arXiv subjects
Publications and source records attributed to Mahamadi Warma.
We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval $(0,1)$ associated with the fractional Laplace operator $(-\partial_x^2)^s$, where $0 0$ if and only if $\frac 12<s<1$.
We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator $ (-\partial_x^2)^s$ (with $0<s<1$) on the interval $(-1,1)$. Our control function is localized in an open set $\mathcal O$ in the exterior of $(-1,1)$, that is, $\mathcal O\subset (\mathbb{R} \setminus (-1,1))$. We show that there exists a minimal (strictly positive) time $T_{\rm min}$ such that the fractional heat dynamics can be controlled from any initial datum in $L^2(-1,1)$ to a positive trajectory through the action of an exterior positive control, if and only if $\frac 12<s<1$. In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results.
Let $(-Δ)_c^s$ be the realization of the fractional Laplace operator on the space of continuous functions $C_0(\mathbb{R})$, and let $(-Δ_h)^s$ denote the discrete fractional Laplacian on $C_0(\mathbb{Z}_h)$, where $0 0$. We show that solutions of fractional order semi-linear Cauchy problems associated with the discrete operator $(-Δ_h)^s$ on $C_0(\mathbb{Z}_h)$ converge to solutions of the corresponding Cauchy problems associated with the continuous operator $(-Δ)_c^s$. In addition, we obtain that the convergence is uniform in $t$ in compact subsets of $[0,\infty)$. We also provide numerical simulations that support our theoretical results.
In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian $(-Δ)^s$ ($0 1/2$. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical results.
This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic PDEs of order $s \in (0,1)$. There are several mathematical tools that are developed during the process to study this problem, for instance, the characterization of the dual of the fractional order Sobolev spaces and well-posedness of fractional PDEs with measure-valued datum. These tools are widely applicable. We show well-posedness of the optimal control problem and derive the first order optimality conditions. Notice that the adjoint equation is a fractional PDE with measure as the right-hand-side datum. We use the characterization of the fractional order dual spaces to study the regularity of the state and adjoint equations. We emphasize that the classical case ($s=1$) was considered by E. Casas in \cite{ECasas_1986a} but almost none of the existing results are applicable to our fractional case.
In this paper we introduce a new notion of optimal control, or source identification in inverse, problems with fractional parabolic PDEs as constraints. This new notion allows a source/control placement outside the domain where the PDE is fulfilled. We tackle the Dirichlet, the Neumann and the Robin cases. For the fractional elliptic PDEs this has been recently investigated by the authors in \cite{HAntil_RKhatri_MWarma_2018a}. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the source/control either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control in the exterior. We introduce the notions of weak and very-weak solutions to the parabolic Dirichlet problem. We present an approach on how to approximate the parabolic Dirichlet solutions by the parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.
Let $\Om\subset\RR^N$ a bounded domain with a Lipschitz continuous boundary. We study the controllability of the space-time fractional diffusion equation \begin{equation*} \begin{cases} \mathbb D_t^αu+(-Δ)^su=0\;\;&\mbox{ in }\;(0,T)\timesΩ\\ u=g &\mbox{ in }\;(0,T)\times(\RR^N\setminusΩ)\\ u(0,\cdot)=u_0&\mbox{ in }\;Ω, \end{cases} \end{equation*} where $u=u(t,x)$ is the state to be controlled and $g=g(t,x)$ is the control function which is localized in a subset $\mathcal O$ of $\Omc$. Here, $0<α\le 1$, $0 0$ be real numbers. After giving an explicit representation of solutions, we show that the system is always approximately controllable for every $T>0$, $u_0\in L^2(Ω)$ and $g\in \mathcal D((0,T)\times\mathcal O)$ where $\mathcal O\subset(\RR^N\setminus\bOm)$ is any open set. The results obtained are sharp in the sense that such a system is never null controllable if $0<α<1$. The proof of our result is based on a new unique continuation principle for the eigenvalues problem associated with the fractional Laplace operator subject to the zero exterior boundary condition that we have established.
Let $Ω\subset\R^N$ be an arbitrary open set and denote by $(e^{-t(-Δ)_{\RR^N}^s})_{t\ge 0}$ (where $0<s<1$) the semigroup on $L^2(\RR^N)$ generated by the fractional Laplace operator. In the first part of the paper we show that if $T$ is a self-adjoint semigroup on $L^2(Ω)$ satisfying a fractional Gaussian estimate in the sense that $|T(t)f|\le e^{-t(-Δ)_{\RR^N}^s}|f|$, $0\le t \le 1$, $f\in L^2(Ω)$, then $T$ defines a bounded holomorphic semigroup of angle $\fracπ{2}$ that interpolates on $L^p(Ω)$, $1\le p<\infty$. Using a duality argument we prove that the same result also holds on the space of continuous functions. In the second part, we apply the above results to realization of fractional order operators with the exterior Dirichlet conditions.
We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset $ω(t)$ which is moving with a constant velocity $c\in\mathbb{R}$, the main result of the paper states that the equation is null controllable in a sufficiently large time $T$ and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.
