arXiv subjects

S. E. Chorfi

Publications and source records attributed to S. E. Chorfi.

At least 19 recordsLinked to original sources

arXiv2026

The Biharmonic Heat Equation with General Dynamic Boundary Conditions

In this work, we initiate the study of the biharmonic heat equation in a spatial bounded domain subject to dynamic boundary conditions involving the bi-Laplace-Beltrami operator on the boundary. The boundary heat equation is coupled to the interior one via a normal derivative term. By combining the sesquilinear form method and semigroup theory, we establish substantial qualitative properties of the fourth-order parabolic equation; in particular, the self-adjointness of the associated operator, compactness of its resolvent, and further spectral properties. We also investigate the generation of a $C_0$-semigroup and analyze its main properties: analyticity, compactness, eventual positivity, and eventual $L^\infty$-contractivity.

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arXiv2026

Maximal regularity for time-fractional Schrödinger equations and application to nonlinear equations

We study the maximal regularity problem for abstract time-fractional Schrödinger equations $\partial_t^α(u-u_0) -\mathrm{i} A u=f$, with a fractional derivative $\partial_t^α$ of order $α\in (0,1)$. We assume that $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. First, we prove the maximal $L^2$-regularity by leveraging properties of Mittag-Leffler functions with an imaginary argument. Compared to existing results for the subdiffusion equations, our proof avoids using the complete monotonicity of Mittag-Leffler functions, which seems difficult to prove within the setting of an imaginary argument. Then, we prove the maximal $L^p$-regularity for $p\in (1,\infty)$ using the operator-valued version of Mikhlin's multiplier theorem. Finally, we apply the maximal regularity results to prove the local well-posedness of quasilinear and semilinear time-fractional Schrödinger equations.

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arXiv2026

Backward problem for a degenerate viscous Hamilton-Jacobi equation: stability and numerical identification

This work is devoted to the analysis of the backward problem for a viscous Hamilton-Jacobi equation with degenerate diffusion and a general Hamiltonian that is not necessarily quadratic. First, we focus on linear degenerate parabolic equations in the nondivergence setting. We prove the conditional stability of the backward problem using Carleman estimates. Then, by a linearization technique, we prove similar results for the nonlinear viscous Hamilton-Jacobi equation. Regarding numerical identification, we first investigate the linear degenerate equation with noisy data using the adjoint state method, combined with a Conjugate Gradient algorithm, to solve the associated minimization problem. Finally, the numerical identification for the nonlinear viscous Hamilton-Jacobi equation is investigated by the Van Cittert iteration. Numerical tests are presented to show the performance of the proposed algorithms.

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arXiv2026

Lipschitz stability for an inverse source problem of the wave equation with kinetic boundary conditions

In this paper, we present a refined approach to establish a global Lipschitz stability for an inverse source problem concerning the determination of forcing terms in the wave equation with mixed boundary conditions. It consists of boundary conditions incorporating a dynamic boundary condition and Dirichlet boundary condition on disjoint subsets of the boundary. The primary contribution of this article is the rigorous derivation of a sharp Carleman estimate for the wave system with a dynamic boundary condition. In particular, our findings complete and drastically improve the earlier results established by Gal and Tebou [SIAM J. Control Optim., 55 (2017), 324-364]. This is achieved by using a different weight function to overcome some relevant difficulties. As for the stability proof, we extend to dynamic boundary conditions a recent argument avoiding cut-off functions. Finally, we also show that our developed Carleman estimate yields a sharp boundary controllability result.

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arXiv2025

Inverse problems for time-fractional Schrödinger equations

We study some inverse problems for time-fractional Schrödinger equations involving the Caputo derivative of fractional order $α\in (0,1)$. We prove refined uniqueness results from sets of positive Lebesgue measure for various problems by weakening the regularity of initial data.

