发表机构
School of Information and Intelligent Science, Donghua University（东华大学信息与智能科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对受边界和分布式输入延迟的单向一阶PDE-ODE耦合系统，设计含Volterra与Fredholm型积分算子的backstepping变换，证明核方程适定性与变换可逆性，建立闭环系统指数稳定性，仿真验证方法有效性。
AI 中文摘要
本文研究了单向一阶双曲型偏微分方程（PDE）与常微分方程（ODE）耦合的系统，该系统用于建模受边界和分布式输入延迟影响的两相对流输运过程。针对定义在两个空间域上的系统，我们设计了一组包含Volterra型和Fredholm型积分算子的backstepping变换，该变换引入了八个核函数：其中五个与PDE状态相关，三个与ODE状态相关。由于分布式延迟变量跨越两个空间域，其变换必须对所有状态变量进行积分，从而产生具有奇异初始条件的三维核函数，这些初始条件涉及Dirac delta函数，作为采样算子建立不同维度变量之间的映射。通过运用特征线法和逐次逼近法，我们证明了核方程的适定性。在此基础上，我们证明了Fredholm变换的可逆性，并在延迟补偿控制器下，建立了闭环系统在$L^{\infty}$范数下的指数稳定性。给出了线性示例和非线性螺杆挤出模型的仿真结果，以证明所提方法的有效性。
英文摘要
The paper considers a system of unidirectional first-order hyperbolic partial differential equations (PDEs) coupled with ordinary differential equations (ODEs), modeling two-phase advective transport processes subject to both boundary and distributed input delays. To address the system defined over two spatial domains, we design a set of three backstepping transformations involving both Volterra- and Fredholm-type integral operators, which introduce eight kernel functions: five associated with the PDE states and three with the ODE states. Since the distributed delayed variable spans the two spatial domains, its transformation must integrate over all state variables, leading to three-dimensional kernel functions with singular initial conditions involving Dirac delta functions that serve as sampling operators to establish mappings between variables of different dimensions. By employing the method of characteristics and successive approximations, we prove the well-posedness of the kernel equations. On this basis, we prove the invertibility of the Fredholm transformation and establish the exponential stability of the closed-loop system in the $L^{\infty}$ norm under the delay-compensated controller. Simulation results for a linear example and a nonlinear screw extrusion model are presented to demonstrate the effectiveness of the proposed method.
Comments20 pages, 27 figures. Extended version of the journal paper containing complete technical proofs
