AI 中文总结
研究带源项的多维连续性方程稳定化，采用局域内部通量反馈，通过有限时间特征阻尼不等式等条件实现误差指数稳定，还讨论反馈通量实现并以二维示例验证。
AI 中文摘要
我们研究有界域上带源项的多维连续性方程的稳定化问题，采用局域内部通量反馈的方法。该反馈通过通量的散度来规定，其选取方式使得相对于参考剖面的误差满足带有局域阻尼的输运方程。主要的几何条件是有限时间特征阻尼不等式，要求相关特征在时间范围内积累均匀的阻尼量。在关联局域阻尼与输运场压缩放大的增益条件下，该条件可使误差在L²空间中呈现指数稳定性。类李雅普诺夫的入口条件确保了受支撑限制的轨迹族上的特征阻尼。加权李雅普诺夫泛函提供了微分李雅普诺夫准则以及针对加性扰动的输入到状态（ISS）估计。我们还讨论了反馈通量的椭圆型右逆实现，并将特征阻尼论证扩展到依赖于状态非线性的速度场。最后通过一个二维示例说明几何条件、增益条件和实现条件。
英文摘要
We study stabilization of multidimensional continu- ity equations with source terms on bounded domains by means of localized interior flux feedback. The feedback is prescribed through the divergence of the flux and is chosen so that the error with respect to a reference profile satisfies a transport equation with localized damping. The main geometric condition is a finite-time characteristic damping inequality, requiring relevant characteristics to accumulate a uniform amount of damping over a time horizon. This condition is shown to yield exponential stability in L2 of the error, under a gain condition relating localized damping to compressive amplification of the transport field. Lyapunov-type entrance conditions ensure characteris- tic damping on support-restricted families of trajectories. A weighted Lyapunov functional provides a differential Lyapunov criterion and an input-to-state (ISS) estimate with respect to additive perturbations. We also discuss elliptic right-inverse realizations of the feedback flux and extend the characteristic damping argument to velocity fields depending nonlinearly on the state. A two-dimensional example finally illustrates the geometric, gain, and realization conditions.
