AI 中文总结
研究非自治随机抛物方程的有限维反馈镇定，通过局部指标型执行器的有限线性组合及斜投影构造反馈律，在一定假设下证明闭环系统适定性，实现指数均方镇定及几乎必然镇定，数值实验展示相关影响。
AI 中文摘要
我们研究由Q-维纳驱动的非线性非自治随机抛物方程的有限维反馈镇定，涵盖加性和乘性扰动。控制由局部指标型执行器的有限线性组合给出，其支撑作为构造的一部分被选择，且总测度可任意小。反馈律通过向合适的有限维子空间的斜投影构造。在变分盖尔范德三元组框架下，我们在标准强制性、增长性和全局利普希茨假设下证明闭环系统的适定性。通过适当选择执行器配置和反馈强度，我们建立随机动力学的指数均方镇定，对于纯乘性噪声，建立几乎必然镇定。一个完全离散的三层实现补充了理论结果。数值实验说明了执行器数量、噪声强度和非线性效应对闭环镇定行为的影响。
英文摘要
We investigate finite-dimensional feedback stabilization for nonlinear nonautonomous stochastic parabolic equations driven by $Q$-Wiener, covering both additive and multiplicative perturbations. The control is given by a finite linear combination of localized indicator-type actuators whose supports are selected as part of the construction and may have arbitrarily small total measure. The feedback law is constructed by means of oblique projections onto suitable finite-dimensional subspaces. Within the variational Gelfand triple framework, we prove well-posedness of the closed-loop system under standard coercivity, growth, and global Lipschitz assumptions. By appropriately choosing the actuator configuration and feedback strength, we establish exponential mean-square stabilization of the stochastic dynamics and, for pure multiplicative noise, almost-sure stabilization. A fully discrete three-layer implementation complements the theoretical results. Numerical experiments illustrate the influence of number of actuators, noise intensity, and nonlinear effects on the closed-loop stabilization behavior.
Comments38 pages, 7 figures. Code available at https://doi.org/10.5281/zenodo.21220182
