Controllability problem for the wave equation with an external load with the impulse control

Abstract

Controllability problems for partial differential equations are being investigated nowadays by a number of mathematicians. In most cases the controllability problems have been investigated where as a control function the initial or boundary conditions has been chosen. But in some cases the controls introduced in the boundary conditions can not be realized from a practical point of view. Therefore we will consider a controllability problem of lumped sources for processes described by wave (for example, string) equations in the prelimit regular space of the functions of bounded q-integral p-variation in the present paper. In this article we build the controls introduced in the function of an external load, which solve the problem of approximate null-controllability and null-controllability. The aim of research is the formulation of the problem definition of the criteria of approximate null-controllability and null-controllability. The applications of this problems may be present in such industries of physics and technology: oscillations of the building constructions (slabs, plates, beams, masts); wave processes in the physics of electromagnetic waves and radio waves (diffraction, etc.); processes in spacecraft design, aircraft; geophysical processes (processes in the Earth’s crust, oceans, nature of earthquakes, tsunamis, etc.); seismic (waves in solids); methods of the medical diagnosis; and others. The controllability problem for the string equation with an external load in the prelimit regular space of the functions of bounded q-integral p-variation is considered in this article. The results may be a basis for the investigation of controllability problems of wave equations of higher dimensions with and without an external load in the prelimit regular space of the functions of bounded q-integral p-variation, i. e. when considering solutions on plane and for some closed regions.