Partial rapid stabilization of linear distributed systems (Q2714173)

The problem (*) \(x'=Ax+Bu\), \(x(0)=x_0\) is studied, where \(A\) is a densely defined, closed linear operator in a Hilbert space \(H\), and \(B\) is a bounded linear operator from another Hilbert space \(G\) into \(D(A^*)'\) (the dual space of the domain of the adjoint of \(A\)). For a given \(T>0\), the state \(x_0\in H\) is said to be controllable if there is a control function \(u\in L^2(0,T;G)\) such that the solution of (*) satisfies \(x(T)=0\). The main result gives a control of the form \(u=F(x)\) so that it stabilizes rapidly the controllable component of the solutions. As an application the wave equation with Dirichlet boundary control is considered.