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**Uniform exponential energy decay of wave equations in a bounded region with \(L_ 2(0,\infty;L_ 2(\Gamma))\)-feedback control in the Dirichlet boundary conditions.**
*(English)*
Zbl 0629.93047

The authors consider the much studied problem of the wave equation in dimension \(n\geq 2\) with boundary control. Here the control enters as the nonhomogeneous boundary condition of Dirichlet type and they seek a suitable linear feedback operator depending directly on the velocity such that the resulting closed-loop system becomes exponentially stable. They prove that for “smooth” boundaries, the control law \(u(t)=(\partial /\partial \gamma)(A^{-1}\omega (t))\) produces asymptotically stable closed-loop solutions; \(\omega\) (t) is the velocity and A is the Laplacian with homogeneous boundary conditions. To obtain the desired exponential stability extra convexity assumptions must be made on the domain. This result has as an immediate consequence a sharper result or exact boundary controllability for the wave equation in higher dimensions considered in the natural state space.

Reviewer: R.Curtain

### MSC:

93D15 | Stabilization of systems by feedback |

35L05 | Wave equation |

93C20 | Control/observation systems governed by partial differential equations |

93B05 | Controllability |

35B37 | PDE in connection with control problems (MSC2000) |

### Keywords:

wave equation; boundary control; boundary condition of Dirichlet type; feedback; exponential stability; exact boundary controllability
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\textit{I. Lasiecka} and \textit{R. Triggiani}, J. Differ. Equations 66, 340--390 (1987; Zbl 0629.93047)

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