We perform an asymptotic analysis, with respect to the viscosity parameter epsilon > 0 of the solution of the viscous Burgers equation u(t)(epsilon)+u(epsilon)u(x)(epsilon)-epsilon u(xx)(epsilon)=0,(x, t)is an element of ,(x, t)is an element of Rx(0,T),u(epsilon)(x,0) =phi(x) We assume that is smooth on R except at x = 0 bounded and decreasing, and that lim(x -> 0-)phi(x)>lim(x -> 0+)phi(x) When & tends to zero, the solution u exhibits a shock layer of size O(E) in the vicinity of a shock curve s(t) starting from x = 0 We denote by u the entropy solution of the corresponding inviscid Burgers equation, i.e. when epsilon = 0 and we assume that there is only one shock curve to the inviscid Burgers equation. Using the method of matched asymptotic expansions, we construct an explicit approximation Pe satisfying parallel to u(epsilon)-P-epsilon parallel to L-infinity(0,T;L-2(R))=& Oscr;(epsilon(3/2))and parallel to u(epsilon)-P-epsilon parallel to L-2(0,T;H1(R))=& Oscr;(epsilon(1/2)) . The approximation takes into account the initial and internal layers, and is uniform, with no truncation term to distinguish the approximation zones. As far as we know, this approximation is original. We also show that parallel to u(epsilon)-u(0)parallel to L-infinity(0,T;L-loc(1)(R))=& Oscr;(epsilon), including the regions containing the shock. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, Al training, and similar technologies.

In order to design a road de-icing device by heating, we consider in a two dimensional setting the optimal control of an advection–diffusion equation with a nonlinear boundary condition of the Stefan-Boltzmann type. The problem models the heating of a road during a winter period to keep positive its surface temperature above a given threshold. The heating device is performed through the circulation of a coolant in a porous layer of the road. We prove the well-posedeness of the nonlinear optimal control problem, subject to unilateral constraints on the control and the state, set up a gradient based algorithm then discuss some numerical results associated with real data obtained from experimental measurements. The study, initially developed in a one dimensional simpler setting in [1], aims to quantify the minimal energy to be provided to keep the road surface without frost or snow.

For autonomous driving applications, we investigate the fog Droplet Size Distribution (DSD) identification from spectral radiation measurements in the range 350 nm - 2500 nm by inverting the 1D stationary radiative transfer equation (RTE). This distribution together with Lorenz-Mie scattering theory allow to compute the optical properties (scattering coefficient, absorption coefficient, and scattering phase function). The inverse problem is based on a cost function minimization by using a gradient descent-based algorithm. In order to calculate the gradient, the method proposed in this work relies on the introduction of an adjoint problem of the RTE which can contain an inscattering term. The work is then not restricted to the Beer-Lambert case as often encountered in the literature.We present some numerical results obtained on synthetic forescattering and backscattering measurements. The robustness of the reconstruction is studied numerically by adding several noise levels to the measurements.

This paper addresses the boundary null-controllability of the semi linear heat equation partial derivative(ty) - partial derivative(xxy) + f(y) = 0, (x, t) is an element of (0, 1) x (0, T). Assuming that the function f is an element of C-1(R) satisfies lim sup(|r|->+infinity) |f(r)|/(|r| ln(3/2) |r | ) <= beta for some beta > 0 small enough and that the initial datum belongs to L-infinity(0, 1), we prove the global null-controllability using the Schauder fixed point theorem and a linearization for which the term f(y) is seen as a right side of the equation. Then, assuming that f satisfies lim sup(|r|->infinity) |f '(r)|/ ln(3/2) |r | <= beta for some beta small enough, we show that the fixed point application is contracting yielding a constructive method to approximate boundary controls for the semilinear equation. The crucial technical point is a regularity property of a state-control pair for a linear heat equation with L-2 right hand side obtained by using a global Carleman estimate with boundary observation. Numerical experiments illustrate the results. The arguments developed can notably be extended to the multi-dimensional case.

