Trajectory and final controllability in hyperbolic and pseudohyperbolic systems with generalized actions
The authors consider a hyperbolic system: \[ \partial^2u/ \partial t^2+A(u)= f(h) \] in a cylindrical domain \(Q=(0,T)\times \Omega\), where \(h\) is a control parameter, and \(A\) is an elliptic operator of the form: \[ A(u)=\Sigma_{ij} \partial/ \partial x_i(a_{ij} (x)\partial u/\partial x_j)+\Sigma_ia_i(x)\partial u/\partial x_i+a(x)u. \] All coefficients are continuously differentiable, except \(a\), which is allowed to be only continuous. In their previous work the authors utilized the concept of negative norms, which otherwise satisfy all requirements for a norm, such as for example the Cauchy-Schwarz inequality. They derive a sequence of inequalities valid in these negative norms. A major project of this paper consists in finding a dense imbedding of the space of functions which satisfy their requirements and are of the class \(C_0^\infty (Q)\), into the space \(L_2(Q)\). Using these techniques the authors prove controllability (subject to a rather general choice of the quality functional), which is valid for a class of hyperbolic and pseudohyperbolic systems of equations similar to the one displayed above.
scientific article; zbMATH DE number 1981958