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arXiv2025

Forward and backward problems for abstract time-fractional Schrödinger equations

We investigate forward and backward problems associated with abstract time-fractional Schrödinger equations $\mathrm{i}^ν\partial_t^αu(t) + A u(t)=0$, $α\in (0,1)\cup (1,2)$ and $ν\in\{1,α\}$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. This kind of equation, which incorporates the Caputo time-fractional derivative of order $α$, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the forward problems in two scenarios: ($ν=1,\,$ $α\in (0,1)$) and ($ν=α,\,$ $α\in (0,1)\cup (1,2)$). Then, we prove well-posedness and stability results for the backward problems depending on the two cases $ν=1$ and $ν=α$. Our approach employs the solution's eigenvector expansion along with the properties of the Mittag-Leffler functions, including the distribution of zeros and asymptotic expansions. Finally, we conclude with a discussion of some open problems.

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arXiv2025

Logarithmic convexity of evolution equations and application to inverse problems

We review some results on the logarithmic convexity for evolution equations, a well-known method in inverse and ill-posed problems. We start with the classical case of self-adjoint operators. Then, we analyze the case of analytic semigroups. In this general case, we give an explicit estimate, which will be used to study inverse problems for initial data recovery. We illustrate our abstract result by an application to the Ornstein-Uhlenbeck equations. We discuss both analytic and non-analytic semigroups. We conclude with recent results for the time-fractional evolution equations with the Caputo derivative of order $0<α<1$. We start with symmetric evolution equations. Then, we show that the results extend to the non-symmetric case for diffusion equations, provided that a gradient vector field generates the drift coefficient. Finally, some open problems will be mentioned.

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arXiv2025

Controllability and Inverse Problems for Hyperbolic and Dispersive Equations with Dynamic Boundary Conditions

This review examines classical and recent results on controllability and inverse problems for hyperbolic and dispersive equations with dynamic boundary conditions. We aim to illustrate the applicability of Carleman estimates to establish exact controllability of such equations and derive Lipschitz stability estimates for inverse problems of source terms and coefficients with general dynamic boundary conditions. We highlight the challenges associated with dynamic boundary conditions compared to classical static ones. Finally, we conclude with a discussion of open problems and future research directions.

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arXiv2024

Uniqueness for the Schrödinger equation with an inverse square potential and application to controllability and inverse problems

In this paper, we prove a sharp uniqueness result for the singular Schrödinger equation with an inverse square potential. This will be done without assuming geometrical restrictions on the observation region. The proof relies on a recent technique transforming the Schrödinger equation into an elliptic equation. We show that this technique is still applicable for singular equations. In our case, substantial difficulties arise when dealing with singular potentials of cylindrical type. Using the uniqueness result, we show the approximate controllability of the equation using a distributed control. The uniqueness result is also applied to prove the uniqueness for an inverse source problem.

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arXiv2024

Stability of backward inverse problems for degenerate mean-field game systems

We investigate inverse backward-in-time problems for a class of second-order degenerate Mean-Field Game (MFG) systems. More precisely, given the final datum $(u(\cdot, T),m(\cdot, T))$ of a solution to the one-dimensional mean-field game system with a degenerate diffusion coefficient, we aim to determine the intermediate states $(u(\cdot,t_{0}),m(\cdot,t_{0}))$ for any $t_{0} \in [0, T)$, i.e., the value function and the mean distribution at intermediate times, respectively. We prove conditional stability estimates under suitable assumptions on the diffusion coefficient and the initial state $(u(\cdot,0),m(\cdot,0))$. The proofs are based on Carleman's estimates with a simple weight function. We first prove a Carleman estimate for the Hamilton-Jacobi-Bellman (HJB) equation. A second Carleman estimate will be derived for the Fokker-Planck (FP) equation. Then, by combining the two estimates, we obtain a Carleman estimate for the mean-field game system, leading to the stability of the backward problems.

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arXiv2024

Controllability and Inverse Problems for Parabolic Systems with Dynamic Boundary Conditions

This review surveys previous and recent results on null controllability and inverse problems for parabolic systems with dynamic boundary conditions. We aim to demonstrate how classical methods such as Carleman estimates can be extended to prove null controllability for parabolic systems and Lipschitz stability estimates for inverse problems with dynamic boundary conditions of surface diffusion type. We mainly focus on the substantial difficulties compared to static boundary conditions. Finally, some conclusions and open problems will be mentioned.

