Uniform Energy Decay Rates for Euler–Bernoulli Equations with Feedback Operators in the Dirichlet/Neumann Boundary Conditions

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The uniform stabilization problem is studied for the Euler–Bernoulli equation (though the methods apply also to the corresponding nonconstant coefficient case) defined in a smooth, bounded domain $\Omega $ of $R^n $, with suitable dissipative boundary feedback operators. These either are active in both the Dirichlet and Neumann boundary conditions, or are active in only the Dirichlet and inactive in the Neumann boundary condition. The uniform stabilization results presented are fully consistent with recently established exact controllability and optimal regularity theories for the solutions, which in fact motivate the choices of functional spaces in the first place. In particular, these uniform stabilization results require no geometrical conditions on $\Omega $ in the case of active Dirichlet/ Neumann feedback operators, and require some geometrical conditions on $\Omega $ in the case of an active feedback operator only in the Dirichlet boundary condition, as is the case of recent exact controllability theories [I. Lasiecka and R. Triggiani, SIAM J. Control Optim., 27 (1989), pp. 330–373]. Moreover, the forms of the dissipative feedback controls are natural consequences of (i) the type of boundary conditions selected; (ii) the choice that the control in the lowest boundary condition be $L_2 $ in time and space.