From internal to pointwise control for the 1D heat equation and minimal control time Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set omega_varepsilon(x_0varepsilon, x_0varepsilon ), in the limit varepsilonrightarrow 0, where x_0in (0,1). It is known that depending on arithmetic properties of x_0, there may exist a minimal time T_0 of pointwise control at x_0 of the heat equation. Besides, for any varepsilon fixed, the heat equation is controllable with control set omega_varepsilon in any time Tu003e0. We relate these two phenomena. We show that the observability constant on omega_varepsilon does not converge to 0 as varepsilonrightarrow 0 at the same speed when Tu003eT_0 (in which case it is comparable to varepsilon12) or Tu003cT_0 (in which case it converges faster to 0). We also describe the behavior of optimal L2 nullcontrols on omega_varepsilon in the limit varepsilon rightarrow 0.

Robust error based noncollocated output tracking control for a heat equation Abstract   In this paper, an output tracking problem for a heat equation is considered, where all possible disturbances produced from an exosystem and a systematic uncertainty are considered and the performance output is noncollocated with control. The objective is twofold: to look at how the internal model principle works for the output tracking of PDEs; and to see how to design a robust tracking error feedback control for PDEs. To this purpose, we first select a frozen case with specially selected frozen coefficients of the disturbances. For this frozen system, we design a feedforward control by solving simply regulator equation and an infinitedimensional extended state observer in terms of tracking error only which gives an estimation of both states of the frozen plant and exosystem. An observerbased tracking error feedback control is then designed for the frozen system, which is shown to be in line with the internal model principle. As a result, the system is shown to be robust to system uncertainty and disturbances in all channels. The numerical simulations validate the results.