In this paper we consider state-feedback global stabilization of a semilinear 1D heat equation with a nonlinearity exhibiting a linear growth bound. We study both non-local and boundary control via a modal decomposition approach. For both cases, we suggest a direct Lyapunov method applied to the full-order closed-loop system. The nonlinear terms are compensated by using Parseval’s inequality, leading to efficient and constructive linear matrix inequality (LMI) conditions for obtaining the controller dimension and gain. For non-local control we provide sufficient conditions that guarantee global stabilization for any linear growth bound via either linear or nonlinear controller, provided the number of actuators is large enough. We prove that the nonlinear controller achieves at least the same performance as the linear one. For the case of boundary control, we employ a multi-dimensional dynamic extension, whereas in the numerical example we manage with a larger linear growth bound. The introduced direct Lyapunov approach gives tools for a variety of robust control problems for semilinear parabolic PDEs.

在本文中，我们考虑具有呈现线性增长界限的非线性的半线性一维热方程的状态反馈全局稳定性。我们通过模态分解方法研究非局部和边界控制。对于这两种情况，我们建议将直接 Lyapunov 方法应用于全阶闭环系统。非线性项通过使用 Parseval 不等式进行补偿，从而产生高效且建设性的线性矩阵不等式 (LMI) 条件以获得控制器维度和增益。对于非局部控制，我们提供了足够的条件来保证通过线性或非线性控制器的任何线性增长边界的全局稳定，前提是执行器的数量足够大。我们证明非线性控制器至少实现了与线性控制器相同的性能。对于边界控制的情况，我们采用多维动态扩展，而在数值示例中，我们使用更大的线性增长边界进行管理。引入的直接 Lyapunov 方法为半线性抛物线 PDE 的各种稳健控制问题提供了工具。