It is shown that an explicit oblique projection nonlinear feedback controller is able to stabilize semilinear parabolic equations, with time-dependent dynamics and with a polynomial nonlinearity. The actuators are typically modeled by a finite number of indicator functions of small subdomains. No constraint is imposed on the sign of the polynomial nonlinearity. The norm of the initial condition can be arbitrarily large, and the total volume covered by the actuators can be arbitrarily small. The number of actuators depends on the operator norm of the oblique projection, on the polynomial degree of the nonlinearity, on the norm of the initial condition, and on the total volume covered by the actuators. The range of the feedback controller coincides with the range of the oblique projection, which is the linear span of the actuators. The oblique projection is performed along the orthogonal complement of a subspace spanned by a suitable finite number of eigenfunctions of the diffusion operator. For rectangular domains, it is possible to explicitly construct/place the actuators so that the stability of the closed-loop system is guaranteed. Simulations are presented, which show the semiglobal stabilizing performance of the nonlinear feedback.

中文翻译：

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非自治半线性抛物线型系统的半全局指数镇定

结果表明，显式斜投影非线性反馈控制器能够稳定半线性抛物方程，具有时变动力学和多项式非线性特性。致动器通常由小子域的有限数量的指标函数建模。多项式非线性的符号没有限制。初始条件的范数可以任意大，并且致动器所覆盖的总体积可以任意小。致动器的数量取决于斜投影的算子范数，非线性的多项式程度，初始条件的范数以及致动器所覆盖的总体积。反馈控制器的范围与斜投影的范围一致，斜投影的范围是执行器的线性范围。倾斜投影是沿着由扩散算子的适当有限数量的本征函数所跨越的子空间的正交补码执行的。对于矩形区域，可以显式构造/放置执行器，从而确保闭环系统的稳定性。仿真结果表明了非线性反馈的半全局稳定性能。