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Eduardo Casas

Publications and source records attributed to Eduardo Casas.

At least 19 recordsLinked to original sources
arXiv2026

Optimal Control of a Class of Reaction-Diffusion Systems with Pointwise State Constraints

We study optimal control problems governed by a system of semilinear reaction-diffusion equations. The assumptions on the governing system are formulated so that our results apply to several models, including the Brusselator, Schnakenberg, and Gray--Scott systems. We prove the existence and uniqueness of a solution to the governing system without imposing any non-negativity assumption on the data. This allows us to study unconstrained, control-constrained, and state-constrained optimal control problems. For all these problems, we derive first-order necessary and second-order sufficient optimality conditions. The control acts on the activator equation through the boundary. Numerical examples demonstrate that both the inhibitor and the activator can be efficiently controlled.

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arXiv2026

Infinite Horizon Optimal Control Problems with Discount Factors

This paper is dedicated to the analysis of infinite horizon optimal control problems subject to semilinear parabolic equations with constraints on the controls and discounted cost functionals. The discount factors on the cost and the state components are allowed to differ from each other. First-order as well as second-order optimality conditions are derived and the importance of allowing different discount factors for the second-order analysis for the class of nonlinearities under consideration is demonstrated. Finally convergence and rate of convergence for the approximation of the infinite horizon problem by a family of finite horizon problems is proven.

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arXiv2026

Boundary bilinear control of semilinear parabolic PDEs: quadratic convergence of the SQP method

We analyze a bilinear control problem governed by a semilinear parabolic equation. The control variable is the Robin coefficient on the boundary. First-order necessary and second-order sufficient optimality conditions are derived. A sequential quadratic programming algorithm is then proposed to compute local solutions. Starting the iterations from an initial point in an $L^2$-neighborhood of the local solution we prove stability and quadratic convergence of the algorithm in $L^p$ ($p < \infty$) and $L^\infty$ assuming that the local solution satisfies a no-gap second-order sufficient optimality condition and a strict complementarity condition.

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arXiv2026

Second-Order Conditions for Infinite-Horizon Semilinear Parabolic Control Problems without Tikhonov Regularization

We consider semilinear parabolic optimal control problems subject to Neumann boundary conditions, control constraints, and an infinite time horizon. The control constraints are pointwise in time, but they can be pointwise or integral in the space variable. Crucially, the optimal control problem does not include a Tikhonov regularization in the cost functional, which provides a major difficulty in the extension of the classical finite-horizon theory to infinite-horizon optimal control problems. As a consequence of our findings, we establish a sufficient second-order optimality condition and prove that local optimal states of the finite-horizon problems approximate local optimal states to the infinite-horizon problem as the horizon tends to infinity.

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arXiv2025

Quadratic convergence of an SQP method for some optimization problems with applications to control theory

We analyze a sequential quadratic programming algorithm for solving a class of abstract optimization problems. Assuming that the initial point is in an $L^2$ neighborhood of a local solution that satisfies no-gap second-order sufficient optimality conditions and a strict complementarity condition, we obtain stability and quadratic convergence in $L^q$ for all $q\in[p,\infty]$ where $p\geq 2$ depends on the problem. Many of the usual optimal control problems of partial differential equations fit into this abstract formulation. Some examples are given in the paper. Finally, a computational comparison with other versions of the SQP method is presented.

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arXiv2024

Error estimates for the discretization of bilinear control problems governed by semilinear elliptic PDEs

This paper studies an optimal control problem governed by a semilinear elliptic equation, in which the control acts in a multiplicative or bilinear way as the reaction coefficient of the equation. We focus on the numerical discretization of the problem. The discretization is carried out by using the finite element method, with piecewise constant functions for the control and continuous piecewise linear functions for the state and the adjoint state. We first prove convergence of the solutions of the discrete problems to solutions of the continuous problem. We also demonstrate that strict local solutions of the continuous problem can be approximated by local solutions of the discrete problems. Next we obtain an error estimate of order $O(h)$ for the difference between continuous and discrete locally optimal controls. To obtain this result we assume no-gap second order sufficient optimality conditions. As it is usual in this kind of discretization, a superconvergence phenomenon of order $O(h^2)$ is observed in numerical experiments for the error estimates of the state and adjoint state. The last part of the paper is dedicated to explain this behaviour. A numerical experiment confirming these results is included.

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arXiv2024

Bilinear control of semilinear elliptic PDEs: Convergence of a semismooth Newton method

In this paper, we carry out the analysis of the semismooth Newton method for bilinear control problems related to semilinear elliptic PDEs. We prove existence, uniqueness and regularity for the solution of the state equation, as well as differentiability properties of the control to state mapping. Then, first and second order optimality conditions are obtained. Finally, we prove the superlinear convergence of the semismooth Newton method to local solutions satisfying no-gap second order sufficient optimality conditions as well as a strict complementarity condition.

