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Constructive exact control of semilinear 1D wave equations by a least-squares approach
TL;DR: In this paper, a constructive proof and algorithm for the controllability of semilinear 1D wave equations with Dirichlet boundary conditions is presented. But the proof is based on the Leray-Schauder fixed point theorem, which is not constructive.
Abstract: It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation ∂tty − ∂xxy + g(y) = f 1ω, with Dirichlet boundary conditions, is exactly controllable in H 1 0 (0, 1) ∩ L 2 (0, 1) with controls f ∈ L 2 ((0, 1) × (0, T)), for any T > 0 and any nonempty open subset ω of (0, 1), assuming that g ∈ C 1 (R) does not grow faster than β|x| ln 2 |x| at infinity for some β > 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 does not grow faster than β ln 2 |x| at infinity for some β > 0 small enough and that g is uniformly Holder continuous on R with exponent s ∈ [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.
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TL;DR: A constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilinear equation, at least with order 1 + p after a finite number of iterations is designed.
Abstract: The null distributed controllability of the semilinear heat equation $\partial_t y-\Delta y + g(y)=f \,1_{\omega}$ assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{\vert r\vert\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$ has been obtained by Fern\'andez-Cara and Zuazua in 2000. The proof based on a non constructive fixed point theorem makes use of precise estimates of the observability constant for a linearized heat equation.
Assuming that $g^\prime$ is bounded and uniformly H\"older continuous on $\mathbb{R}$ with exponent $p\in (0,1]$, we design a constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations. The method is based on a least-squares approach and coincides with a globally convergent damped Newton methods: it guarantees the convergence whatever be the initial element of the sequence. Numerical experiments in the one dimensional setting illustrate our analysis.
7 citations
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TL;DR: The null distributed controllability of the semilinear heat equation was shown to be super linear with a rate equal to 1 + s in this paper, where s is the initial element of the sequence.
Abstract: 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 $g(s)/(\vert s\vert \log^{3/2}(1+\vert s\vert))\rightarrow 0$ as $\vert s\vert \rightarrow \infty$ and that $g^\prime\in L^\infty_{loc}(\mathbb{R})$ has been obtained by Fernandez-Cara and Zuazua in 2000. The proof based on a fixed point argument makes use of precise estimates of the observability constant for a linearized heat equation. It does not provide however an explicit construction of a null control. Assuming that $g^\prime\in W^{s,\infty}(\mathbb{R})$ for one $s\in (0,1]$, 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, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In particular, after a finite number of iterations, the convergence is super linear with a rate equal to $1+s$. Numerical experiments in the one dimensional setting support our analysis.
4 citations
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TL;DR: In this article, the authors present a constructive proof for the controllability of the semilinear heat equation in the one-dimensional setting with order 1 + p. The proof is based on a non-constructive fixed point argument and uses precise estimates of the observability constant for a linearized heat equation.
Abstract: 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 $g\in C^1(\mathbb{R})$ satisfies the
growth condition $\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert
r\vert)=0$ has been obtained by Fern\'andez-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 $\beta \ln^{3/2}\vert r\vert$ at infinity for $\beta>0$ small enough and that $g^\prime$ is uniformly H\"older continuous on $\mathbb{R}$ with exponent $p\in [0,1]$, 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.
3 citations
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TL;DR: In this paper, the null controllability problem for the wave equation was considered, and a stabilized finite element method formulated on a global, unstructured spacetime mesh was analyzed.
Abstract: We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.
TL;DR: In this article , the exact controllability of the semilinear wave equation with respect to potentials was proved based on a non-constructive fixed point argument, which makes use of precise estimates of the observability constant for a linearized wave equation.
