On the optimal control of the Schlögl-model

Buchholz, R.; Engel, Harald; Kammann, Eileen; Tröltzsch, Fredi

doi:10.1007/s10589-013-9550-y

Cited by 54 publications

(72 citation statements)

References 23 publications

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“…[4]. In this way, an equation with monotone nonlinearity is obtained where the known results on existence, uniqueness, and regularity [16,Theorem 5.5] or in [5] can be applied.…”

Section: Well Posedness Of State Equation and Control Problemmentioning

confidence: 99%

See 1 more Smart Citation

Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

Casas

Tröltzsch

2015

Vietnam J. Math.

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Second order sufficient optimality conditions are considered for a simplified class of semilinear parabolic equations with quadratic objective functional including distributed and terminal observation. Main emphasis is laid on problems where the objective functional does not include a Tikhonov regularization term. Here, standard second order conditions cannot be expected to hold. For this case, new second order conditions are established that are based on different types of critical cone. Depending on the choice of this cone, the second order conditions are sufficient for local minima that are weak or strong in the sense of calculus of variations.

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“…[4]. In this way, an equation with monotone nonlinearity is obtained where the known results on existence, uniqueness, and regularity [16,Theorem 5.5] or in [5] can be applied.…”

Section: Well Posedness Of State Equation and Control Problemmentioning

confidence: 99%

“…Then the following differentiability properties can be proved, see [4] or the more general version [9, Theorem 2.2].…”

Section: Well Posedness Of State Equation and Control Problemmentioning

confidence: 99%

Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems

Casas

Tröltzsch

2015

Vietnam J. Math.

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“…Several control strategies have been developed for purposeful manipulation of wave dynamics as the application of closed-loop or feedback-mediated control loops with and without delays [8][9][10][11] and open-loop control that includes external spatio-temporal forcing [10,[12][13][14], optimal control [15][16][17], and control by imposed geometric constraints and heterogeneities on the medium [18,19]. While feedback-mediated control relies on continuously monitoring of the system's state, open-loop control is based on a detailed knowledge of the system's dynamics and its parameters.…”

Section: Introductionmentioning

confidence: 99%

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

Ryll

Löber

Martens

et al. 2016

Understanding Complex Systems

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Abstract. This work deals with the position control of selected patterns in reactiondiffusion systems. Exemplarily, the Schlögl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control to so-called sparse optimal control that results in very localized control signals and allows the analysis of second order optimality conditions.

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“…In Buchholz et al (2013), the well-posedness of the system governed by (3) is studied for homogeneous Neumann boundary conditions. It is shown that for initial data in L ∞ (0, L), the system has a unique weak solution that is continuous for t > 0.…”

Section: Existence and Uniqueness Of The Solutionsmentioning

confidence: 99%

“…The traveling wave solutions connect the two stable constant stationary states. The problem to steer associated wave fronts to rest by distributed optimal control methods was considered in Buchholz, Engel, Kammann, and Tröltzsch (2013) for the Schlögl model and in Casas, Ryll, and Tröltzsch (2013) for the FitzHugh-Nagumo system, where spiral waves occur. In the present paper, we propose a boundary control law that stabilizes the system exponentially fast to a desired orbit.…”

Section: Introductionmentioning

confidence: 99%