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Register a new userConstructive 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 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 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 $vert svert rightarrow infty$ and that $g^primein L^infty_{loc}(mathbb{R})$ has been obtained by Zuazua in the nineties. The proof based on a Leray-Schauder fixed point argument makes use of precise estimates of the observability constant for a linearized wave equation. It does not provide however an explicit construction of a null control. Assuming that $g^primein L^infty_{loc}(mathbb{R})$, that $sup_{a,bin mathbb{R},a eq b} vert g^prime(a)-g^{prime}(b)vert/vert a-bvert^r<infty $ for some $rin (0,1]$ and that $g^prime$ satisfies the growth condition $vert g^prime(s)vert/log^{2}(vert svert)rightarrow 0$ as $vert svert rightarrow infty$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach guarantees the convergence whatever the initial element of the sequence may be. In particular, after a finite number of iterations, the convergence is super linear with rate $1+r$. This general method provides a constructive proof of the exact controllability for the semilinear wave equation.
In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{o}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
In this paper, we study approximate and exact controllability of the linear difference equation $x(t) = sum_{j=1}^N A_j x(t - Lambda_j) + B u(t)$ in $L^2$, with $x(t) in mathbb C^d$ and $u(t) in mathbb C^m$, using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$, and some suitable matrix coefficients. When $Lambda_1, dotsc, Lambda_N$ are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. This paper focuses on providing characterizations of approximate and exact controllability without the commensurability assumption. In the case of two-dimensional systems with two delays, we obtain an explicit characterization of approximate and exact controllability in terms of the parameters of the problem. In the general setting, we prove that approximate controllability from zero to constant states is equivalent to approximate controllability in $L^2$. The corresponding result for exact controllability is true at least for two-dimensional systems with two delays.
It is well-known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the Geometric Control Condition (GCC), which stipulates that any geodesic ray meets the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $Delta=-sum_{i=1}^m X_i^*X_i$ on a manifold $M$ such that $text{Lie}(X_1,ldots,X_m)=TM$ but $text{Span}(X_1,ldots,X_m)subsetneq TM$, we show that for any $T_0>0$ and any measurable subset $omegasubset M$ such that $Mbackslash omega$ has nonempty interior, the wave equation with subelliptic Laplacian $Delta$ is not observable on $omega$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on spiraling geodesics (for the associated sub-Riemannian distance) spending a long time in $Mbackslash omega$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space.
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