arXiv subjects

Francisco Fuica

Publications and source records attributed to Francisco Fuica.

At least 19 recordsLinked to original sources

arXiv2026

On quantitative sufficient second-order optimality conditions for elliptic optimal control problems

In this paper, a quantitative condition for optimality for distributed optimal control problems with box-constraints that are subject to a semilinear elliptic equation is considered. An important property of the investigated optimal control problems is the absence of a Tikhonov regularization. It is well known that at a given control, the second variation is a quadratic form in the linearized state, and its curvature coefficient function may vanish or change sign. We present a quantitative condition that implies coercivity with respect to the $L^2$-norm of the linearized-states. As a consequence, the stability of the second-order condition under perturbations of the states and the tracking data is shown.

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arXiv2026

Error estimates for an unregularized optimal control problem for the stationary Navier-Stokes equations

We consider an unregularized optimal control problem subject to the steady-state Navier-Stokes equations. We derive the existence of optimal solutions and prove first- and second-order optimality conditions. To approximate solutions to the optimal control problem, we consider the variational discretization scheme. We analyze convergence properties of the discretization and prove a priori error estimates for locally optimal controls that are nonsingular and which satisfy a growth condition which implies a bang-bang structure. We also propose a residual-type a posteriori error estimator that accounts for the discretization of the state and adjoint equations, and prove suitable reliability properties for such an error estimator.

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arXiv2025

A posteriori error estimates for a bang-bang optimal control problem

We propose and analyze a posteriori error estimates for a control-constrained optimal control problem with bang-bang solutions. We consider a solution strategy based on the variational approach, where the control variable is not discretized; no Tikhonov regularization is made. We design, for the proposed scheme, a residual-type a posteriori error estimator that can be decomposed as the sum of two individual contributions related to the discretization of the state and adjoint equations. We explore reliability and efficiency properties of the aforementioned error estimator. We illustrate the theory with numerical examples.

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arXiv2025

Error estimates for a multiobjective optimal control of a pointwise tracking problem

We analyze a pointwise tracking multiobjective optimal control problem subject to the Poisson problem and bilateral control constraints. To approximate Pareto optimal points and the Pareto front numerically, we consider two different finite element-based scalarization techniques, namely the weighted-sum method and the reference point method, where in both methods many scalar-constrained optimization problems have to be solved. We prove a priori error estimates for both scalarizations. The underlying subproblems of either method are solved with a Barzilai-Borwein gradient method. Numerical experiments illustrate the accuracy of the proposed method.

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arXiv2025

Adaptive finite element method for an unregularized semilinear optimal control problem

We devise an a posteriori error estimator for an affine optimal control problem subject to a semilinear elliptic PDE and control constraints. To approximate the problem, we consider a semidiscrete scheme based on the variational discretization approach. For this solution technique, we design an a posteriori error estimator that accounts for the discretization of the state and adjoint equations, and prove, under suitable local growth conditions of optimal controls, reliability and efficiency properties of such error estimator. A simple adaptive strategy based on the devised estimator is designed and its performance is illustrated with numerical examples.

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arXiv2025

A space-time finite element method for parabolic obstacle problems

We propose and analyze a general framework for space-time finite element methods that is based on least-squares finite element methods for solving a first-order reformulation of the thick parabolic obstacle problem. Discretizations based on simplicial and prismatic meshes are studied and we show a priori error estimates for both. Convergence rates are derived for sufficiently smooth solutions. Reliable a posteriori bounds are provided and used to steer an adaptive algorithm. Numerical experiments including a one-phase Stefan problem and an American option pricing problem are presented.

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arXiv2024

Mixed finite element methods for elliptic obstacle problems

Mixed variational formulations for the first-order system of the elastic membrane obstacle problem and the second-order system of the Kirchhoff--Love plate obstacle problem are proposed. The force exerted by the rigid obstacle is included as a new unknown. A priori and a posteriori error estimates are derived for both obstacle problems. The a posteriori error estimates are based on conforming postprocessed solutions. Numerical experiments conclude this work.

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arXiv2024

Error estimates for a bilinear optimal control problem of Maxwell's equations

We consider a control-constrained optimal control problem subject to time-harmonic Maxwell's equations; the control variable belongs to a finite-dimensional set and enters the state equation as a coefficient. We derive existence of optimal solutions, and analyze first- and second-order optimality conditions. We devise an approximation scheme based on the lowest order Nédélec finite elements to approximate optimal solutions. We analyze convergence properties of the proposed scheme and prove a priori error estimates. We also design an a posteriori error estimator that can be decomposed as the sum two contributions related to the discretization of the state and adjoint equations, and prove that the devised error estimator is reliable and locally efficient. We perform numerical tests in order to assess the performance of the devised discretization strategy and the a posteriori error estimator.

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arXiv2024

Bilinear optimal control for the fractional Laplacian: analysis and discretization

We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic partial differential equation (PDE) as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized, and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.

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arXiv2023

Fractional, semilinear, and sparse optimal control: a priori error bounds

In this work, we use the integral definition of the fractional Laplace operator and study a sparse optimal control problem involving a fractional, semilinear, and elliptic partial differential equation as state equation; control constraints are also considered. We establish the existence of optimal solutions and first and second order optimality conditions. We also analyze regularity properties for optimal variables. We propose and analyze two finite element strategies of discretization: a fully discrete scheme, where the control variable is discretized with piecewise constant functions, and a semidiscrete scheme, where the control variable is not discretized. For both discretization schemes, we analyze convergence properties and a priori error bounds.

