Optimal control of fractional diffusion with Dirac measures

Enrique Otarola; Abner J. Salgado

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Optimal control of fractional diffusion with Dirac measures

Enrique Otarola, Abner J. Salgado

arXiv:2608.27776v1Aug 27, 2026math.OCmath.NA

TL;DR

The paper studies optimal control of spectral fractional diffusion forced by finite combinations of Dirac measures, with source amplitudes as controls. It proves well-posedness and uniqueness of the optimizer, derives adjoint-based optimality conditions, and develops a finite-element discretization with a priori error bounds. The analysis also identifies a nonoptimal convergence rate associated with relying on state L2-error estimates.

Problem
The paper addresses PDE-constrained optimization for spectral fractional diffusion with a linear combination of Dirac measures as forcing and source amplitudes as controls.
Method
It combines adjoint-based optimal control analysis with a finite-element discretization, leaving the finite-dimensional control variable undiscretized.
Results
The paper proves existence and uniqueness of the optimal solution, derives first-order optimality conditions, and establishes a priori finite-element error bounds.
Takeaways & Limitations
The framework provides a finite-dimensional control formulation for singular sources together with computable optimality conditions and discretization error estimates.
Takeaways & Limitations
The derived rate is not optimal because the argument uses L2(Ω)-error estimates rather than local L∞-estimates, which are unavailable for these fractional finite-element approximations.

Abstract

from arXiv · show

We study a PDE-constrained optimization problem for an elliptic equation with the spectral fractional Laplacian and a linear combination of Dirac measures as the forcing term; the controls are the amplitudes of these singular sources. We prove existence and uniqueness of an optimal solution and derive first-order optimality conditions. We then propose a discretization based on finite elements. Since the set of admissible controls is finite dimensional, the control variable itself does not require discretization. We conclude by deriving a priori error bounds

1. Introduction

The paper formulates an optimal control problem for spectral fractional diffusion driven by finitely many Dirac sources, whose amplitudes are bounded controls. The setting extends the analysis to three dimensions under a regularity condition ensuring well-defined point evaluations.

The analysis covers d ∈ {2, 3} and assumes s > d/4, which ensures continuity of the adjoint state and well-defined point evaluations.
The state equation uses the spectral fractional Laplacian with homogeneous Dirichlet boundary conditions and a singular forcing term.
The objective combines tracking of a desired state in L2(Ω) with regularization controlled by α > 0.
The control vector consists of the amplitudes of Dirac sources supported on a finite set D, with componentwise lower and upper bounds.
The formulation is motivated by applications such as active sound control, where source amplitudes act as controls.

2. Notation and preliminaries

This section establishes the spectral and functional-analytic framework for the fractional problem. It introduces eigenfunction-based notation, fractional spaces, and the measure duality used later.

Finite Radon measures M(Ω) are identified with the dual of C0(Ω), with ⟨·,·⟩ denoting the associated duality pairing.
The Dirichlet Laplacian has a countable family of eigenpairs whose eigenfunctions form orthonormal and orthogonal bases in the stated spaces.
The spectral fractional Laplacian is defined using the eigenfunction representation for s ∈ (0,1).
The spaces H^r(Ω) are identified with classical fractional Sobolev spaces using equivalent norms in this convex-domain setting.

3. Fractional diffusion under a measure-valued right-hand side

The paper gives a weak formulation and well-posedness theory for the spectral fractional diffusion equation with a measure-valued right-hand side. The analysis uses a regularity range enabled by s > d/4.

The state equation is the fractional PDE (−∆)^s u = µ in Ω with µ ∈ M(Ω).
Choosing θ ∈ (d/2 − s, s) is possible because s > d/4, and yields the regularity inequalities required in the analysis.
A weak solution u ∈ H^{s−θ}(Ω) satisfies A(u,v) = ⟨µ,v⟩ for all v ∈ H^{s+θ}(Ω).
Every measure µ ∈ M(Ω) defines a bounded linear functional on H^{s+θ}(Ω), making the weak right-hand side well defined.
For every µ ∈ M(Ω), the problem has a unique solution u ∈ H^{s−θ}(Ω) satisfying the theorem’s stated bounds.

4. The optimal control problem

The optimal control problem is reduced through a linear control-to-state map, yielding a unique optimizer and pointwise adjoint-based optimality conditions. The section also establishes the regularity needed to evaluate adjoint values at source points.

4.1. Existence and uniqueness of an optimal control.: The control problem minimizes the cost functional subject to the weak formulation of the fractional state equation.
4.1. Existence and uniqueness of an optimal control.: The bounded linear control-to-state map S sends q to the unique state Sq, and the reduced functional j(q) = J(Sq,q) is well defined.
4.1. Existence and uniqueness of an optimal control.: The optimal control problem has a unique solution (q̄,ū) in H^{s−θ}(Ω) × Qad.
4.2. Optimality conditions.: The adjoint problem is posed through a bilinear form B with right-hand side given by the state-tracking residual.
4.2. Optimality conditions.: The adjoint belongs to H^{2s}(Ω) and embeds into C0(Ω), so p̄(z) is well defined at every source point.
4.2. Optimality conditions.: The first-order condition characterizes optimality through the adjoint values and regularized control amplitudes.

5. Discretization

The paper discretizes the fractional PDE with finite elements, regularizes measure-valued sources, and transfers the resulting state approximation to the finite-dimensional optimal control problem. It derives existence, uniqueness, and a priori error bounds for the discrete optimizer and state.

5.1. A finite element method for fractional PDEs.: The finite element method uses continuous piecewise linear functions on a quasiuniform family of conforming simplicial meshes.The discrete space consists of functions vanishing on the boundary.
5.1. A finite element method for fractional PDEs.: Measure-valued right-hand sides are regularized by convolution with a scaled smooth mollifier before applying the finite element scheme.The mollifier is nonnegative, compactly supported in the unit ball, and has unit integral.
5.1. A finite element method for fractional PDEs.: The fractional PDE discretization chooses Y ≂ 2^s| log h|, K ≂ Y/h, and ε ≂ h, yielding the estimate stated in Theorem 4.Theorem 4 applies these coupled truncation, spectral, and regularization choices to measure-valued data.
5.2. A finite element method for the optimal control problem.: The discrete control-to-state map S_h is linear and uniformly bounded from R^ℓ into L2(Ω), while the admissible control remains finite dimensional.Because the regularized measure depends linearly on q, each discrete state problem is linear in the control amplitudes.
5.2. A finite element method for the optimal control problem.: The discrete reduced problem has a unique optimal solution, characterized by the stated first-order condition.Continuity and strict convexity of the discrete functional on compact, convex Q_ad ensure existence and uniqueness.