Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFrom internal to pointwise control for the 1D heat equation and minimal control time
160
0
0.0
(
0
)
Authors
Cyril Letrouit
Ask ChatGPT about the research
No Arabic abstract
Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $omega_{varepsilon}=(x_{0}-varepsilon, x_{0}+varepsilon )$, in the limit $varepsilonrightarrow 0$, where $x_{0}in (0,1)$. 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 $T>0$. 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 $T>T_{0}$ (in which case it is comparable to $varepsilon^{1/2}$) or $T<T_{0}$ (in which case it converges faster to $0$). We also describe the behavior of optimal $L^{2}$ null-controls on $omega_{varepsilon}$ in the limit $varepsilon rightarrow 0$.
rate research
Read More
We discuss reachable states for the Hermite heat equation on a segment with boundary $L^2$-controls. The Hermite heat equation corresponds to the heat equation to which a quadratic potential is added. We will discuss two situations: when one endpoint of the segment is the origin and when the segment is symmetric with respect to the origin. One of the main results is that reachable states extend to functions in a Bergman space on a square one diagonal of which is the segment under consideration, and that functions holomorphic in a neighborhood of this square are reachable.
Let $u(t,x)$ be a solution of the heat equation in $mathbb{R}^n$. Then, each $k-$th derivative also solves the heat equation and satisfies a maximum principle, the largest $k-$th derivative of $u(t,x)$ cannot be larger than the largest $k-$th derivative of $u(0,x)$. We prove an analogous statement for the solution of the heat equation on bounded domains $Omega subset mathbb{R}^n$ with Dirichlet boundary conditions. As an application, we give a new and fairly elementary proof of the sharp growth of the second derivatives of Laplacian eigenfunction $-Delta phi_k = lambda_k phi_k$ with Dirichlet conditions on smooth domains $Omega subset mathbb{R}^n$.
The goal of this work is to compute a boundary control of reaction-diffusion partial differential equation. The boundary control is subject to a constant delay, whereas the equation may be unstable without any control. For this system equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed. To do that we decompose the infinite-dimensional system into two parts: one finite-dimensional unstable part, and one stable infinite-dimensional part. An finite-dimensional delay controller is computed for the unstable part, and it is shown that this controller succeeds in stabilizing the whole partial differential equation. The proof is based on a an explicit form of the classical Artstein transformation, and an appropriate Lyapunov function. A numerical simulation illustrate the constructive design method.
The exact distributed controllability of the semilinear heat equation $partial_{t}y-Delta y + g(y)=f ,1_{omega}$ posed over multi-dimensional and bounded domains, assuming that $gin C^1(mathbb{R})$ satisfies the growth condition $limsup_{rto infty} g(r)/(vert rvert ln^{3/2}vert rvert)=0$ has been obtained by Fernandez-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that $g^prime$ does not grow faster than $beta ln^{3/2}vert rvert$ at infinity for $beta>0$ small enough and that $g^prime$ is uniformly Holder continuous on $mathbb{R}$ with exponent $pin [0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations.
We consider the damped wave equation on a manifold with imperfect geometric control. We show the sub-exponential energy decay estimate in cite{Chr-NC-erratum} is optimal in the case of one hyperbolic periodic geodesic. We show if the equation is overdamped, then the energy decays exponentially. Finally we show if the equation is overdamped but geometric control fails for one hyperbolic periodic geodesic, then nevertheless the energy decays exponentially.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
