Convergence Results for Optimal Control Problems Governed by Elliptic Quasivariational Inequalities

Sofonea, Mircea; Tarzia, Domingo A.

doi:10.1080/01630563.2020.1772288

Cited by 10 publications

(7 citation statements)

References 27 publications

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“…A proof of the theorem can be found in Sofonea and Tarzia [25]. Note that the existence and uniqueness part in this theorem is a direct consequence of Theorem 1.…”

Section: Preliminariesmentioning

confidence: 97%

“…First, it is easy to see that the set K given by equation (31) satisfies condition (2). Next, we use assumptions ( 24) and (25) to see that the operator A defined by (32) satisfies the inequalities:…”

Section: Unique Weak Solvabilitymentioning

confidence: 99%

“…In Section 2, we present the mathematical preliminaries we need in the analysis of our contact problem. These preliminaries concern existence, uniqueness, and convergence results obtained in Sofonea and Matei [23], Sofonea and Tarzia [25] in the study of elliptic quasivariational inequalities. In Section 3, we present the physical setting of the spring-rods system we consider, then we list the mechanical assumptions and state the corresponding mathematical model.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Modelling, analysis, and numerical simulation of a spring-rods system with unilateral constraints

Ochal,

Prządka,

Sofonea

et al. 2023

Mathematics and Mechanics of Solids

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In this paper, we consider a mathematical model which describes the equilibrium of two elastic rods attached to a nonlinear spring. We derive the variational formulation of the model which is in the form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the problem, then we state and prove some convergence results, for which we provide the corresponding mechanical interpretation. Next, we turn to the numerical approximation of the problem based on a finite element scheme. We use a relaxation method to solve the discrete problems that we implement on the computer. Using this method, we provide numerical simulations which validate our convergence results.

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“…A proof of the theorem can be found in Sofonea and Tarzia [25]. Note that the existence and uniqueness part in this theorem is a direct consequence of Theorem 1.…”

Section: Preliminariesmentioning

confidence: 97%

Section: Unique Weak Solvabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Modelling, analysis, and numerical simulation of a spring-rods system with unilateral constraints

Ochal,

Prządka,

Sofonea

et al. 2023

Mathematics and Mechanics of Solids

Self Cite

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“…Very recently, Du et al [14] obtained another fascinating q-integral identity and obtained various q-analogues of certain integral inequalities. For more information about quantum calculus and its applications, see [15][16][17][18][19][20][21][22][23][24][25][26].…”

Section: Definition 2 ([1]mentioning

confidence: 99%

Multi-Parameter Quantum Integral Identity Involving Raina’s Function and Corresponding q-Integral Inequalities with Applications

et al. 2022

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Convexity performs its due role in the theoretical field of inequalities according to the nature and conduct of the properties it displays. A correlation connectivity, which is visible between the two variables symmetry and convexity, enhances its importance. In this paper, we derive a new multi-parameter quantum integral identity involving Raina’s function. Applying this generic identity as an auxiliary result, we establish some new generalized quantum estimates of certain integral inequalities pertaining to the class of Rs-convex functions. Moreover, we give quantum integral inequalities for the product of Rs1- and Rs2-convex functions as well as another quantum result for a function that satisfies a special condition. In order to demonstrate the efficiency of our main results, we offer many important special cases for suitable choices of parameters and finally for Rs-convex functions that are absolute-value bounded.

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“…The results in [12,27,28,31,37] concern the analysis of hemivariational inequalities and are based on properties of the subdifferential in the sense of Clarke, defined for locally Lipschitz functions, which may be nonconvex. Results in the study of optimal control for variational and hemivariational inequalities have been discussed in several works, including [2,3,4,8,19,23,25,26,29,32,33,34,35,36] and, more recently, in [36,39].…”

Section: Introductionmentioning

confidence: 99%

Tykhonov Well-posedness of a Heat Transfer Problem with Unilateral Constraints

Sofonea,

Tarzia

2021

Preprint

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We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain D ⊂ IR d and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by P. We associate to Problem P an optimal control problem, denoted by Q. Then, using appropriate Tykhonov triples, governed by a nonlinear operator G and a convex K, we provide results concerning the well-posedness of problems P and Q. Our main results are Theorems 14 and 18, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem P, constructed with particular choices of G and K. We prove that Theorems 14 and 18 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results.

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Convergence Results for Optimal Control Problems Governed by Elliptic Quasivariational Inequalities

Cited by 10 publications

References 27 publications

Modelling, analysis, and numerical simulation of a spring-rods system with unilateral constraints

Modelling, analysis, and numerical simulation of a spring-rods system with unilateral constraints

Multi-Parameter Quantum Integral Identity Involving Raina’s Function and Corresponding q-Integral Inequalities with Applications

Tykhonov Well-posedness of a Heat Transfer Problem with Unilateral Constraints

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