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arXiv2024

Boundary null controllability of the heat equation with Wentzell boundary condition and Dirichlet control

We consider the linear heat equation with a Wentzell-type boundary condition and a Dirichlet control. Such a boundary condition can be reformulated as one of dynamic type. First, we formulate the boundary controllability problem of the system within the framework of boundary control systems, proving its well-posedness. Then we reduce the question to a moment problem. Using the spectral analysis of the associated Sturm-Liouville problem and the moment method, we establish the null controllability of the system at any positive time $T$. Finally, we approximate minimum energy controls by a penalized HUM approach. This allows us to validate the theoretical controllability results obtained by the moment method.

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arXiv2024

Stability of backward-in-time semilinear coupled parabolic systems

We consider backward problems for semilinear coupled parabolic systems in bounded domains. We prove conditional stability estimates for linear and semilinear systems of strongly coupled parabolic equations involving general semilinearities. The proof of the stability estimates relies on a modified method by Carleman estimates incorporating the simple weight function $e^{λt}$ with a sufficiently large parameter $λ$.

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arXiv2024

Logarithmic convexity of non-symmetric time-fractional diffusion equations

We consider a class of diffusion equations with the Caputo time-fractional derivative $\partial_t^αu=L u$ subject to the homogeneous Dirichlet boundary conditions. Here, we consider a fractional order $0<α< 1$ and a second-order operator $L$ which is elliptic and non-symmetric. In this paper, we show that the logarithmic convexity extends to this non-symmetric case provided that the drift coefficient is given by a gradient vector field. Next, we perform some numerical experiments to validate the theoretical results in both symmetric and non-symmetric cases. Finally, some conclusions and open problems will be mentioned.

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arXiv2023

Numerical impulse controllability for parabolic equations by a penalized HUM approach

This work presents a comparative study to numerically compute impulse approximate controls for parabolic equations with various boundary conditions. Theoretical controllability results have been recently investigated using a logarithmic convexity estimate at a single time based on a Carleman commutator approach. We propose a numerical algorithm for computing the impulse controls with minimal $L^2$-norms by adapting a penalized Hilbert Uniqueness Method (HUM) combined with a Conjugate Gradient (CG) method. We consider static boundary conditions (Dirichlet and Neumann) and dynamic boundary conditions. Some numerical experiments based on our developed algorithm are given to validate and compare the theoretical impulse controllability results.

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arXiv2023

Stability estimates for initial data in general Ornstein-Uhlenbeck equations

We consider the inverse problem of determining initial data in general Ornstein-Uhlenbeck equations on the Euclidean space from partial measurement localized on the so-called thick sets. Using the logarithmic convexity technique and recent observability results, we prove new stability estimates of logarithmic rate for large classes of initial data. Such stability estimates are crucial when dealing with the numerical reconstruction of initial data. Our analysis covers both cases: the analytic Ornstein-Uhlenbeck semigroup on $L^2\left(\mathbb R^N, \mathrm{d}μ\right)$ with invariant measure $μ$, and the non-analytic Ornstein-Uhlenbeck semigroup on $L^2\left(\mathbb R^N, \mathrm{d} x\right)$ with the Lebesgue measure. We treat general equations with a diffusion matrix and the Ornstein-Uhlenbeck equation with fractional diffusion in the latter case. This allows us to extend some recent results and simplify some parts of the proof.

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arXiv2023

Logarithmic stability estimates for initial data in Ornstein-Uhlenbeck equation on $L^2$-space

In this paper, we continue the investigation on the connection between observability and inverse problems for a class of parabolic equations with unbounded first order coefficients. We prove new logarithmic stability estimates for a class of initial data in the Ornstein-Uhlenbeck equation posed on $L^2\left(\mathbb{R}^N\right)$ with respect to the Lebesgue measure. The proofs combine observability and logarithmic convexity results that include a non-analytic semigroup case. This completes the picture of the recent results obtained for the analytic Ornstein-Uhlenbeck semigroup on $L^2$-space with invariant measure.

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arXiv2022

The backward problem for time fractional evolution equations

In this paper, we consider the backward problem for fractional in time evolution equations $\partial_t^αu(t)= A u(t)$ with the Caputo derivative of order $0<α\le 1$, where $A$ is a self-adjoint and bounded above operator on a Hilbert space $H$. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag-Leffler functions. Then we prove conditional stability estimates of Hölder type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results.

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