Observability Inequality, Log-Type Hausdorff Content and Heat Equations

Observability on lattice points for heat equations and applications

This article concerns some quantitative versions of unique continuation known as observability inequalities. One of them is a lower bound on the spectral projectors of the Dirichlet Laplacian which generalizes the unique continuation of an eigenfunction from any open set Omega. Another one is equivalent to the interior null-controllability in time T of the heat equation with Dirichlet condition (the input function is a source in (0,T) x Omega). On a compact Riemannian manifolds, these inequalities are known to hold for arbitrary T and Omega. This article states and links these observability inequalities on a complete non-compact Riemannian manifold, and tackles the quite open problem of finding which Omega and T ensure their validity. It proves that it is sufficient for Omega to be the exterior of a compact set (for arbitrary T), but also illustrates that this is not necessary. It provides a necessary condition saying that there is no sequence of balls going infinitely far away from Omega without shrinking in a generalized sense (depending on T) which also applies when the distance to Omega is bounded.

Abstract. This paper is mainly concerned with the observability inequalities for heat equations with time-dependent Lipschtiz potentials. The observability inequality for heat equations asserts that the total energy of a solution is bounded above by the energy localized in a subdomain with an observability constant. For a bounded measurable potential V = V (x,t), the factor in the observability constant arising from the Carleman estimate is best known to be exp(C∥V ∥2∞/3) (even for time-independent potentials). In this paper, we show that, for Lipschtiz potentials, this factor can be replaced by exp(C(∥∇V ∥1∞/2 + ∥∂tV ∥1∞/3)), which improves the previous bound exp(C∥V ∥2∞/3) in some typical scenarios. As a consequence, with such a Lipschitz potential, we obtain a quantitative regular control in a null controllability problem. In addition, for the one-dimensional heat equation with some time-independent bounded measurable potential V = V (x), we obtain the observability inequality with optimal constant on arbitrary measurable subsets of positive measure both in space and time.

This article presents two observability inequalities for the heat equation over Ω × (0, T). In the first one, the observation is from a subset of positive measure in Ω × (0, T), while in the second, the observation is from a subset of positive surface measure on ∂Ω × (0, T). We will provide some applications for the above-mentioned observability inequalities, the bang-bang property for the minimal time control problems and the bang-bang property for the minimal norm control problems, and also establish new open problems related to observability inequalities and the aforementioned applications.

This paper presents two observability inequalities for the heat equation over \Omega\times (0,T) . In the first one, the observation is from a subset of positive measure in \Omega\times (0,T) , while in the second, the observation is from a subset of positive surface measure on \partial\Omega\times (0,T) . It also proves the Lebeau-Robbiano spectral inequality when \Omega is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

This article presents two observability inequalities for the heat equation over Ω× (0,T). In the first one, the observation is from a subset of positive measure in Ω× (0,T), while in the second, the observation is from a subset of positive surface measure on ∂Ω× (0,T). We will provide some applications for the above-mentioned observability inequalities, the bang-bang property for the minimal time, time optimal and minimal norm control problems, and also establish new open problems related to observability inequalities and the aforementioned applications.

Observable set, observability, interpolation inequality and spectral inequality for the heat equation in [formula omitted

We study the controllability of the pair (deformation/velocity of deformation) for a viscoelastic body. The idea is that existing controllability results for the memoryless wave equation can be lifted to the system with memory. As we have seen, the component \( w \) can also be interpreted as the temperature of thermodynamic systems with memory, so that we get exact controllability of the temperature (at a suitable time \( T \)), a property that cannot hold for the standard heat equation, derived from Fourier law. In this chapter, we show the use of moment methods in the study of controllability of viscoelastic materials, or thermodynamical systems with memory (a different proof based on the observation inequality is in Chap. 6). The final section shows an application of controllability to a source identification problem.

The quadratic control problem is considered for the semilinear heat equation with Lipschitz nonlinearity and associated cost function.It is proved that the value function is locally Lipschitz using by the observability inequality and then can be characterized as the unique positive viscosity solution of the corresponding Hamilton-Jacobi equation.Consequently the optimal feedback law is obtained.

For the heat equation on a bounded subdomain $Ω$ of $\mathbb{R}^d$, we investigate the optimal shape and location of the observation domain in observability inequalites. A new decomposition of $L^2(\mathbb{R}^d)$ into heat packets allows us to remove the randomisation procedure and assumptions on the geometry of $Ω$ in previous works. The explicit nature of the heat packets gives new information about the observability constant in the inverse problem.

This paper has been conceived as an overview on the controllability properties of some relevant (linear and nonlinear) parabolic systems. Specifically, we deal with the null controllability and the exact controllability to the trajectories. We try to explain the role played by the observability inequalities in this context and the need of global Carleman estimates. We also recall the main ideas used to overcome the difficulties motivated by nonlinearities. First, we considered the classical heat equation with Dirichlet conditions and distributed controls. Then we analyze recent extensions to other linear and semilinear parabolic systems and/or boundary controls. Finally, we review the controllability properties for the Stokes and Navier–Stokes equations that are known to date. In this context, we have paid special attention to obtaining the necessary Carleman estimates. Some open questions are mentioned throughout the paper. We hope that this unified presentation will be useful for those researchers interested in the field.

Previous article Next article Time-Optimal Control of Solutions of Operational Differenital EquationsH. O. FattoriniH. O. Fattorinihttps://doi.org/10.1137/0302005PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Einar Hille and , Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957xii+808 MR0089373 0392.46001 Google Scholar[2] S. Bochner and , A. E. Taylor, Linear functionals on certain spaces of abstractly valued functions, Ann. of Math. (2), 39 (1938), 913–944 MR1503445 CrossrefGoogle Scholar[3] R. Bellman, , I. Glicksberg and , O. Gross, On the “bang-bang” control problem, Quart. Appl. Math., 14 (1956), 11–18 MR0078516 0073.11501 CrossrefGoogle Scholar[4] R. S. Phillips, Perturbation theory for semi-groups of linear operators, Trans. Amer. Math. Soc., 74 (1953), 199–221 MR0054167 CrossrefISIGoogle Scholar[5] Tosio Kato, On linear differential equations in Banach spaces, Comm. Pure Appl. Math., 9 (1956), 479–486 MR0086986 0070.34602 CrossrefISIGoogle Scholar[6] L. S. Pontryagin, , V. G. Boltyanskii, , R. V. Gamkrelidze and , E. F. Mishchenko, The mathematical theory of optimal processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962viii+360 MR0166037 0102.32001 Google Scholar[7] Ju. V. Egorov, Optimal control in a Banach space, Dokl. Akad. Nauk SSSR, 150 (1963), 241–244, translated in Soviet Mathematics, 4 (1963). 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Insensitizing controls for a forward stochastic heat equation

In this paper, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is locally null-controllable for any arbitrary small time. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we extend in a nonlinear setting the strategy introduced in [18] that consists in constructing odd controls for the linear heat equation. The proof relies on three main steps. First, we obtain from the classical L 2 parabolic Carleman estimate, conjugated with maximal regularity results, a weighted L p observability inequality for the nonhomogeneous heat equation. Secondly, we perform a duality argument, close to the well-known Hilbert Uniqueness Method in a reflexive Banach setting, to prove that the heat equation perturbed by a source term is null-controllable thanks to odd controls. Finally, the nonlinearity is handled with a Schauder fixed-point argument.

Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient