### RICAM Events

#### Group Seminar

##### OOC: Riccati based feedback stabilization to trajectories for parabolic equations
Speaker: Duy Phan-Duc (RICAM)
Date: April 22, 2016 10:30
Location: SP2 451

The problem which we address here is the {\em local} exponential stabilization to trajectories for parabolic systems, for $t \in (0, +\infty)$ in the form \begin{align*}% \partial_t y - \nu \Delta y + F(y) + \nabla \cdot G(y) + f +\textstyle\sum\limits_{i=1}^M u_i\Phi_i&= 0;\quad &&y|_\Gamma = g;%\label{sys-y-cont_i}\\ % % % % \end{subequations} % % % \begin{subequations}\label{sys-y-cont_b} % \begin{align} \intertext{or in the form } \partial_t y - \nu \Delta y + F(y) + \nabla \cdot G(y) + f&= 0;\quad &&y|_\Gamma = g +\textstyle\sum\limits_{i=1}^M u_i\Psi_i,%\label{sys-y-cont_b} % \end{align} %\end{subequations} \end{align*} where $F$ and $G$ may be nonlinear functions and vector functions respectively, and $u:~(0,+\infty) \longrightarrow \mathbb{R}^M$ is a control function. That is, given a positive constant $\lambda0$ and a solution $\hat{y}(t)=\hat{y}(t,\cdot)$ of the uncontrolled system (with~$u = 0$), our goal is to find $u$ such that the solution $y(t):= y(t,\,\cdot)$ of the system, supplemented with the initial condititon \begin{align*}%\label{InitCond} y(0):= y(0,x) = y_0(x), \end{align*} is defined on $\left[0, + \infty \right)$ 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}

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