We address the exact boundary controllability of the semilinear wave equation $\partial_{tt}y-\Delta y + f(y)=0$ posed over a bounded domain $\Omega$ of $\mathbb{R}^d$. Assuming that $f$ is continuous and satisfies the condition $\limsup_{\vert r\vert\to \infty} \vert f(r)\vert /(\vert r\vert \ln^p\vert r\vert)\leq \beta$ for some $\beta$ small enough and some $p\in [0,3/2)$, we apply the Schauder fixed point theorem to prove the uniform controllability for initial data in $L^2(\Omega)\times H^{-1}(\Omega)$. Then, assuming that $f$ is in $\mathcal{C}^1(\mathbb{R})$ and satisfies the condition $\limsup_{\vert r\vert\to \infty} \vert f^\prime(r)\vert/\ln^p\vert r\vert\leq \beta$, we apply the Banach fixed point theorem and exhibit a strongly convergent sequence to a state-control pair for the semilinear equation.

The exact distributed controllability of the semi-linear wave equation partial derivative(tt)y -Delta y + g(y) = f1(omega) posed over multi-dimensional and bounded domains, assuming that g is an element of C-1(R) satisfies the growth condition lim sup(| r|->infinity) g(r)/(|r| ln(1/2) |r|) = 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 the derivative of g does not grow faster than beta ln(1/2) |r| at infinity for beta > 0 small enough and is uniformly Holder continuous on R with exponent s is an element of (0, 1], we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semi-linear equation, at least with order 1+ s after a finite number of iterations. Numerical experiments in the two-dimensional case illustrate the results. This work extends to a multi-dimensional case, enriches with additional results and completes with some numerical experiments the study in 2021 by Munch and Tr ' elat devoted to the one-dimensional situation.

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.

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.

We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection–diffusion equation y t ε + M ( x , t ) y x ε − ε y x x ε = 0, ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ), supplemented with Dirichlet boundary conditions. For small values of ε, the solution y ε exhibits a boundary layer of size O ( ε ) in the neighborhood of x = 1 (assuming M > 0) and an internal layer of size O ( ε 1 / 2 ) in the neighborhood of the characteristic starting from the point ( 0 , 0 ). Assuming that these layers interact each other after a finite time T > 0 and using the method of matched asymptotic expansions, we construct an explicit approximation P ε satisfying ‖ y ε − P ε ‖ L ∞ ( 0 , T ; L 2 ( 0 , 1 ) ) = O ( ε 1 / 2 ). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors.

We analyze a least-squares approach in order to approximate weak solutions of the 2D-Navier Stokes system. In a first part, we consider the steady case and introduce a quadratic functional based on a weak norm of the state equation. We construct a minimizing sequence for the functional which converges strongly to a solution of the equation. After a finite number of iterates related to the value of the viscosity constant, the convergence is quadratic, from any initial guess. We then apply iteratively the analysis on the backward Euler scheme associated to the unsteady Navier-Stokes equation and prove the convergence of the iterative process uniformly with respect to the time discretization. In a second part, we reproduce the analysis for the unsteady case by introducing a space-time least-squares functional. The method turns out to be related to the globally convergent damped Newton approach applied to the Navier-Stokes operator, in contrast to standard Newton method used to solve the weak formulation of the Navier-Stokes system. Numerical experiments illustrate our analysis.

The exact distributed controllability of the semilinear heat equation partial derivative(t)y - Delta y + f(y) = v 1 omega posed over multi-dimensional and bounded domains, assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup broken vertical bar r broken vertical bar ->infinity vertical bar f(r)vertical bar/(vertical bar r vertical bar ln3/2 vertical bar r vertical bar) < Q for some Q small enough 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. Under the same assumption, by introducing a different fixed point application, we present a different and somewhat simpler proof of the exact controllability, which is not based on the cost of observability of the heat equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup vertical bar r vertical bar ->infinity vertical bar f0(r)vertical bar/ ln3/2 vertical bar r vertical bar < Q for some Q small enough, we show that the above fixed point application is contracting yielding a constructive method to compute the controls for the semilinear equation. Numerical experiments illustrate the results.

