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The anti-disturbance property of a closed-loop system of 1-d wave equation with boundary control matched disturbance. (English) Zbl 1524.35075

It is considered the so called anti-disturbance problem for the boundary controlled and perturbed \(1D\) wave equation \[ \displaylines{ \displaystyle{w_{tt}-w_{xx}=0\;,\;x\in(0,1)\;,\;t>0}\cr \displaystyle{w(0,t)=0\;,\;w_x(1,t)=u(t)+d(t)}\cr \displaystyle{w(x,0)=w_0(x)\;,\;w_t(x,0)=w_1(x)}} \] where \(u(t)\) is the input control signal and \(d(t)\) is a uniformly bounded unknown disturbance. The feedback control signal is of the type D(erivative) controller, i.e. \(u(t)=-kw_t(1,t)\). The paper tackles the closed loop dynamics. In particular, it deals with choosing the gain \(k>0\) in order to have the bound of the closed loop solution small for any initial data \(\{w_0(x),w_1(x)\}\). A problem of parameter optimization is stated and solved, thus reminding of optimum controller tuning.

MSC:

35B35 Stability in context of PDEs
93C05 Linear systems in control theory
93C20 Control/observation systems governed by partial differential equations
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