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arXiv2024

Semismooth Newton Method for Boundary Bilinear Control

We study a control-constrained optimal control problem governed by a semilinear elliptic equation. The control acts in a bilinear way on the boundary, and can be interpreted as a heat transfer coefficient. A detailed study of the state equation is performed and differentiability properties of the control-to-state mapping are shown. First and second order optimality conditions are derived. Our main result is the proof of superlinear convergence of the semismooth Newton method to local solutions satisfying no-gap second order sufficient optimality conditions as well as a strict complementarity condition.

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arXiv2023

New assumptions for stability analysis in elliptic optimal control problems

This paper is dedicated to the stability analysis of the optimal solutions of a control problem associated with a semilinear elliptic equation. The linear differential operator of the equation is neither monotone nor coercive due to the presence of a convection term. The control appears only linearly, or even it can not appear in an explicit form in the objective functional. Under new assumptions, we prove Lipschitz stability of the optimal controls and associated states with respect to perturbations in the equation and the objective functional as well as with respect to the Tikhonov regularization parameter.

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arXiv2022

A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations

In this paper, we study optimal control problems of semilinear elliptic and parabolic equations. A tracking cost functional, quadratic in the control and state variables, is considered. No control constraints are imposed. We prove that the corresponding state equations are well-posed for controls in $L^2$. However, it is well-known that in the $L^2$ framework the mappings involved in the control problem are not Frechet differentiable in general, which makes any analysis of the optimality conditions challenging. Nevertheless, we prove that every $L^2$ optimal control belongs to $L^\infty$, and consequently standard optimality conditions are available.

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arXiv2022

Critical cones for sufficient second order conditions in PDE constrained optimization

In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.

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arXiv2021

Infinite Horizon Optimal Control Problems for a Class of Semilinear Parabolic Equations

Infinite horizon open loop optimal control problems for semilinear parabolic equations are investigated. The controls are subject to a cost-functional which promotes sparsity in time. The focus is put on deriving first order optimality conditions. This is achieved without relying on a well-defined control-to-state mapping in a neighborhood of minimizers. The technique of proof is based on the approximation of the original problem by a family of finite horizon problems. The optimality conditions allow to deduce sparsity properties of the optimal controls in time.

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arXiv2021

Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space

This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form $\|u(t)\|_{L^1(Ω)} \le γ$ for $t \in (0,T)$. This limits the total control that can be applied to the system at any instant of time. The $L^1$-norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to $γ$ is investigated on the basis of different second order conditions.

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arXiv2021

Well-Posedness of Evolutionary Navier-Stokes Equations with Forces of Low Regularity on Two-Dimensional Domains

Existence and uniqueness of solutions to the Navier-Stokes equation in dimension two with forces in the space $L^q( (0,T); \mathbf{W}^{-1,p}(Ω))$ for $p$ and $q$ in appropriate parameter ranges are proven. The case of spatially measured-valued inhomogeneities is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions with $1 < p, q < \infty$ arbitrary.

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arXiv2020

Optimal Control of the 2D Evolutionary Navier-Stokes Equations with Measure Valued Controls

In this paper, we consider an optimal control problem for the two-dimensional evolutionary Navier-Stokes system. Looking for sparsity, we take controls as functions of time taking values in a space of Borel measures. The cost functional does not involve directly the control but we assume some constraints on them. We prove the well-posedness of the control problem and derive necessary and sufficient conditions for local optimality of the controls.

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arXiv2020

Analysis of control problems of nonmontone semilinear elliptic equations

In this paper we study optimal control problems governed by a semilinear elliptic equation. The equation is nonmonotone due to the presence of a convection term, despite the monotonocity of the nonlinear term. The resulting operator is neither monotone nor coervive. However, by using conveniently a comparison principle we prove existence and uniqueness of solution for the state equation. In addition, we prove some regularity of the solution and differentiability of the relation control-to-state. This allows us to derive first and second order conditions for local optimality.

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arXiv2020

Analysis of Optimal Control Problems with an $L^0$ Term in the Cost Functional

In this paper, we investigate optimal control problems subject to a semilinear elliptic partial differential equations. The cost functional contains a term that measures the size of the support of the control, which is the so-called $L^0$-norm. We provide necessary and sufficient optimality conditions of second-order. The sufficient second-order condition is obtained by analyzing a partially convexified problem. Interestingly, the structure of the problem yields second-order conditions with different bilinear forms for the necessary and for the sufficient condition.

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