Boundary and distributed controllability of the damped wave equation: Reduction of control time
The authors continue the study of the controllability problem for the damped wave equation (see the papers of the authors' references (1-7)). Also, one considers the one-dimensional wave equation which represents the vibration of a string with spatially nonhomogeneous damping modulus of elasticity and density coefficients. The equation is defined on a finite interval with linear first-order nonselfadjoint boundary conditions containing damping terms. Section 2 contains auxiliary propositions and a formulation of the main results. All new results are proven in Section 3 and 4. In this paper, the authors study two extensions. First, they consider the boundary control combined with the aforementioned distributed control, and second they examine more general distributed control. By combining in this note the boundary and the distributed controls, it is shown that it is possible to reduce the time interval. The authors consider the wave equation \[ u_{tt}+ 2d(x) u_t+ Lu= F(x,t),\quad x\in[0,a],\quad t>0,\tag{1} \] where \(L\) is the Sturm-Liouville operator defined on smooth functions \(\varphi\) such that \[ L\varphi= -{1\over\rho(x)} {d\over dx} \Biggl(p(x) {d\varphi\over dx}\Biggr)+ q(x)\varphi. \] Equation (1) describes the forced motion of a string with density \(\rho(x)\), with modulus of elasticity \(p(x)\) and subject to an external harmonic force with rigidity \(q(x)\) and in the presence of viscous damping \(2d(x)\). The function \(u(x,t)\) satisfies the initial conditions \[ u(x,0)= u^0(x),\quad u_t(x,0)= u^1(x) \] and the boundary conditions \[ (u_x+ ku_t)(0,t)= 0,\quad (u_x+ hu_t)(a,t)= \varphi(t),\quad h,k\in\mathbb{C}\cup\{\infty\}. \]
scientific article; zbMATH DE number 1398593
scientific article