Observability for heat equations
نویسنده
چکیده
This talk describes di¤erent approaches to get the observability for heat equations without the use of Carleman inequalities. Contents 1 The heat equation and observability 2 2 Our motivation 3 3 Our strategy 4 3.1 Proof of "Hölder continuous dependence from one point in time" ) "Sum of Laplacian eigenfunctions" . . . . . . . . . . . . . . . 4 3.2 Proof of "Hölder continuous dependence from one point in time" ) "Observability" . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Proof of "Hölder continuous dependence from one point in time" ) "Re ned Observability" . . . . . . . . . . . . . . . . . . . . . 7 4 What I hope 9 4.1 Logarithmic convexity method . . . . . . . . . . . . . . . . . . . 9 4.2 Weighted logarithmic convexity method . . . . . . . . . . . . . . 10 5 What I can do 12 5.1 The frequency function . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2 The frequency function with weight . . . . . . . . . . . . . . . . . 14 5.3 The heat equation with space-time potential . . . . . . . . . . . . 19 This talk was done when the author visited School of Mathematics & Statistics, Northeast Normal University, Changchun, China, (July 4-21, 2011).
منابع مشابه
On observability of parallel - ow di usive heat exchanger equations with boundary output
This paper is concerned with the observability problem of a parallelow twouid heat exchanger equation with di usion term. First, the case where two uid temperatures are measured at the outlet is considered. It is shown that the observed system with the measurements becomes observable on any interval of time through a concrete series expression of the solution. Next, the two cases where each one...
متن کاملNull Controllability of Some Systems of Two Backward Stochastic Heat Equations with One Control Force
In this paper, we establish the null controllability for system coupled by two backward stochastic parabolic equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system. 2000 Mathematics Subject Classification. Primary 93B05; Secondary 93B07
متن کاملControllability of Partial Differential Equations and Its Semi-discrete Approximations
In these notes we analyze some problems related to the controllability and observability of partial differential equations and its space semidiscretizations. First we present the problems under consideration in the classical examples of the wave and heat equations and recall some well known results. Then we analyze the 1−d wave equation with rapidly oscillating coefficients, a classical problem...
متن کاملObservability of Switched Differential-Algebraic Equations
We study observability of switched differentialalgebraic equations (DAEs) for arbitrary switching. We present a characterization of observability and, a related property called, determinability. These characterizations utilize the results for the single-switch case recently obtained by the authors. Furthermore, we study observability conditions when only the mode sequence of the switching signa...
متن کاملA Deterministic Optimal Design Problem for the Heat Equation
For the heat equation on a bounded subdomain Ω of R, we investigate the optimal shape and location of the observation domain in observability inequalities. A new decomposition of L(R) into heat packets allows us to remove the randomization procedure and assumptions on the geometry of Ω in previous works. The explicit nature of the heat packets gives new information about the observability const...
متن کامل