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Mahamadi Warma

Publications and source records attributed to Mahamadi Warma.

At least 19 recordsLinked to original sources
arXiv2026

Reconstruction algorithms for the fractional Laplacian and applications to inverse problems

We introduce two reconstruction schemes that enable the recovery of a function in the entire Euclidean space $\mathbb{R}^n$ from local data $(u|_W, [(-\Delta)^s u]|_W)$, where $W$ is an arbitrarily small nonempty open subset of $\mathbb R^n$ and $(-\Delta)^s$ denotes the fractional Laplace operator of order $s\in (0,1)$. These procedures rely crucially on the weak Unique Continuation Property (UCP) for the fractional Laplacian. We apply these schemes to two distinct inverse problems. Following the seminal work from Ghosh et al., the first one concerns the recovery of a potential (Calder\'on-type problem) from the fractional Schr\"odinger equation under nonlocal Robin-type exterior conditions. The second one involves recovering the solution of the space-fractional heat equation in $\mathbb{R}^n$ from localized time-dependent measurements within a ball. To tackle these problems, we introduce new analytical tools such as a generalized weak Kelvin transform and a fractional Robin-to-Robin map. Finally, we provide numerical simulations for one of the reconstruction methods, illustrating the stability issues and the severe ill-posedness inherent to such inverse problems.

math.AP↗
arXiv2026

A unified view of nonlinear, nonlocal operators and qualitative properties of associated elliptic and parabolic problems

We put together a general framework to deal with elliptic and parabolic equations associated with (nonlinear) nonlocal (fractional order) operators. Many well-known nonlocal operators enter into our framework, and in addition one may introduce many other, new nonlocal operators that have not yet been considered in the literature. We use the abstract theory of (nonlinear) semigroups generated by subgradients of proper, lower semicontinuous and convex functionals on Hilbert spaces to build a rigorous and applicable framework that works for many classical elliptic operators but also nonlocal or sometimes fractional operators. After recalling the notion of a nonlinear semigroup generated by subgradients and $j$-subgradients of the associated energy functions, we introduce a general class of (nonlinear) nonlocal elliptic type operators and define rigorously subgradients and $j$-subgradients of such functionals that generate (nonlinear) submarkovian semigroups and hence, the abstract Cauchy problem associated with these subgradients and/or $j$-subgradients is wellposed. The existence and the qualitative properties of solutions to these Cauchy problems and the corresponding semigroups are investigated. More precisely, we show some comparison and maximum principles, submarkovian, domination, ultracontractivity properties, and some H\"older type estimates for these semigroups of operators. These results are usually useful in several branches of pure and applied partial differential equations. We finish the paper by giving several examples of nonlocal operators in Euclidean spaces, graphs, metric random walk spaces, fractional Brownian motions, and L\'evy flights, that fit in our general framework.

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arXiv2025

Superdiffusive fractional dynamics: Unveiling regularity results in systems with general positive self-adjoint operators

We investigate the following fractional order in time Cauchy problem \begin{equation*} \begin{cases} \mathbb{D}_{t}^{α}u(t)+Au(t)=f(u(t)), & 1<α<2, \\ u(0)=u_{0},\,\,\,u^{\prime }(0)=u_{1}. & \end{cases}% \end{equation*}% where $\mathbb{D}_{t}^{α}u(\cdot )$ is the Caputo time-fractional derivative of order $α\in (1, 2)$ of the function $u$. Such problems are increasingly used in concrete models in applied sciences, notably phenomena with memory effects. We obtain results on existence and regularity of weak and strong energy solutions assuming that $A$ is any positive self-adjoint operator in a Hilbert space, when the nonlinearity $f\in C^{1}({\mathbb{R}}) $ satisfies suitable growth conditions. Our aim is to obtain regularity results {without} assuming that the operator $A$ has compact resolvent readily extending our recent results from our previous paper \cite{AGKW}. Examples of operators $A$ are considered, mainly differential operators such as Schrödinger operators, as well as various nonlocal operators.

math.AP↗
arXiv2025

Boundary observation and control for fractional heat and wave equations

We establish boundary null controllability of the heat equation driven by the integral fractional Laplacian $(-\Delta)^s$ on a bounded smooth domain for every $T>0$ and every fractional exponent $s\in(1/2,1)$. The control acts through the singular boundary trace naturally associated with the fractional Dirichlet problem. The main ingredient is a frequency-dependent boundary observability inequality for the associated fractional wave equation, obtained by combining multiplier arguments with the fractional Pohozaev identity. In contrast with the classical wave equation, the observability time deteriorates with the spectral cutoff, reflecting the slow propagation of high-frequency fractional waves. We transfer this estimate to the parabolic problem by transmutation and combine the resulting low-frequency controllability estimates with the high-frequency dissipation of the fractional heat semigroup through a Lebeau-Robbiano iteration. The balance between these two mechanisms yields null controllability precisely in the range $s>1/2$.

