论文标题
四阶和二阶抛物线方程的随机耦合系统的可控性结果
Controllability results for stochastic coupled systems of fourth- and second-order parabolic equations
论文作者
论文摘要
在本文中，我们研究了耦合四阶和二阶抛物线方程的随机系统的一些可控性和可观察性问题。主要目标是控制两个方程，只有一个位于四阶方程漂移的控制器。我们分析了两种情况：一方面，我们研究了线性向后系统的可控性，其中仅通过一阶项制作耦合。关键点是对热方程式使用合适的Carleman估计值，而四阶操作员的重量相同，以推断出伴随系统的可观察性不平等。另一方面，我们研究了正向方程的简化非线性耦合模型的可控性。这种情况众所周知，很难解决，遵循了最近引入的方法，并依赖于随机设置中众所周知的源术语方法以及截断程序的适应。这种方法为随机系统提供了可控性的新概念。
In this paper, we study some controllability and observability problems for stochastic systems coupling fourth- and second-order parabolic equations. The main goal is to control both equations with only one controller localized on the drift of the fourth-order equation. We analyze two cases: on one hand, we study the controllability of a linear backward system where the couplings are made only through first-order terms. The key point is to use suitable Carleman estimates for the heat equation and the fourth-order operator with the same weight to deduce an observability inequality for the adjoint system. On the other hand, we study the controllability of a simplified nonlinear coupled model of forward equations. This case, which is well-known to be harder to solve, follows a methodology that has been introduced recently and relies on an adaptation of the well-known source term method in the stochastic setting together with a truncation procedure. This approach gives a new concept of controllability for stochastic systems.
