Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Casas, Eduardo; Tröltzsch, Fredi

doi:10.1051/cocv/2010025

Cited by 15 publications

(5 citation statements)

References 13 publications

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“…These equations are not of monotone type. In the elliptic case, we mention [12], [13], [21], [22]. We only know two papers for this type of nonlinearity in the parabolic case: [5] and [31].…”

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confidence: 99%

Analysis and optimal control of some quasilinear parabolic equations

Casas¹,

Chrysafinos²

2018

Mathematical Control &Amp; Related Fields

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In this paper, we consider optimal control problems associated with a class of quasilinear parabolic equations, where the coefficients of the elliptic part of the operator depend on the state function. We prove existence, uniqueness and regularity for the solution of the state equation. Then, we analyze the control problem. The goal is to get first and second order optimality conditions. To this aim we prove the necessary differentiability properties of the relation control-to-state and of the cost functional.

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confidence: 99%

Analysis and optimal control of some quasilinear parabolic equations

Casas¹,

Chrysafinos²

2018

Mathematical Control &Amp; Related Fields

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“…Similar problems were considered by Casas [2] and Chryssoverghi and Kokkinis [3]. In the eld of nite element approximations for optimal controls governed by PDEs, we refer the readers to the papers [4][5][6][7][8][9][10] and the references therein. is present paper is mainly motivated by the work of [2] where the author considered the following state equation with Our main goal is to generalize the results in [2] to the case of p-Laplacian.…”

Section: Remark 2 Since 3 < < +∞mentioning

confidence: 99%

Discretization of Optimal Control Problems Governed by p-Laplacian Elliptic Equations

Luan

2019

Journal of Applied Mathematics

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“…3 we study the approximation of the state equation by finite elements. The reader is referred to [32] for the linear case or [8,16,22] for the case of nonmonotone but coercive quasilinear equations. In the quasilinear case, the discrete equation has at least one solution for every h, which easily follows from an application of Brouwer's fixed point theorem and the coercivity of the operator.…”

Section: Introductionmentioning

confidence: 99%

“…In all these references, the equations were coercive and monotone. Optimal control problems governed by quasi-linear elliptic equations have been studied in [8,16,17,20]. In these works, the equations were coercive but not monotone.…”

Section: Introductionmentioning

confidence: 99%

Numerical approximation of control problems of non-monotone and non-coercive semilinear elliptic equations

2021

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We analyze the numerical approximation of a control problem governed by a non-monotone and non-coercive semilinear elliptic equation. The lack of monotonicity and coercivity is due to the presence of a convection term. First, we study the finite element approximation of the partial differential equation. While we can prove existence of a solution for the discrete equation when the discretization parameter is small enough, the uniqueness is an open problem for us if the nonlinearity is not globally Lipschitz. Nevertheless, we prove the existence and uniqueness of a sequence of solutions bounded in $$L^\infty (\varOmega )$$ L ∞ ( Ω ) and converging to the solution of the continuous problem. Error estimates for these solutions are obtained. Next, we discretize the control problem. Existence of discrete optimal controls is proved, as well as their convergence to solutions of the continuous problem. The analysis of error estimates is quite involved due to the possible non-uniqueness of the discrete state for a given control. To overcome this difficulty we define an appropriate discrete control-to-state mapping in a neighbourhood of a strict solution of the continuous control problem. This allows us to introduce a reduced functional and obtain first order optimality conditions as well as error estimates. Some numerical experiments are included to illustrate the theoretical results.

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Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Cited by 15 publications

References 13 publications

Analysis and optimal control of some quasilinear parabolic equations

Analysis and optimal control of some quasilinear parabolic equations

Discretization of Optimal Control Problems Governed by p-Laplacian Elliptic Equations

Numerical approximation of control problems of non-monotone and non-coercive semilinear elliptic equations

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