On stabilizability of evolution partial differential equations on $\mathbb{R}^n\times [0,+\infty)$ by time-delayed feedback controls

The problem of stabilizability by time-delayed feedback control is investigated for an evolution partial differential equation on $\mathbb R^n\times [0,+\infty)$. We use the Tarski–Seidenberg theorem and its corollaries to obtain some estimates of semi-algebraic functions on semi-algebraic sets and obtain estimates of the real parts of quasipolynomial zeros. These estimates make it possible to apply the Fourier transform method to investigate the stabilizability problem. We utilise some results of the theory of ordinary differential-difference equations to study a “dual” system obtained from the original system with time-delayed feedback control by applying the Fourier transform. We also give some examples of stabilizable and non-stabilizable systems.