Group Seminar: OOC: Riccati based feedback stabilization to trajectories for parabolic equations
The problem which we address here is the {\em local} exponential stabilization to trajectories for parabolic systems, for $t∈(0,+∞)$ in the form
$∂ty−νΔy+F(y)+∇⋅G(y)+f+∑i=1MuiΦi=0;y|Γ=g;\intertextorintheform∂ty−νΔy+F(y)+∇⋅G(y)+f=0;y|Γ=g+∑i=1MuiΨi,$
where $F$ and $G$ may be nonlinear functions and vector functions respectively, and $u: (0,+∞)⟶RM$ is a control function. That is, given a positive constant $λ0$ and a solution $y^(t)=y^(t,⋅)$ of the uncontrolled system (with$~u=0$), our goal is to find $u$ such that the solution $y(t):=y(t,⋅)$ of the system, supplemented with the initial condititon
$y(0):=y(0,x)=y0(x)$,
is defined on $[0,+∞)$ and satisfies, for a suitable Banach space$~X$, \begin{equation*}%\label{goal} \left| y(t) - \hat{y} (t) \right|^2_{X} \le C \mathrm{e}^{-\lambda t} \left| y(0) - \hat{y} (0) \right|^2_{X},\quad\mbox{provided}\quad|y(0) - \hat{y} (0)|_{X}