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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 controls $fin L^2((0,1)times(0,T))$, for any $T>0$ and any nonempty open subset $omega$ of $(0,1)$, assuming that $gin mathcal{C}^1(R)$ does not grow faster than $betavert xvert ln^{2}vert xvert$ at infinity for some $beta>0$ small enough. The proof, based on the Leray-Schauder fixed point theorem, is however not constructive. In this article, we design a constructive proof and algorithm for the exact controllability of semilinear 1D wave equations. Assuming that $g^prime$ does not grow faster than $beta ln^{2}vert xvert$ at infinity for some $beta>0$ small enough and that $g^prime$ is uniformly Holder continuous on $R$ with exponent $sin[0,1]$, we design a least-squares algorithm 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.
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
The exact distributed controllability of the semilinear wave equation $y_{tt}-y_{xx} + g(y)=f ,1_{omega}$, assuming that $g$ satisfies the growth condition $vert g(s)vert /(vert svert log^{2}(vert svert))rightarrow 0$ as $vert svert rightarrow infty$
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}
The null distributed controllability of the semilinear heat equation $y_t-Delta y + g(y)=f ,1_{omega}$, assuming that $g$ satisfies the growth condition $g(s)/(vert svert log^{3/2}(1+vert svert))rightarrow 0$ as $vert svert rightarrow infty$ and that
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. Thi