An Extension of Pontryagin’s Principle for State-Constrained Optimal Control of Semilinear Elliptic Equations and Variational Inequalities

Bonnans, Frédéric; Casas, Eduardo

doi:10.1137/s0363012992237777

Cited by 72 publications

(40 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same results can be obtained using other related but different techniques; for instance, see the results in [4] and [9]. However, we would like to emphasize two features of the strategy we have chosen:…”

supporting

confidence: 52%

Some optimal control problems for a two-phase field model of solidification

2009

View full text Add to dashboard Cite

In this paper we deal with some optimal control problems for a solidification phase field model of metallic alloys. The model allows crystallizations of two kinds, each one described by its own phase field. Accordingly, the state is the triplet (τ, u, v), where τ is the temperature and u and v are phase field functions. The optimality conditions for the optimal control problems considered in this work are obtained by using the Dubovitskii-Milyutin formalism.

show abstract

supporting

confidence: 52%

Some optimal control problems for a two-phase field model of solidification

2009

View full text Add to dashboard Cite

show abstract

“…Recall that problem (9) also admits a unique solution u * ∈ U ad which satisfies (3). Since with the previous proposition the requirements of Theorem 2.4 are satisfied for the setting of this section one immediately gets (9) and (11), respectively.…”

Section: Application To Control Of Elliptic Equationsmentioning

confidence: 70%

A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case

Hinze

2005

Comput Optim Applic

459

411

View full text Add to dashboard Cite

A new discretization concept for optimal control problems with control constraints is introduced which utilizes for the discretization of the control variable the relation between adjoint state and control. Its key feature is not to discretize the space of admissible controls but to implicitly utilize the first order optimality conditions and the discretization of the state and adjoint equations for the discretization of the control. For discrete controls obtained in this way an optimal error estimate is proved. The application to control of elliptic equations is discussed. Finally it is shown that the new concept is numerically implementable with only slight increase in program management. A numerical test confirms the theoretical investigations.

show abstract

“…Then (12) gives q i η i = |η i | ≥ 0. (14)-(18) ⇒ (11)-(13): Due to (15) and (18) we have λ i q i ≤ 0, whenever |q i | = 1, which, in view of (19), implies (13). Because of (15), we have…”

Section: Differentiability For a Finite-dimensional VI Of The Second mentioning

confidence: 91%

“…For problems involving variational inequalities of the second kind, only weak results have been obtained in the past, due to the very general structure (see, e.g., [4,5,13,14]). In [15], a special class of problems was investigated, where a richer structure of the nondifferentiability was exploited.…”

Section: Introductionmentioning

confidence: 99%

Strong Stationarity Conditions for a Class of Optimization Problems Governed by Variational Inequalities of the Second Kind

Reyes

Meyer

2015

J Optim Theory Appl

View full text Add to dashboard Cite

We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primaldual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional problems and for problems in function spaces, under suitable assumptions on the active set. A characterization of Bouligand and strong stationary points is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems.

show abstract

An Extension of Pontryagin’s Principle for State-Constrained Optimal Control of Semilinear Elliptic Equations and Variational Inequalities

Cited by 72 publications

References 13 publications

Some optimal control problems for a two-phase field model of solidification

Some optimal control problems for a two-phase field model of solidification

A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case

Strong Stationarity Conditions for a Class of Optimization Problems Governed by Variational Inequalities of the Second Kind

Contact Info

Product

Resources

About