Locally distributed control and damping for the conservative systems

In this paper we note the equivalence between exact controllability and exponential stabilizability for an abstract conservative system with bounded control. This enables us to establish a frequency domain characterization for the exact controllability/uniform exponential decay property of second-order elastic systems, such as the wave equation and the Petrovsky equation, with (locally) distributed control/damping. A piecewise multiplier method for frequency domain is introduced. For several classes of PDEs on regions which are not necessarily smooth, we obtain a sufficient condition for the subregion on which the application of central/damping will yield the exact controllability/uniform exponential decay property. This result provides useful information for designing the location of controllers/dampers for distributed systems with a law of conservation.