In this paper we consider the optimal control of semilinear fractional PDEs with both spectral and integral fractional diffusion operators of order $2s$ with $s \in (0,1)$. We first prove the boundedness of solutions to both semilinear fractional PDEs under minimal regularity assumptions on domain and data. We next introduce an optimal growth condition on the nonlinearity to show the Lipschitz continuity of the solution map for the semilinear elliptic equations with respect to the data. We further apply our ideas to show existence of solutions to optimal control problems with semilinear fractional equations as constraints. Under the standard assumptions on the nonlinearity (twice continuously differentiable) we derive the first and second order optimality conditions.
Very recently M. Warma has shown that for nonlocal PDEs associated with the fractional Laplacian, the classical notion of controllability from the boundary does not make sense and therefore it must be replaced by a control that is localized outside the open set where the PDE is solved. Having learned from the above mentioned result, in this paper we introduce a new class of source identification and optimal control problems where the source/control is located outside the observation domain where the PDE is satisfied. The classical diffusion models lack this flexibility as they assume that the source/control is located either inside or on the boundary. This is essentially due to the locality property of the underlying operators. We use the nonlocality of the fractional operator to create a framework that now allows placing a source/control outside the observation domain. We consider the Dirichlet, Robin and Neumann source identification or optimal control problems. These problems require dealing with the nonlocal normal derivative (that we shall call interaction operator). We create a functional analytic framework and show well-posedness and derive the first order optimality conditions for these problems. We introduce a new approach to approximate, with convergence rate, the Dirichlet problem with nonzero exterior condition. The numerical examples confirm our theoretical findings and illustrate the practicality of our approach.
We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. In the first part, we show that if $0 0$, then there is no control function $g$ such that the following system \begin{equation*} \begin{cases} u_{tt} + (-Δ)^{s} u + δ(-Δ)^{s} u_{t}=0 & \mbox{ in }\; Ω\times(0,T),\\ u=gχ_{\mathcal O\times (0,T)} &\mbox{ in }\; (\Omc)\times (0,T) ,\\ u(\cdot,0) = u_0, u_t(\cdot,0) = u_1 &\mbox{ in }\; Ω, \end{cases} \end{equation*} is exact or null controllable at time $T>0$. In the second part, we prove that for every $δ\ge 0$ and $0 0$ and $g\in \mathcal D(\mathcal O\times(0,T))$, where $\mathcal O\subset\Omc$ is any non-empty open set.
In this paper 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*}% The fractional in time derivative is taken in the classical Caputo sense. In the scientific literature such equations are sometimes dubbed as fractional-in time wave equations or super-diffusive equations. We obtain results on existence and regularity of local and global weak solutions assuming that $A$ is a nonnegative self-adjoint operator with compact resolvent in a Hilbert space and with a nonlinearity $f\in C^{1}({\mathbb{R}}% )$ that satisfies suitable growth conditions. Further theorems on the existence of strong solutions are also given in this general context.
We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-Δ)^s$ on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in the Besov space $B^{2s}_{p,2,\textrm{loc}}(Ω)$, while for $2\leq p<\infty$ we show that the solutions belong to $W^{2s,p}_{\textrm{loc}}(Ω)$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.
We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.
In this paper we study optimal control problems with either fractional or regional fractional $p$-Laplace equation, of order $s$ and $p\in [2,\infty)$, as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the involved operator. To overcome the degeneracy of both fractional $p$-Laplacians, we introduce a regularization for both operators. We show existence and uniqueness of solution to the regularized state equations and existence of solution to the regularized optimal control problems. We also prove several auxiliary results for the regularized problems which are of independent interest. We conclude with the convergence of the regularized solutions.
Let $Ω\subset\mathbb R^N$ be a bounded open set with Lipschitz continuous boundary $Γ$. Let $γ>0$, $δ\ge 0$ be real numbers and $β$ a nonnegative measurable function in $L^\infty(Γ)$. Using some suitable Carleman estimates, we show that the linear heat equation $\partial_tu - γΔu = 0$ in $Ω\times(0,T)$ with the non-homogeneous general dynamic boundary conditions $\partial_tu_Γ -δΔ_Γu_Γ+ γ\partial_νu + βu_Γ = g$ on $Γ\times(0,T)$ is always null controllable from the boundary for every $T>0$ and initial data $(u_0,u_{Γ,0})\in L^2(Ω)\times L^2(Γ)$.
Let $A$ be a densely defined closed, linear $ω$-sectorial operator of angle $θ\in [0,\fracπ{2})$ on a Banach space $X$ for some $ω\in\mathbb R$. We give an explicit representation (in terms of some special functions) and study the precise asymptotic behavior as time goes to infinity of solutions to the following diffusion equation with memory: $\displaystyle u'(t)=Au(t)+(κ\ast Au)(t), \, t >0$, $u(0)=u_0$, associated with the (possible) singular kernel $κ(t)=αe^{-βt}\frac{t^{μ-1}}{Γ(μ)},\;\;t>0$, where $α\in\R$, $α\ne 0$, $β\ge 0$ and $0<μ\le 1$.