We consider forced Lur’e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay and partial differential equations are known to belong to this class of infinite-dimensional systems. We present refinements of recent incremental input-to-state stability results (Guiver in SIAM J Control Optim 57:334–365, 2019) and use them to derive convergence results for trajectories generated by Stepanov almost periodic inputs. In particular, we show that the incremental stability conditions guarantee that for every Stepanov almost periodic input there exists a unique pair of state and output signals which are almost periodic and Stepanov almost periodic, respectively. The almost periods of the state and output signals are shown to be closely related to the almost periods of the input, and a natural module containment result is established. All state and output signals generated by the same Stepanov almost periodic input approach the almost periodic state and the Stepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficient conditions guaranteeing incremental input-to-state stability and the existence of almost periodic state and Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion.

我们考虑强迫线性系统，其中线性动态分量是一个无限维的适定系统。已知许多物理动机的延迟和偏微分方程都属于此类无限维系统。我们对最近的增量输入到状态稳定性结果进行了改进（Guiver in SIAM J Control Optim 57：334–365，2019），并使用它们来得出由Stepanov近似周期输入生成的轨迹的收敛结果。特别是，我们证明了增量稳定性条件保证了对于每个Stepanov几乎周期性的输入，都存在一对唯一的状态和输出信号对，它们分别是近似周期性的和Stepanov近似周期性的。状态和输出信号的几乎周期与输入的几乎周期密切相关，并建立了自然的模块容纳结果。当时间到达无穷大时，同一Stepanov几乎周期性的输入所生成的所有状态和输出信号在适当的意义上分别接近于近似周期性的状态和Stepanov近似周期性的输出。足以保证输入到状态稳定性的充分条件以及几乎周期性的状态和Stepanov几乎周期性的输出信号的存在使人联想到众所周知的绝对稳定性标准（例如复杂的Aizerman猜想和圆标准）中的条件。