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arXiv2024

Bilinear optimal control for a fractional diffusive equation

We consider a bilinear optimal control for an evolution equation involving the fractional Laplace operator of order $0<s<1$. We first give some existence and uniqueness results for the considered evolution equation. Next, we establish some weak maximum principle results allowing us to obtain more regularity of our state equation. Then, we consider an optimal control problem which consists to bring the state of the system at final time to a desired state. We show that this optimal control problem has a solution and we derive the first and second order optimality conditions. Finally, under additional assumptions on the initial datum and the given target, we prove that local uniqueness of optimal solutions can be achieved.

math.OC↗
arXiv2024

Exterior Nonlocal Variational Inequalities

This paper introduces a new class of variational inequalities where the obstacle is placed in the exterior domain that is disjoint from the observation domain. This is carried out with the help of nonlocal fractional operators. The need for such novel variational inequalities stems from the fact that the classical approach only allows placing the obstacle either inside the observation domain or on the boundary. A complete analysis of the continuous problem is provided. Additionally, perturbation arguments to approximate the problem are discussed.

math.AP↗
arXiv2023

Optimal control of a class of semilinear fractional elliptic equations

In this paper, a class of semilinear fractional elliptic equations associated to the spectral fractional Dirichlet Laplace operator is considered. We establish the existence of optimal solutions as well as a minimum principle of Pontryagin type and the first order necessary optimality conditions of associated optimal control problems. Second order conditions for optimality are also obtained for $L^{\infty}$ and $L^2-$ local solutions under some structural assumptions.

math.OC↗
arXiv2023

Memory approximate controllability properties for higher order Hilfer time fractional evolution equations

In this paper we study the approximate controllability of fractional partial differential equations associated with the so-called Hilfer type time fractional derivative and a non-negative selfadjoint operator $A$ with a compact resolvent on $L^2(Ω)$, where $Ω\subset\RR^N$ ($N\geq 1$) is an open set. More precisely, we show that if $0\leν\le 1$, $1<μ\le 2$ and $Ω\subset\RR^N$ is an open set, then the system \begin{equation*} \begin{cases} \D^{μ,ν}_tu+Au=fχ_ω\;\;&\mbox{ in }\;Ω\times(0,T),\\ (I_t^{(1-ν)(2-μ)}u)(\cdot,0)=u_0 &\mbox{ in }\;Ω,\\ (\partial_tI_t^{(1-ν)(2-μ)}u)(\cdot,0)=u_1 &\mbox{ in }\;Ω, \end{cases} \end{equation*} is memory approximately controllable for any $T>0$, $u_0\in D(A^{1/μ})$, $u_1\in L^2(Ω)$ and any non-empty open set $ω\subsetΩ$. The same result holds for every $u_0\in D(A^{1/2})$ and $u_1\in L^2(Ω)$.

math.AP↗
arXiv2022

Introducing and solving generalized Black-Scholes PDEs through the use of functional calculus

We introduce some families of generalized Black--Scholes equations which involve the Riemann-Liouville and Weyl space-fractional derivatives. We prove that these generalized Black--Scholes equations are well-posed in $(L^1-L^\infty)$-interpolation spaces. More precisely, we show that the elliptic type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial operators and sectorial operators in an abstract setting. Such a connection extends some known results in the topic to a wider family of both operators and the functions involved.

math.AP↗
arXiv2022

Optimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data

We consider parabolic equations on bounded smooth open sets $\Om\subset \R^N$ ($N\ge 1$) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator $\mathscr{L} \coloneqq - Δ+ (-Δ)^{s}$ ($0<s<1$). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic problems with smooth, and then with singular boundary-exterior data. Secondly, we show the existence of optimal solutions of associated optimal control problems, and we characterize the optimality conditions. This is the first time that such topics have been presented and studied in a unified fashion for mixed local-nonlocal PDEs with singular data.

