Penalization of Dirichlet optimal control problems

Casas, Eduardo; Mateos, Mariano; Raymond, Jean‐Pierre

doi:10.1051/cocv:2008049

Cited by 51 publications

(39 citation statements)

References 15 publications

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“…By integrating the square of the previous estimate on ]0, T [ and using (9), we obtain the desired estimate.…”

Section: The State Equationmentioning

confidence: 99%

“…This penalization is the key argument for the numerical analysis of the discretization that we intend to perform in a forthcoming work. The Robin penalization issue has been considered for elliptic problems for the linear case in [6] and for the nonlinear case in [9]; we also refer to [16][17][18] for the extension to the Navier-Stokes system. Moreover, and in a different spirit, the Dirichlet control may be viewed as a limit of a Robin control, for small ε, which seems currently used in thermics (see [26]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dirichlet boundary control for a parabolic equation with a final observation I: A space–time mixed formulation and penalization

Belgacem

Bernardi

Fekih

2011

Asymptotic Analysis

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We are interested in the optimal control problem of a parabolic equation with no state constraints, where the quadratic cost functional involves a final observation and the control variable is a Dirichlet boundary condition. Practical considerations lead us to use Dirichlet controls with no more regularity than square integrability, which arise some technical difficulties in the mathematical analysis. After setting the state equation and the related adjoint equation we fit the two coupled equations into a mixed space-time variational framework. The resulting saddle-point problem turns out to be well posed. We then use a Robin penalization on the Dirichlet control which enables us to re-transcript the mixed problem in a form better suited to numerical computations. We analyze and establish the convergence when the penalty parameter tends to zero, first without additional smoothness assumptions on the optimal control and then for smooth controls.

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“…By integrating the square of the previous estimate on ]0, T [ and using (9), we obtain the desired estimate.…”

Section: The State Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dirichlet boundary control for a parabolic equation with a final observation I: A space–time mixed formulation and penalization

Belgacem

Bernardi

Fekih

2011

Asymptotic Analysis

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“…Applying the logarithm, this is equivalent to 2 ln(μ n ) < μ 2 n (T − τ ) − ln (2). The right-hand side satisfies, for n ≥ 4,…”

Section: Some Preparatory Inequalities Frommentioning

confidence: 99%

“…For elliptic Dirichlet boundary control problems, this issue was discussed by Casas et al [2], who proved convergence and error estimates w.r. to the penalization parameter that corresponds to α. In the control of parabolic PDEs, the optimality conditions for Dirichlet controls can also be achieved by passing to the limit, α = β → ∞, in the optimality conditions for the penalized Robin problem (P α ) (see Arada et al [1]).…”

Section: Application-approximation Of Dirichlet Boundary Controlsmentioning

confidence: 99%

Some aspects of reachability for parabolic boundary control problems with control constraints

Dhamo¹,

Tröltzsch²

2009

Comput Optim Appl

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“…Since ϕ r (u) = 0 on Γ, we have that ∂ n ϕ r (u)(χ j ) = 0 (see [9,Lemma A.2] and [6, §4]) and ∂ n ϕ r (u) ∈ C(Γ). This compatibility condition is enough (cf.…”

mentioning

confidence: 99%

Dirichlet control of elliptic state constrained problems

Mateos

Neitzel

2015

Comput Optim Appl

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We study a state constrained Dirichlet optimal control problem and derive a priori error estimates for its finite element discretization. Additional control constraints may or may not be included in the formulation. The pointwise state constraints are prescribed in the interior of a convex polygonal domain. We obtain a priori error estimates for the L 2 (Γ)-norm of order h 1−1/p for pure state constraints and h 3/4−1/(2p) when additional control constraints are present. Here, p is a real number that depends on the largest interior angle of the domain. Unlike in e.g. distributed or Neumann control problems, the state functions associated with L 2 -Dirichlet control have very low regularity, i.e. they are elements of H 1/2 (Ω). By considering the state constraints in the interior we make use of higher interior regularity and separate the regularity limiting influences of the boundary on the one-hand, and the measure in the right-hand-side of the adjoint equation associated with the state constraints *

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Penalization of Dirichlet optimal control problems

Cited by 51 publications

References 15 publications

Dirichlet boundary control for a parabolic equation with a final observation I: A space–time mixed formulation and penalization

Dirichlet boundary control for a parabolic equation with a final observation I: A space–time mixed formulation and penalization

Some aspects of reachability for parabolic boundary control problems with control constraints

Dirichlet control of elliptic state constrained problems

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