Carleman Inequality Research Articles

Hierarchical Stackelberg Control for a Two-Stroke Linear System in Population Dynamics

We study a unique hierarchical control problem for a two-stoke linear system, adjoint to an age and space dependent population dynamics problem. Using Stackelberg's method, we introduce two levels of control: a boundary control to achieve optimal flow regulation and a distributed control for null controllability. Our approach employs Carleman inequalities to address non-homogeneous Dirichlet boundary conditions, leading to new insights in controlling invasive species populations. These results highlight the applicability of hierarchical controls in ecological systems, providing a robust framework for future studies in control theory and population dynamics.

Inverse Problems for One-Dimensional Fluid-Solid Interaction Models

AbstractWe consider a one-dimensional fluid-solid interaction model governed by the Burgers equation with a time varying interface. We discuss the inverse problem of determining the shape of the interface from Dirichlet and Neumann data at one endpoint of the spatial interval. In particular, we establish unique results and some conditional stability estimates. For the proofs, we use and adapt some lateral estimates that, in turn, rely on appropriate Carleman and interpolation inequalities.

Hierarchical null controllability of a semilinear degenerate parabolic equation with a gradient term

ABSTRACT In this paper, we apply a hierarchical strategy to a semilinear weakly degenerate parabolic equation with non-linearity that depends on the solution of the system and the spatial derivative of the solution. We use the Stackelberg–Nash strategy with one leader aiming to drive the solution to zero and two followers intended to solve a Nash equilibrium corresponding to a bi-objective optimal control problem. Since the system is semilinear, the functionals are not convex in general. To overcome this difficulty, we first prove the existence and uniqueness of the Nash quasi-equilibrium, which is a weaker formulation of the Nash equilibrium. Next, with additional conditions, we establish the equivalence between the concepts of Nash quasi-equilibrium and Nash equilibrium. We establish a suitable Carleman inequality for the adjoint system and then an observability inequality. Based on this observability inequality, we prove the null controllability of the linearized system. Then, using Kakutani's fixed point Theorem, we demonstrate the null controllability of the main system.

Insensitizing controls for the micropolar fluids

In this paper, we investigate the existence of insensitizing controls for the micropolar fluids in a bounded domain with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. The study of insensitizing controls is essential to solve a stability problem, which means that we look for controls such that some functionals of the velocity fields (the so‐called sentinels) are insensitive to the small perturbations of initial data. The problem of insensitizing controls is transformed into a suitable controllability problem for a cascade system. Our proof relies on a new global Carleman inequality and the inverse mapping theorem.

Carleman inequalities and unique continuation for the polyharmonic operators

We obtain a complete characterization of Lp−Lq Carleman estimates with weight ev⋅x for the polyharmonic operators. Our result extends the Carleman inequalities for the Laplacian due to Kenig–Ruiz–Sogge. Consequently, we obtain new unique continuation properties of higher order Schrödinger equations relaxing the integrability assumption on the solution spaces.

Exact controllability to the trajectories of the one-phase Stefan problem

This paper deals with the boundary exact controllability to the trajectories of the one-phase Stefan problem in one spatial dimension. This is a free-boundary problem that models solidification and melting processes. We prove the local exact controllability to (smooth) trajectories. To this purpose, we first reformulate the problem as the local null controllability of a coupled PDE-ODE system with distributed controls. Then, a new Carleman inequality for the adjoint of the linearized PDE-ODE system, coupled on the boundary through nonlocal in space and memory terms, is presented. This leads to the null controllability of an appropriate linear system. Finally, the result is obtained via local inversion, by using Lyusternik-Graves' Theorem. As a byproduct of our approach, we find that some parabolic equations which contains memory terms located on the boundary are null-controllable.

