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Escape function conditions for the observation, control, and stabilization of the wave equation

机译：波动方程的观测，控制和稳定的逃逸函数条件

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For the linear wave equation with time-invariant coefficients on a connected compact Riemannian manifold (Omega, g) with C-3 boundary, the geodesics condition of Bardos, Lebeau, and Rauch [SIAM J. Control Optim., 30 (1992), pp. 1024 1065] is characterized in terms of escape functions, which are some Lyapunov functions on the phase space S*(&UOmega;) over bar (the unit sphere cotangent bundle). Differentiable escape functions yield a sufficient condition which is slightly less sharp but does not refer to geodesics. The escape function condition yields a straightforward geometric proof that the geodesics condition holds in the situations where first order differential multiplier methods apply. Using microlocal control results, it allows us to generalize some control results (that were obtained by multiplier methods) to variable coefficients and lower order terms. It also allows us to prove, in some class of simple situations (e.g., in R-2 with constant coefficients), that no first order differential multiplier method can reach the optimal control time or control regions. [References: 34]
机译：对于具有C-3边界的紧紧黎曼流形（Omega，g）上具有时不变系数的线性波动方程，Bardos，Lebeau和Rauch的测地条件[SIAM J. Control Optim。，30（1992）， [pp.1024 1065]以逃逸函数为特征，逃逸函数是棒（单位球面正切束）上相空间S *（ωU）上的一些李雅普诺夫函数。可微分的逃逸函数会产生一个足够的条件，该条件略微不那么尖锐，但不涉及测地线。逸出函数条件产生了直接几何证明，即在应用一阶微分乘子方法的情况下，测地线条件成立。使用微局部控制结果，它使我们可以将一些控制结果（通过乘数方法获得）概括为可变系数和低阶项。它还使我们能够证明，在某些简单情况下（例如，在具有恒定系数的R-2中），没有任何一阶微分乘法器方法可以达到最佳控制时间或控制区域。 [参考：34]

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