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Proportional integral regulation control of a one-dimensional semilinear wave equation. (English) Zbl 1481.35266

Authors’ abstract: This paper is concerned with the proportional integral (PI) regulation control of the left Neumann trace of a one-dimensional semilinear wave equation. The control input is selected as the right Neumann trace. The control design goes as follows. First, a preliminary (classical) velocity feedback is applied in order to shift all but a finite number of the eivenvalues of the underlying unbounded operator into the open left half-plane. We then leverage the projection of the system trajectories into an adequate Riesz basis to obtain a truncated model of the system capturing the remaining unstable modes. The controller is computed by applying a classical PI control design scheme to this truncated model. Local stability of the resulting closed-loop infinite-dimensional system is obtained through the study of an adequate Lyapunov function. Finally, an estimate assessing the set point tracking performance of the left Neumann trace is derived.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
93C20 Control/observation systems governed by partial differential equations
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