Observability and control of parabolic equations on networks with loops

Network theory can be useful for studying complex systems such as those that arise, for example, in physical sciences, engineering, economics and sociology. In this paper, we prove the observability of parabolic equations on networks with loops. By using a novel Carleman inequality, we find that the observability of the entire network can be achieved under certain hypothesis about the position of the observation domain. The main difficulty we tackle, due to the existence of loops, is to avoid entering into a circular fallacy, notably in the construction of the auxiliary function for the Carleman inequality. The difficulty is overcome with a careful treatment of the boundary terms on the junctions. Finally, we use the observability to prove the null controllability of the network and to obtain the Lipschitz stability for an inverse problem consisting on retrieving a stationary potential in the parabolic equation from measurements on the observation domain.

An inverse problem for a hyperbolic system in a bounded domain

In this Note we consider a two-by-two hyperbolic system defined on a bounded domain. Using Carleman inequalities, we obtain a Lipschitz stability result for the four spatially varying coefficients with measurements of only one component, given two sets of initial conditions.

A Carleman inequality on product manifolds and applications to rigidity problems

Abstract In this article, we prove a Carleman inequality on a product manifold M × R M\times {\mathbb{R}} . As applications, we prove that (1) a periodic harmonic function on R 2 {{\mathbb{R}}}^{2} that decays faster than all exponential rate in one direction must be constant 0, (2) a periodic minimal hypersurface in R 3 {{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rate in one direction must be a hyperplane, and (3) a periodic translator in R 3 {{\mathbb{R}}}^{3} that has an end asymptotic to a hyperplane faster than all exponential rates in one direction must be a translating hyperplane.

Stackelberg–Nash Null Controllability for a Non Linear Coupled Degenerate Parabolic Equations

The main purpose of this paper is to apply the notion of hierarchical control to a coupled degenerate non linear parabolic equations. We use the Stackelberg–Nash strategy with one leader and two followers. The followers solve a Nash equilibrium corresponding to a bi-objective optimal control problem and the leader a null controllability problem. Since the considered problem is non linear, the associated cost is non-convex. We first prove the existence, uniqueness and the characterization of the Nash quasi-equilibrium, which is a weak formulation of the Nash equilibrium because the cost associated to the non linear problem is non-convex. Next, we show that under suitable conditions, the Nash quasi-equilibrium is equivalent to the Nash equilibrium. Finally using some Carleman inequalities that we established, and the Kakutani’s fixed point Theorem, we brough the states of our system to the rest at final time T.

Sharp, double inequalities bounding the function (1+x)^(1/x) and a refinement of Carleman's inequality

Sharp, double inequalities bounding the function (1+x)^(1/x) and a refinement of Carleman's inequality

Carleman inequality for a class of super strong degenerate parabolic operators and applications

In this paper, we present a new Carleman estimate for the adjoint equations associated to a class of super strong degenerate parabolic linear problems. Our approach considers a standard geometric imposition on the control domain, which can not be removed in general. Additionally, we also apply the aforementioned main inequality in order to investigate the null controllability of two nonlinear parabolic systems. The first application is concerned a global null controllability result obtained for some semilinear equations, relying on a fixed point argument. In the second one, a local null controllability for some equations with nonlocal terms is also achieved, by using an inverse function theorem.

A New Carleman Inequality for a Linear Schrödinger Equation on Some Unbounded Domains

A New Carleman Inequality for a Linear Schrödinger Equation on Some Unbounded Domains

Stabilization of the damped plate equation under general boundary conditions

We consider a damped plate equation on an open bounded subset of ℝ d , or a smooth manifold, with boundary, along with general boundary operators fulfilling the Lopatinskiĭ-Šapiro condition. The damping term acts on an internal region without imposing any geometrical condition. We derive a resolvent estimate for the generator of the damped plate semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first through microlocal estimates, then local estimates, and finally a global estimate.

Bilinear local controllability to the trajectories of the Fokker–Planck equation with a localized control

This work is devoted to the control of the Fokker-Planck equation, posed on a bounded domain of R^d (d>0). More precisely, the control is the drift force, localized on a small open subset. We prove that this system is locally controllable to regular nonzero trajectories. Moreover, under some conditions on the reference control, we explain how to reduce the number of controls around the reference control. The results are obtained thanks to a linearization method based on a standard inverse mapping procedure and the fictitious control method. The main novelties of the present article are twofold. Firstly, we propose an alternative strategy to the standard fictitious control method: the algebraic solvability is performed and used directly on the adjoint problem. Secondly, we prove a new Carleman inequality for the heat equation with one order space-varying coefficients: the right-hand side is the gradient of the solution localized on a subset (rather than the solution itself), and the left-hand side can contain arbitrary high derivatives of the solution.

