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تسجيل مستخدم جديدConstructive proof of the exact controllability for semi-linear wave equations
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The exact distributed controllability of the semilinear wave equation $partial_{tt}y-Delta y + g(y)=f ,1_{omega}$ posed over multi-dimensional and bounded domains, assuming that $gin C^1(mathbb{R})$ satisfies the growth condition $limsup_{rto infty} g(r)/(vert rvert ln^{1/2}vert rvert)=0$ has been obtained by Fu, Yong and Zhang in 2007. The proof based on a non constructive Leray-Schauder fixed point theorem makes use of precise estimates of the observability constant for a linearized wave equation. Assuming that $g^prime$ does not grow faster than $beta ln^{1/2}vert rvert$ at infinity for $beta>0$ small enough and that $g^prime$ is uniformly Holder continuous on $mathbb{R}$ with exponent $sin (0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+s$ after a finite number of iterations.
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It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation $partial_{tt}y-partial_{xx}y + g(y)=f 1_{omega}$, with Dirichlet boundary conditions, is exactly controllable in $H^1_0(0,1)cap L^2(0,1)$ with con
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