math.AP↗
arXiv2021

Control and numerical approximation of fractional diffusion equations

The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain $Ω$. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the $L^2$ and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset $ω\subsetΩ$. Secondly, we move our attention to the exterior control problem, in which the control region $\mathcal O\subsetΩ^c$ is located outside $Ω$. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on $\partialΩ$. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation.

math.AP↗
arXiv2021

A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs

We consider optimal control of fractional in time (subdiffusive, i.e., for $% 0<γ<1$) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we{~\textsf{first show}} the existence and regularity of solutions to the forward and the associated {\textsf{backward (adjoint)}} problems. In the second part, we prove existence of optimal {\textsf{controls% }} and characterize the associated {\textsf{first order}} optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the\ semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces.

math.OC↗
arXiv2021

Optimal Control, Numerics, and Applications of Fractional PDEs

This article provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems.

math.OC↗
arXiv2020

Exponential Turnpike property for fractional parabolic equations with non-zero exterior data

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Control problems play a key role in engineering, economics and sciences. To be more precise, in climate sciences, often times, relevant problems are formulated in long time scales, so that, the problem of possible asymptotic behaviors when the time-horizon goes to infinity becomes natural. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question arises: Do time averages of optimal controls and trajectories converge to the stationary optimal controls and states as the time-horizon goes to infinity? This question is very closely related to the so-called turnpike property that shows that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In the present paper we deal with heat equations with non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the fractional Laplace operator $(-Δ)^s$ ($0<s<1$). We prove the turnpike property for the nonlocal Robin optimal control problem and the exponential turnpike property for both Dirichlet and nonlocal Robin optimal control problems.

math.AP↗
arXiv2020

Moreau-Yosida regularization for optimal control of fractional PDEs with state constraints: parabolic case

This paper considers optimal control of fractional parabolic PDEs with both state and control constraints. The key challenge is how to handle the state constraints. Similarly, to the elliptic case, in this paper, we establish several new mathematical tools in the parabolic setting that are of wider interest. For example, existence of solution to the fractional parabolic equation with measure data on the right-hand-side. We employ the Moreau-Yosida regularization to handle the state constraints. We establish convergence, with rate, of the regularized optimal control problem to the original one. Numerical experiments confirm what we have proven theoretically.

math.AP↗
arXiv2020

Approximate and mean approximate controllability properties for Hilfer time-fractional differential equations

We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator $A_B$ with a compact resolvent on $L^2(Ω)$, where $Ω\subset\mathbb{R}^N$ ($N\ge 1$) is a bounded open set. More precisely, we show that if $0\leν\le 1$, $0<μ\le 1$ and $Ω\subset\mathbb R^N$ is a bounded open set, then the system $$\mathbb D_t^{μ,ν} u+A_Bu=f|_ω\;\; \mbox{ in }\; Ω\times (0,T),\,\, (\mathbb I_t^{(1-ν)(1-μ)}u)(\cdot,0)=u_0 \mbox{ in }\;Ω,$$ is approximately controllable for any $T>0$, $u_0\in L^2(Ω)$ and any non-empty open set $ω\subsetΩ$. In addition, if the operator $A_B$ has the unique continuation property, then the system is also mean approximately controllable. The operator $A_B$ can be the realization in $L^2(Ω)$ of a symmetric, non-negative uniformly elliptic second order operator with Dirichlet or Robin boundary conditions, or the realization in $L^2(Ω)$ of the fractional Laplace operator $(-Δ)^s$ ($0<s<1$) with the Dirichlet exterior condition, $u=0$ in $\mathbb R^N\setminusΩ$, or the nonlocal Robin exterior condition, $\mathcal N^su+βu=0$ in $\mathbb R^N\setminus\overlineΩ$.

math.AP↗
arXiv2020

Realization of the fractional Laplacian with nonlocal exterior conditions via forms method

Let $Ω\subset\RR^n$ ($n\ge 1$) be a bounded open set with a Lipschitz continuous boundary. In the first part of the paper, using the method of bilinear forms, we give a rigorous characterization of the realization in $L^2(Ω)$ of the fractional Laplace operator $(-Δ)^s$ ($0<s<1$) with the nonlocal Neumann and Robin exterior conditions. Contrarily to the classical local case $s=1$, it turns out that the nonlocal (Robin and Neumann) exterior conditions are incorporated in the form domain. We show that each of the above operators generates a strongly continuous submarkovian semigroup which is also ultracontractive. In the second part, we show that the semigroup corresponding to the nonlocal Robin exterior condition is always sandwiched between the fractional Dirichlet semigroup and the fractional Neumann semigroup.

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