Abstract: The exact controllability of the semilinear wave equation $$y_{tt}-y_{xx}+ f(y)=0$$ , $$x\in (0,1)$$ assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f(r)\vert /(\vert r\vert \ln ^{p}\vert r\vert )\leqslant \beta $$ for some $$\beta $$ small enough and $$p=2$$ has been obtained by Zuazua (Ann Inst H Poincaré Anal Non Linéaire 10(1):109–129, 1993). The proof based on a non-constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized wave equation. Under the above asymptotic assumption with $$p=3/2$$ , by introducing a different fixed point application, we present a simpler proof of the exact boundary controllability which is not based on the cost of observability of the wave equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f^\prime (r)\vert /\ln ^{3/2}\vert r\vert \leqslant \beta $$ for some $$\beta $$ small enough, we show that the above fixed point application is contracting yielding a constructive method to approximate the controls for the semilinear equation. Numerical experiments illustrate the results. The results can be extended to the multi-dimensional case and for nonlinearities involving the gradient of the solution.
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Book•
17 Apr 2007
TL;DR: In this article, the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions are studied, with a focus on specific phenomena due to nonlinearities.
Abstract: This book presents methods to study the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions. The emphasis is put on specific phenomena due to nonlinearities. In particular, many examples are given where nonlinearities turn out to be essential to get controllability or stabilization. Various methods are presented to study the controllability or to construct stabilizing feedback laws. The power of these methods is illustrated by numerous examples coming from such areas as celestial mechanics, fluid mechanics, and quantum mechanics. The book is addressed to graduate students in mathematics or control theory, and to mathematicians or engineers with an interest in nonlinear control systems governed by ordinary or partial differential equations.
993 citations
"Constructive exact control of semil..." refers background in this paper
...o of F(see [7] where divergence of the sequence is shown for large data). The controllability of nonlinear partial differential equations has attracted a large number of works in the last decades (see [4] and references therein). However, as far as we know, few are concerned with the approximation of exact controls for nonlinear partial differential equations, and the construction of convergent control...
[...]
01 Jan 1988
TL;DR: Controle optimal de systemes. (Systemes couples) as mentioned in this paper : Cas of systemes soumis a des perturbations. Controlabilite exacte and perturbation singulieres.
Abstract: Controle optimal de systemes. Deuxieme partie : Cas de systemes soumis a des perturbations. (Systemes couples. Controlabilite exacte et penalisation. Controlabilite exacte et perturbations singulieres.- Perturbations des modes d'action sur les systemes - Perturbations des domaines.- Homogeneisation.Systemes a memoire).
755 citations
TL;DR: In this paper, it was shown that the system is controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term f(y) is such that |f(s)| grows slower than |s|log3/2(1+|s|) as | s|→∞.
Abstract: We consider the semilinear heat equation in a bounded domain of Rd , with control on a subdomain and homogeneous Dirichlet boundary conditions. We prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term f(y) is such that |f(s)| grows slower than |s|log3/2(1+|s|) as |s|→∞ . For instance, this condition is fulfilled by any function f growing at infinity like |s|logp(1+|s|) with 1 2 , null controllability does not hold. The problem remains open when f behaves at infinity like |s|logp(1+|s|) , with 3/2≤p≤2 . Results of the same kind are proved in the context of approximate controllability.
371 citations
TL;DR: In this paper, the exact controllability of the semilinear wave equation with Dirichlet boundary conditions is studied and a method of proof based on the HUM (Hilbert Uniqueness Method) and on a fixed point technique is presented.
Abstract: The exact controllability of the semilinear wave equation y″ – yxx + f(y) = h in one space dimension with Dirichlet boundary conditions is studied. We prove that if |f(s)|/|s |log2 |s| → 0 as |s| → ∞, then the exact controllability holds in H 0 1 ( Ω ) × L 2 ( Ω ) with controls h ∈ L2(Ω × (0, T)) supported in any open and non empty subset of Ω. The exact controllability time is that of the linear case where f = 0. Our method of proof is based on HUM (Hilbert Uniqueness Method) and on a fixed point technique. We also show that this result is almost optimal by proving that if f behaves like – s logp(1 + |s|) with p > 2 as |s| → ∞, then the system is not exactly controllable. This is due to blow-up phenomena. The method of proof is rather general and applied also to the wave equation with Neumann type boundary conditions.
226 citations