arXiv subjects

Vu Huu Nhu

Publications and source records attributed to Vu Huu Nhu.

12 recordsLinked to original sources

arXiv2026

Generalized Differentiability and Second-Order Necessary Optimality Conditions for an Elliptic Optimal Control Problem with Exponential Nonlinearity and Discrete Measures

This paper deals with generalized differentiability and second-order necessary optimality conditions for a box-constrained optimal control problem governed by an exponential semilinear elliptic equation with discrete measures as sources, where the control belongs to the space of absolutely summable sequences. The presence of the exponential nonlinearity and discrete measures makes the analysis particularly challenging. In particular, the control-to-state operator may fail to be directionally differentiable. To address this issue, we first establish finite-dimensional directional differentiability of the control-to-state operator; that is, the operator is directionally differentiable along directions contained in finite-dimensional subspaces of the control space. We then introduce a notion of generalized derivative defined as the limit of the associated finite-dimensional directional derivatives as the dimension of these subspaces tends to infinity. Based on this concept, together with estimates for first- and second-order Taylor-type expansions of the exponential Nemytskii operator associated with the control-to-state mapping, we derive first- and second-order generalized differentiability of the reduced objective functional. This leads to first- and second-order necessary optimality conditions for the optimal control problem.

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arXiv2025

Optimal control of a two-dimensional elliptic equation with exponential nonlinearity and Dirac measure data

This work addresses an optimal control problem for a semilinear elliptic equation in two-dimensional space, characterized by an exponential nonlinearity and a singular source term. The source is modeled as a finite linear combination of Dirac measures concentrated at a fixed set of distinct points. The control variable is a finite-dimensional vector whose components represent the masses assigned to these point sources. Due to the interplay between the exponential nonlinearity and the singular measure data, the state equation is generally ill-posed and admits a unique very weak solution only when the largest component of the control vector does not surpass a certain critical threshold. Consequently, the control-to-state operator might be continuously differentiable only on an open subset of the control space. To derive first-order optimality conditions for the original problem, we introduce a family of regularized problems by imposing box constraints on the control variables. These constraints are chosen such that the admissible control sets of the regularized problems lie entirely within the open subset where the control-to-state operator is smooth. By analyzing the optimality systems associated with the regularized problems and passing to the limit, we obtain necessary optimality conditions for the original, unregularized problem.

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arXiv2025

Optimality conditions in terms of Bouligand generalized differentials for a nonsmooth semilinear elliptic optimal control problem with distributed and boundary control pointwise constraints

This paper is concerned with an optimal control problem governed by nonsmooth semilinear elliptic partial differential equations with both distributed and boundary unilateral pointwise control constraints, in which the nonlinear coefficient in the state equation is not differentiable at one point. Therefore, the Bouligand subdifferential of this nonsmooth coefficient in every point consists of one or two elements that will be used to construct the two associated Bouligand generalized derivatives of the control-to-state operator in any admissible control. These Bouligand generalized derivatives appear in a novel optimality condition, which extends the purely primal optimality condition saying that the directional derivative of the reduced objective functional in admissible directions in nonnegative. We then establish the optimality conditions in the form of multiplier existence. There, in addition to the existence of the adjoint states and of the nonnegative multipliers associated with the unilateral pointwise constraints as usual, other nonnegative multipliers exist and correspond to the nondifferentiability of the control-to-state mapping. The latter type of optimality conditions shall be applied to an optimal control satisfying the so-called \emph{constraint qualification} to derive a \emph{strong} stationarity, where the sign of the associated adjoint state does not vary on the level set of the corresponding optimal state at the value of nondifferentiability. Finally, this strong stationarity is also shown to be equivalent to the purely primal optimality condition.

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arXiv2024

Numerical analysis of a nonsmooth quasilinear elliptic control problem: II. Finite element discretization and error estimates

In this paper, we carry out the numerical analysis of a nonsmooth quasilinear elliptic optimal control problem, where the coefficient in the divergence term of the corresponding state equation is not differentiable with respect to the state variable. Despite the lack of differentiability of the nonlinearity in the quasilinear elliptic equation, the corresponding control-to-state operator is of class $C^1$ but not of class $C^2$. Analogously, the discrete control-to-state operators associated with the approximated control problems are proven to be of class $C^1$ only. By using an explicit second-order sufficient optimality condition, we prove a priori error estimates for a variational approximation, a piecewise constant approximation, and a continuous piecewise linear approximation of the continuous optimal control problem. The numerical tests confirm these error estimates.

