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Infinite-dimensional Lur’e systems with almost periodic forcing. (English) Zbl 1455.93097

Summary: 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 [C. Guiver et al., SIAM J. Control Optim. 57, No. 1, 334–365 (2019; Zbl 1405.93204)] 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.

MSC:

93C35 Multivariable systems, multidimensional control systems
93D25 Input-output approaches in control theory
93B52 Feedback control
93C20 Control/observation systems governed by partial differential equations
35B15 Almost and pseudo-almost periodic solutions to PDEs

Citations:

Zbl 1405.93204
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References:

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