Stabilization of parabolic systems via a degenerate nonnegative feedback

A degenerate nonnegative linear operatorB is added to an unbounded self-adjoint linear operatorL in a Hilbert space. The perturbed operator is of the formL + kB, k being a positive parameter. A sufficient condition is sought so that we can choose the minimum eigenvalue ofL + kB arbitrarily large. It is also studied from the numerical analytic viewpoint how the minimum eigenvalue behaves ask increases in relation to an approximation ofB. Modifications of the classical theory of spectrum perturvations are required, since a new type of information is relevant concerning bounds on the multiplicative constant in the familiar formulations of exponential decay. The result is then applied to a new stabilization problem for a class of linear and/or semilinear parabolic differential equations so that their evolutions can be stabilized and at the same time not deviate very much from the initial states for everyt>0, wherekB is considered as a feedback operator.