The exact distributed controllability of the semilinear heat equation partial differential ty-Delta y+g(y) = f 1 omega posed over multi-dimensional and bounded domains, assuming that g is an element of C1(R) satisfies the growth condition lim supr ->infinity g(r)/ (|r| ln3/2 |r | ) = 0 has been obtained by Fernandez-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' does not grow faster than ,Q ln3/2 |r | at infinity for ,Q > 0 small enough and that g' is uniformly Holder continuous on R with exponent p is an element of [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.

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 $f\in L^2((0,1)\times(0,T))$, for any $T>0$ and any nonempty open subset $\omega$ of (0,1), assuming that $g\in \mathcal{C}^1(\mathbb{R})$ does not grow faster than $\beta\vert x\vert \ln^{2}\vert x\vert$ 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 $\beta \ln^{2}\vert x\vert$ at infinity for some $\beta>0$ small enough and that $g^\prime$ is uniformly Hölder continuous on $\mathbb{R}$ with exponent $s\in[0,1]$, 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 Petrowsky type equation y(t)(t)epsilon + ey(xxxx)(epsilon) - y(xx)(epsilon) = 0, epsilon > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order root epsilon occurring at the extremities, these boundary controls get singular as e goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution ye and derive a rigorous second order asymptotic expansion of the control of minimal weighted L-2-norm, with respect to the parameter e. The weight in the norm is chosen to guarantee the smoothness of the control. In particular, we recover and enrich earlier results due to J.-L. Lions in the eighties showing that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation. The asymptotic analysis also provides a robust discrete approximation of the control for any epsilon small enough. Numerical experiments support our study.

We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where a noisy data is known in an interior subset of spacetime. For this problem, we consider a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors. We prove error estimates using the stability properties of the numerical scheme and a continuum observability estimate, based on the sharp geometric control condition by Bardos, Lebeau and Rauch. The order of convergence for our numerical scheme is optimal with respect to stability properties of the continuum problem and the approximation order of the finite element residual. Numerical examples are provided that illustrate the methodology.

The exact distributed controllability of the semilinear wave equation ∂_tty-Δ y + g(y)=f 1_ω posed over multi-dimensional and bounded domains, assuming that g∈ C^1(ℝ) satisfies the growth condition lim sup_r→∞ g(r)/(| r|ln^1/2| r|)=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^' does not grow faster than βln^1/2| r| at infinity for β>0 small enough and that g^' is uniformly Hölder continuous on ℝ with exponent s∈ (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.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Constructive exact controls for some semilinear PDEs Arnaud Munch

This work analyzes a least-squares method in order to solve implicit time schemes associated to the 2D and 3D Navier–Stokes system, introduced in 1979 by Bristeau, Glowinksi, Periaux, Perrier and Pironneau. Implicit time schemes reduce the numerical resolution of the Navier–Stokes system to multiple resolutions of steady Navier–Stokes equations. We first construct a minimizing sequence (by a gradient type method) for the least-squares functional which converges strongly and quadratically toward a solution of a steady Navier–Stokes equation from any initial guess. The method turns out to be related to the globally convergent damped Newton approach applied to the Navier–Stokes operator. Then, we apply iteratively the analysis on the fully implicit Euler scheme and show the convergence of the method uniformly with respect to the time discretization. Numerical experiments for 2D examples support our analysis.

This work is concerned with the null controllability of the one-dimensional wave equation over non-cylindrical distributed domains. The controllability in that case has been obtained by Castro et al. [SIAM J. Control Optim. 52 (2014)] for domains satisfying the usual geometric optic condition. We analyze the problem of optimizing the non-cylindrical support q of the control of minimal L 2 ( q )-norm. In this respect, we prove a uniform observability inequality for a class of domains q satisfying the geometric optic condition. The proof based on the d’Alembert formula relies on arguments from graph theory. Numerical experiments are discussed and highlight the influence of the initial condition on the optimal domains.