eigenfunctions of the system operator for the hyperbolic equation. Hence, we cannot construct the target system and the controller by using the finite-dimensional control theory together. In this paper, using \(C_0\)-semigroups for hyperbolic equations with nonlocal boundary condition, we show that the proposed controller is expressed by an abstract form in a Hilbert space so-called a predictor, and that the predictor makes sense under a condition. Further, for the inverse integral transformation, we obtain an interesting result on its continuity under the same condition. A numerical algorithm is also proposed for solving a non-standard hyperbolic equation appearing in our controller design.


Hyperbolic equation Boundary control Input delay Volterra–Fredholm backstepping transformation Predictor Semigroup 

Mathematics Subject Classification

Primary 93C60 Secondary 93D15 



This research is supported by KAKENHI (Grant-in-Aid for Scientific Research (C) No. 15K04999, (S) No. 15H05729), Japan Society for the Promotion of Science.


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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Graduate School of System InformaticsKobe UniversityNadaJapan

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