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arXiv2023

A pointwise tracking optimal control problem for the stationary Navier--Stokes equations

We study a pointwise tracking optimal control problem for the stationary Navier--Stokes equations; control constraints are also considered. The problem entails the minimization of a cost functional involving point evaluations of the state velocity field, thus leading to an adjoint problem with a linear combination of Dirac measures as a forcing term in the momentum equation, and whose solution has reduced regularity properties. We analyze the existence of optimal solutions and derive first and, necessary and sufficient, second order optimality conditions in the framework of regular solutions for the Navier--Stokes equations. We develop two discretization strategies: a semidiscrete strategy in which the control variable is not discretized, and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For each solution technique, we analyze convergence properties of discretizations and derive a priori error estimates.

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arXiv2023

A DPG method for linear quadratic optimal control problems

The DPG method with optimal test functions for solving linear quadratic optimal control problems with control constraints is studied. We prove existence of a unique optimal solution of the nonlinear discrete problem and characterize it through first order optimality conditions. Furthermore, we systematically develop a priori as well as a posteriori error estimates. Our proposed method can be applied to a wide range of constrained optimal control problems subject to, e.g., scalar second-order PDEs and the Stokes equations. Numerical experiments that illustrate our theoretical findings are presented.

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arXiv2022

An Optimal Control Problem for the Navier--Stokes Equations with Point Sources

We analyze, in two dimensions, an optimal control problem for the Navier--Stokes equations where the control variable corresponds to the amplitude of forces modeled as point sources; control constraints are also considered. This particular setting leads to solutions to the state equation exhibiting reduced regularity properties. We operate under the framework of Muckenhoupt weights, Muckenhoupt-weighted Sobolev spaces, and the corresponding weighted norm inequalities and derive the existence of optimal solutions and first and, necessary and sufficient, second order optimality conditions.

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arXiv2022

A posteriori error estimates for an optimal control problem with a bilinear state equation

We propose and analyze a posteriori error estimators for an optimal control problem that involves an elliptic partial differential equation as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We consider two different strategies to approximate optimal variables: a fully discrete scheme in which the admissible control set is discretized with piecewise constant functions and a semi-discrete scheme where the admissible control set is not discretized; the latter scheme being based on the so-called variational discretization approach. We design, for each solution technique, an a posteriori error estimator and show, in two and three dimensional Lipschitz polygonal/polyhedral domains (not necessarily convex), that the proposed error estimator is reliable and efficient. We design, based on the devised estimators, adaptive strategies that deliver optimal experimental rates of convergence for the performed numerical examples.

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arXiv2021

Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equation

We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. We devise two strategies of discretization to approximate a solution of the optimal control problem: a semidiscrete scheme where the control variable is not discretized -- the so-called variational discretization approach -- and a fully discrete scheme where the control variable is discretized with piecewise constant functions. For both solution techniques, we analyze convergence properties of discretizations and derive error estimates.

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arXiv2021

A posteriori error estimates for a distributed optimal control problem of the stationary Navier-Stokes equations

In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we propose and analyze a posteriori error estimators for an optimal control problem involving the stationary Navier--Stokes equations; control constraints are also considered. We devise two strategies of discretization: a semi discrete scheme where the control variable is not discretized -- the so-called variational discrezation approach -- and a fully discrete scheme where the control is discretized with piecewise quadratic functions. For each solution solution technique, we design an a posteriori error estimator that can be decomposed as the sum of contributions related to the discretization of the state and adjoint equations and, additionally, the discretization of the control variable for when the fully discrete scheme is considered. We prove that the devised error estimators are reliable and also explore local efficiency estimates. Numerical experiments reveal a competitive performance of adaptive loops based on the devised a posteriori error estimators.

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arXiv2020

Error estimates for optimal control problems involving the Stokes system and Dirac measures

The aim of this work is to derive a priori error estimates for finite element discretizations of control--constrained optimal control problems that involve the Stokes system and Dirac measures. The first problem entails the minimization of a cost functional that involves point evaluations of the velocity field that solves the state equations. This leads to an adjoint problem with a linear combination of Dirac measures as a forcing term and whose solution exhibits reduced regularity properties. The second problem involves a control variable that corresponds to the amplitude of forces modeled as point sources. This leads to a solution of the state equations with reduced regularity properties. For each problem, we propose a finite element solution technique and derive a priori error estimates. Finally, we present numerical experiments, in two and three dimensions, that illustrate our theoretical developments.

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arXiv2019

A posteriori error estimates in $\mathbf{W}^{1,p} \times \mathrm{L}^p$ spaces for the Stokes system with Dirac measures

We design and analyze a posteriori error estimators for the Stokes system with singular sources in suitable $\mathbf{W}^{1,p}\times \mathrm{L}^p$ spaces. We consider classical low-order inf-sup stable and stabilized finite element discretizations. We prove, in two and three dimensional Lipschitz, but not necessarily convex polytopal domains, that the devised error estimators are reliable and locally efficient. On the basis of the devised error estimators, we design a simple adaptive strategy that yields optimal experimental rates of convergence for the numerical examples that we perform.

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