The Study For Boundary Control And The Carleman Estimate Of Unstable Heat Equation

In this paper,we study mainly the problem of the boundary control and the Carleman eatimate of the unstable heat equation.Firstly, we'll study the boundary control problem of heat equation with a destabilizing term, and the exponential stability estimate under the given boundary condition.Boundary control is an important part of the modern control theory , which has been emphasized in the control theory field and has been extensively studied and developed . we choice suitable boundary feedback condition ,seek an unsingular invertible coordinate transformation to transfer the system into a new set of coordinates where we can design a control law that achieves stabilization . We will use the Fredlom operator theorem,the variation of constants formula,functional analysis to proof the unsingularity of the coordinate transformation.Later, Designing Lyapunov function,applying semi-group theorem and Sobolev space theorem and some unequalities to turn out the stable control law. And gain the L~2 and H~1 exponential stability estimate under Dirichlet and Neumann boundary control.Secondly,we'll study the Carleman eatimate for the unstable heat equation with BV coefficients.Mainly using the dominated convergence theorem and some inequalities.