Control estimates and well-posedness of an inverse subsidence problem in geomechanics

Abstract

Underground geotechnical or civil engineering works change the primary stressed and strained state of the rock mass and subsequently the fields of displacement and stresses. A significant effect of these changes is the formation of subsidence in the earth’s surface with significant negative economic or other consequences. In a series of recent works, large-scale subsidence in geostructures was modelled using Litwiniszyn’s theory (Einstein–Kolmogorov Differential Equation) and the trap-door mechanism. The solution of the Inverse Subsidence Diffusion–Convection (ISDC) problem – which belongs to the class of backward parabolic problems – provides the base (gravity-directed) displacement using the surface subsidence as ‘initial’ condition. Early well-posedness of the initial and boundary value problems has already been examined (in terms of stability estimates) in previous papers. In this work, study of control estimates and well-posedness of the ISDC problem, using energy method, are discussed.