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arXiv2023

Optimality conditions and Lipschitz stability for non-smooth semilinear elliptic optimal control problems with sparse controls

This paper is concerned with first- and second-order optimality conditions as well as the stability for non-smooth semilinear optimal control problems involving the $L^1$-norm of the control in the cost functional. In addition to the appearance of the $L^1$-norm leading to the non-differentiability of the objective and promoting the sparsity of the optimal controls, the non-smoothness of the nonlinear coefficient in the state equation causes the same property of the control-to-state operator. Exploiting a regularization scheme, we derive $C$-stationarity conditions for any local optimal control. Under a structural assumption on the associated state, we define the curvature functional for the part not including the $L^1$-norm of controls of the objective for which the second-order necessary and sufficient optimality conditions are shown. Furthermore, under a more restrictive structural assumption imposed on the mentioned state, an explicit formulation of the curvature is established and thus the explicit second-order optimality conditions are stated. Finally, the Lipschitz stability of local solutions with respect to the sparsity parameter is shown.

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arXiv2023

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green's first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

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arXiv2021

Lipschitz continuity of solutions and corresponding multipliers to distributed and boundary semilinear elliptic optimal control problems with mixed pointwise control-state constraints

This paper is concerned with the existence and regularity of mininizers as well as of corresponding multipliers to an optimal control problem governed by semilinear elliptic equations, in which mixed pointwise control-state constraints are considered in a quite general form and the controls act simultaneously in the domain and on the boundary. Under standing assumptions, the minimizers and the corresponding multipliers do exist. Furthermore, by applying the bootstrapping technique and establishing some calculation tools for functions in Sobolev spaces of fractional order, the optimal solutions and the associated Lagrange multipliers are shown to be Lipschitz continuous.

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arXiv2021

On the no-gap second-order optimality conditions for a non-smooth semilinear elliptic optimal control

This work is concerned with second-order necessary and sufficient optimality conditions for optimal control of a non-smooth semilinear elliptic partial differential equation, where the nonlinearity is the non-smooth max-function and thus the associated control-to-state operator is in general not Gâteaux-differentiable. In addition to standing assumptions, two main hypotheses are imposed. The first one is the Gâteaux-differentiability at the considered control of the objective functional and it is precisely characterized by the vanishing of an adjoint state on the active set. The second one is a structural assumption on the 'almost' active sets, i.e., the sets of all points at which the values of the interested state are 'close' to the non-differentiability point of the max-function. We then derive a 'no-gap' theory of second-order optimality conditions in terms of an abstract curvature functional, i.e., for which the only change between necessary or sufficient second-order optimality conditions is between a strict and non strict inequality.

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arXiv2020

No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is countably $PC^2$ (continuous and $C^2$ apart from countably many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish "no-gap" second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i. e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided.

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arXiv2019

Bouligand-Levenberg-Marquardt iteration for a non-smooth ill-posed inverse problem

In this paper, we consider a modified Levenberg--Marquardt method for solving an ill-posed inverse problem where the forward mapping is not Gâteaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we derive asymptotic stability estimates that are then used to prove the convergence of the proposed method. This method can be applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Fréchet derivative, and we show that the corresponding Bouligand-Levenberg-Marquardt iteration is an iterative regularization scheme. Numerical examples illustrate the advantage over the corresponding Bouligand-Landweber iteration.

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arXiv2019

Bouligand-Landweber iteration for a non-smooth ill-posed problem

This work is concerned with the iterative regularization of a non-smooth nonlinear ill-posed problem where the forward mapping is merely directionally but not Gâteaux differentiable. Using a Bouligand subderivative of the forward mapping, a modified Landweber method can be applied; however, the standard analysis is not applicable since the Bouligand subderivative mapping is not continuous unless the forward mapping is Gâteaux differentiable. We therefore provide a novel convergence analysis of the modified Landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition. These conditions are verified for an inverse source problem for an elliptic PDE with a non-smooth Lipschitz continuous nonlinearity, showing that the corresponding Bouligand--Landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle. This is illustrated

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arXiv2018

Optimal control of a non-smooth quasilinear elliptic equation

This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a PC^1 function with countably many points of nondifferentiability, we show that both conditions are equivalent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples.

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