90006678

open access

Accepted Manuscript

Boundary stabilization of first-order hyperbolic equations with input delay

Japan Journal of Industrial and Applied Mathematics

36(2)

325-355

Springer Nature

2019-07

2020-08-01

This is a post-peer-review, pre-copyedit version of an article published in Japan Journal of Industrial and Applied Mathematics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s13160-019-00346-6

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 $$C_0$$C0-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.

Semigroup

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