New regularity results and improved error estimates for optimal control problems with state constraints

Casas, Eduardo; Mateos, Mariano; Vexler, Boris

doi:10.1051/cocv/2013084

Cited by 50 publications

(53 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It became clear to us that we could extend this approach to analyze the regularity of the Lagrange multiplier μ ν and the corresponding adjoint state. The reader is also referred to [10], where the authors have obtained recently a similar result for more general pointwise state constraints and a linear state equation.…”

Section: 2mentioning

confidence: 66%

See 1 more Smart Citation

Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls

Casas¹,

Tröltzsch²

2014

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

Abstract. An optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state. The main novelty of the paper is the presence of the L 1 -norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions. Second-order sufficient optimality conditions are analyzed. They are applied to show the convergence of optimal solutions for vanishing L 2 -regularization parameter for the control. The associated convergence rate is estimated.

show abstract

Section: 2mentioning

confidence: 66%

“…Then, u ν and λ ν belong to H 1 0 (Ω). The proof of this theorem is based on the following result that is proved in [16, Theorem 10.1 and equation (2.22)]; see also [10]. …”

Section: 2mentioning

confidence: 99%

Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls

Casas¹,

Tröltzsch²

2014

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Thus (5d) is satisfied. We have proven that (y * , u * , p * , µ * ) is a KKT point of (P ), i.e., (y * , u * , p * , µ * ) solves (5). It remains to show strong convergence of u + n → u * in L 2 (Ω).…”

Section: Convergence Towards Kkt Pointsmentioning

confidence: 92%

“…Assumption 4 (Quadratic growth condition (QGC)). Letū ∈ U ad be a control satisfying the first-order necessary optimality conditions (5). We assume that there exist β > 0 and > 0 such that the quadratic growth condition…”

Section: Convergence Towards Local Solutionsmentioning

confidence: 99%

An augmented Lagrange method for elliptic state constrained optimal control problems

Karl

Wachsmuth

2017

Comput Optim Appl

View full text Add to dashboard Cite

In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.

show abstract

“…Therefore, using [16, Theorems 4.4.3.7 and 5.1.1.4] we obtain (10). Notice that now we do not have the restriction s ≤ 3, since the right hand side of (13) is zero.…”

mentioning

confidence: 94%

Dirichlet control of elliptic state constrained problems

Mateos

Neitzel

2015

Comput Optim Appl

Self Cite

View full text Add to dashboard Cite

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 *

show abstract

New regularity results and improved error estimates for optimal control problems with state constraints

Cited by 50 publications

References 20 publications

Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls

Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls

An augmented Lagrange method for elliptic state constrained optimal control problems

Dirichlet control of elliptic state constrained problems

Contact Info

Product

Resources

About