An inverse problem for Moore–Gibson–Thompson equation arising in high intensity ultrasound

Abstract In this article, we study the inverse problem of recovering a space-dependent coefficient of the Moore–Gibson–Thompson (MGT) equation from knowledge of the trace of the solution on some open subset of the boundary. We obtain the Lipschitz stability for this inverse problem, and we design a convergent algorithm for the reconstruction of the unknown coefficient. The techniques used are based on Carleman inequalities for wave equations and properties of the MGT equation.

Weighted Carleman inequality for fractional gradient

Weighted Carleman inequality for fractional gradient

Retracted] Local Null‐Controllability for Some Quasi‐Linear Phase‐Field Systems with Neumann Boundary Conditions by one Control Force

In this article, we study the local null‐controllability for some quasi‐linear phase‐field systems with homogeneous Neumann boundary conditions and an arbitrary located internal controller in the frame of classical solutions. In order to minimize the number of control forces, we prove the Carleman inequality for the associated linear system. By constructing a sequence of optimal control problems and an iteration method based on the parabolic regularity, we find a qualified control in Hölder space for the linear system. Based on the theory of Kakutani’s fixed point theorem, we prove that the quasi‐linear system is local null‐controllable when the initial datum is small and smooth enough.

Improved Quantitative Regularity for the Navier–Stokes Equations in a Scale of Critical Spaces

We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty L_x^q}<\infty$ where $r=\sqrt{x_1^2+x_2^2}$ and either $q\in(3,\infty)$, or $u$ is axisymmetric and $q\in(2,3]$. Using the strategy of Tao (2019), we obtain improved subcritical estimates for such solutions depending only on the double exponential of the critical norm. One consequence is a double logarithmic lower bound on the blowup rate. We make use of some tools such as a decomposition of the solution that allows us to use energy methods in these spaces, as well as a Carleman inequality for the heat equation suited for proving quantitative backward uniqueness in cylindrical regions.

Quantitative Regularity for the Navier\u2013Stokes Equations Via Spatial Concentration

This paper is concerned with quantitative estimates for the Navier–Stokes equations. First we investigate the relation of quantitative bounds to the behavior of critical norms near a potential singularity with Type I bound Vert uVert _{L^{infty }_{t}L^{3,infty }_{x}}le M. Namely, we show that if T^* is a first blow-up time and (0,T^*) is a singular point then ‖u(·,t)‖L3(B0(R))≥C(M)log(1T∗-t),R=O((T∗-t)12-).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Vert u(\\cdot ,t)\\Vert _{L^{3}(B_{0}(R))}\\ge C(M)\\log \\Big (\\frac{1}{T^*-t}\\Big ),\\,\\,\\,\\,\\,\\,R=O((T^*-t)^{\\frac{1}{2}-}). \\end{aligned}$$\\end{document}We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin (Commun Math Phys 312(3):833–845, 2012), which says that if u is a smooth finite-energy solution to the Navier–Stokes equations on {mathbb {R}}^3times (0,1) with supn‖u(·,t(n))‖L3(R3)<∞andt(n)↑1,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sup _{n}\\Vert u(\\cdot ,t_{(n)})\\Vert _{L^{3}({\\mathbb {R}}^3)}<\\infty \\,\\,\\,\\text {and}\\,\\,\\,t_{(n)}\\uparrow 1, \\end{aligned}$$\\end{document}then u does not blow-up at t=1. To prove our results we develop a new strategy for proving quantitative bounds for the Navier–Stokes equations. This hinges on local-in-space smoothing results (near the initial time) established by Jia and Šverák (2014), together with quantitative arguments using Carleman inequalities given by Tao (2019). Moreover, the technology developed here enables us in particular to give a quantitative bound for the number of singular points in a Type I blow-up scenario.