Pointwise and Internal Controllability for the Wave Equation

Abstract. Problems of internal and pointwise observation and control for the one-dimensional wave equation arise in the simulation of control and identification processes in electrical engineering, flaw detection, and medical tomography. The generally accepted way of modelling sensors and actuators as pointlike objects leads to results which may make no apparent physical sense: they may depend, for instance, on the rationality or irrationality of the location for a point sensor or actuator. We propose a new formulation of sensor (actuator) action, expressed mathematically by using somewhat unconventional spaces for data presentation and processing. For interaction restricted to an interval of length ε , the limit system of observation (or control) now makes sense when ε tends to zero without a sensitive dependence on the precise location of the limiting point.