Boundary geometric control of a heat equation

The present study proposes a design approach of a boundary control for a one-dimensional heat equation. The objective is to control the temperature at a given punctual position. The control law is based on the nonlinear geometric control theory. The idea consists to make the boundary condition homogeneous, by inserting the manipulated variable by means of Dirac delta function into the state equation that describes the spatial-temporal evolution of the temperature. Then, in order to overcome the controllability problem encountered by considering a punctual output in control design, a weighted value of the temperature, along the spatial domain, is considered as a measured output. By calculating the successive derivatives of this measured output, a control law is deduced and a control strategy is proposed in order to meet the desired control objective of the punctual output. The control performance of the proposed strategy is evaluated through numerical simulation by considering the control problem of the temperature of a thin metal rod modelled by a heat equation with a linear source.