AI 中文总结
本文提出一种复合边界反馈控制律，通过解析增益函数和时变Volterra核显式设计，并构造非二次Lyapunov函数和广义Lyapunov泛函，实现了具有时变系数的耦合ODE-抛物型PDE系统的输入到状态稳定性，数值仿真验证了有效性。
AI 中文摘要
本文提出了一种新颖的复合边界反馈控制律，该控制律确保了两个子系统均具有时变系数的耦合ODE-抛物型PDE系统的输入到状态稳定性（ISS）。在控制器设计中，我们通过采用解析预定义增益函数和时变Volterra核函数来显式设计控制律，从而避免直接求解耦合的时变抛物-双曲核方程。在稳定性分析中，为同时应对Dirichlet边界扰动和时变系数的挑战，我们分别采用时变正定矩阵的平方根和超线性函数，在目标系统中为ODE构造非二次Lyapunov函数，为PDE构造广义Lyapunov泛函，从而建立了闭环系统在$L^2$-范数下的ISS。数值仿真验证了所提出控制方案的有效性。
英文摘要
This paper proposes a novel composite boundary feedback control law that ensures input-to-state stability (ISS) for a coupled ODE-parabolic PDE system with time-varying coefficients in both subsystems. In controller design, we circumvent the need to directly solve coupled time-varying parabolic-hyperbolic kernel equations by employing an analytic pre-defined gain function and a time-varying Volterra kernel function to design the control law explicitly. In stability analysis, to address the simultaneous challenges of Dirichlet boundary disturbances and time-varying coefficients, we employ the square root of a time-varying positive definite matrix and a superlinear function to construct a nonquadratic Lyapunov function for the ODE and a generalized Lyapunov functional for the PDE, respectively, in the target system, thereby establishing the ISS in the $L^2$-norm of the closed-loop system. Numerical simulations are presented to illustrate the effectiveness of the proposed control scheme.
