Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

Casas, Eduardo; Mateos, Mariano; Rösch, Arnd

doi:10.1007/s10589-018-9979-0

Cited by 16 publications

(18 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, where the second inequality is obtained by virtue of Lemma 3.1 and integration by parts. Inserting the two obtained estimates into (19) yields the assertion after summation.…”

Section: Yet By Virtue Of Lemma 47 This Impliesmentioning

confidence: 99%

“…[8,13,14,17,22,23] for the former and [11,12,15,16,29,30] for the latter.Error estimates for PDE-constrained optimal control problems involving measures have been presented in [11,30,31,34,35]. For error estimates of further sparsity promoting optimal control problems with PDEs see for example [19,30]. The literature on error estimates for optimal control problems with controls in BV is rather limited.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer¹,

Mannel²,

Neitzel³

et al. 2019

Mathematical Control &Amp; Related Fields

View full text Add to dashboard Cite

Here, u ∈ V := H 1 0 (Ω) is the state and q ∈ Q := BV (Ω) is the control, where BV (Ω) denotes the space of functions of bounded variation (BV) on the interval Ω := (0, 1). The operator A is elliptic and α is a positive real number. The two finite element schemes that will be analyzed are identical in regard to the discretization of state and adjoint state, but they differ in the treatment of the control. In the variational discretization the control is not discretized, while in the second scheme the control is discretized by piecewise constant functions.The significance of the above control problem is given by the use of the BVseminorm q ′ M(Ω) in the objective. This favors piecewise constant controls with only a limited number of jumps, which makes this problem type interesting in many practical applications. The precise functional analytic setting will be provided in the next section.Optimal control problems with BV-controls defined in one space dimension are strongly related to control problems with measures as controls. Both BV optimal control problems and optimal control problems with measures have attracted significant research interest in the recent past, see, e.g. [8,13,14,17,22,23] for the former and [11,12,15,16,29,30] for the latter.Error estimates for PDE-constrained optimal control problems involving measures have been presented in [11,30,31,34,35]. For error estimates of further sparsity promoting optimal control problems with PDEs see for example [19,30]. The literature on error estimates for optimal control problems with controls in BV is rather limited. We are only aware of [14,18]. Error estimates and numerical analysis for inverse problems involving BV-functions are studied in [5,6]. Related discussion of ODE-constrained control problems involving discontinuous functions and their numerical analysis can be found in, e.g., [1,2,10,21,25,26,37,38].The main difficulty in deriving error estimates for the above problem is given by the fact that it lacks certain coercivity properties that are usually employed to obtain error estimates for the controls, for instance by suitably testing the first order necessary optimality conditions. Hence, only error estimates for the state and the adjoint state can be proven in a rather direct manner; these are, however, suboptimal. To obtain an error estimate for the control and also to improve the error estimates for state and adjoint state, we make use of a structural assumption on the Lagrange multiplierΦ arising from the convex subdifferential of the term q ′ M(Ω) . Specifically, we assume thatΦ, which is a C 2 function inΩ, has only finitely many global extreme points and that it exhibits quadratic growth near those points (i.e.,Φ ′′ = 0 near those points; see Assumption 4.4 and Assumption 4.5). Since the jump set of the optimal control is contained in the set of global extreme points ofΦ, see Corollary 1, this assumption implies that the optimal control admits only finitely many jumps, which is a rather typical situation in practice. In addition, it ensures ...

show abstract

“…, where the second inequality is obtained by virtue of Lemma 3.1 and integration by parts. Inserting the two obtained estimates into (19) yields the assertion after summation.…”

Section: Yet By Virtue Of Lemma 47 This Impliesmentioning

confidence: 99%

mentioning

confidence: 99%

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer¹,

Mannel²,

Neitzel³

et al. 2019

Mathematical Control &Amp; Related Fields

View full text Add to dashboard Cite

show abstract

“…In the second, so-called Tikhonov regularization term, a small positive number ν guarantees the existence of an optimal control f opt that minimizes the objective functional (9) for Ω ⊂ R q , q = 1, 2, 3, see Ref. [52]. If the desired state U d is not exactly realizable and ν = 0, then the functional ( 9) would not have a minimum.…”

Section: Optimal Controlmentioning

confidence: 99%

Control of traveling localized spots

Martens

Ryll

Löber

et al. 2021

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

Traveling localized spots represent an important class of self-organized two- dimensional patterns in reaction-diffusion systems. We study open-loop control intended to guide a stable spot along a desired trajectory with desired velocity. Simultaneously, the spot's concentration profile does not change under control. For a given protocol of motion, we first express the control signal analytically in terms of the Goldstone modes and the propagation velocity of the uncontrolled spot. Thus, detailed information about the underlying nonlinear reaction kinetics is unnecessary. Then, we confirm the optimality of this solution by demonstrating numerically its equivalence to the solution of a regularized, optimal control problem. To solve the latter, the analytical expressions for the control are excellent initial guesses speeding-up substantially the otherwise time-consuming calculations.

show abstract

“…In the second, so-called Tikhonov regularization term, a small but finite, positive value ν guarantees the existence of an optimal control f opt that minimizes the objective functional J (9) for Ω ⊂ R q , q = 1, 2, 3, see Ref. [51].…”

Section: Optimal Controlmentioning

confidence: 99%

Control of traveling localized spots

Martens¹,

Ryll²,

Löber³

et al. 2017

Preprint

View full text Add to dashboard Cite

Traveling localized spots represent an important class of selforganized two-dimensional patterns in reaction-diffusion systems. We study openloop control intended to guide a stable spot along a desired trajectory with desired velocity. Simultaneously, the spot's concentration profile does not change under control. For a given protocol of motion, we first express the control signal analytically in terms of the Goldstone modes and the propagation velocity of the uncontrolled spot. Thus, detailed information about the underlying nonlinear reaction kinetics is unnecessary. Then, we confirm the optimality of this solution by demonstrating numerically its equivalence to the solution of a regularized, optimal control problem. To solve the latter, the analytical expressions for the control are excellent initial guesses speeding-up substantially the otherwise timeconsuming calculations.

show abstract

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

Cited by 16 publications

References 25 publications

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Control of traveling localized spots

Control of traveling localized spots

Contact Info

Product

Resources

About