Hybrid optimal control problems for a class of semilinear parabolic equations

Court, Sébastien; Kunisch, Karl; Pfeiffer, Laurent

doi:10.3934/dcdss.2018060

Cited by 6 publications

(4 citation statements)

References 22 publications

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“…Reactiondiffusion equations cover a variety of particular cases with important applications in mathematical physics, and in biological models such as the Schlögl model or the FitzHugh-Nagumo system [13]. The problem of optimal control of reaction-diffusion equations has been recently the topic of many works of (classical) numerical analysis: see, e.g., [9,15,20,22,41,42].…”

Section: Reaction-diffusion Equationsmentioning

confidence: 99%

Guaranteed Optimal Reachability Control of Reaction-Diffusion Equations Using One-Sided Lipschitz Constants and Model Reduction

Coent

Fribourg

2020

Cyber Physical Systems. Model-Based Design

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We show that, for any spatially discretized system of reactiondiffusion, the approximate solution given by the explicit Euler timediscretization scheme converges to the exact time-continuous solution, provided that diffusion coefficient be sufficiently large. By "sufficiently large", we mean that the diffusion coefficient value makes the one-sided Lipschitz constant of the reaction-diffusion system negative. We apply this result to solve a finite horizon control problem for a 1D reactiondiffusion example. We also explain how to perform model reduction in order to improve the efficiency of the method.

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Section: Reaction-diffusion Equationsmentioning

confidence: 99%

Guaranteed Optimal Reachability Control of Reaction-Diffusion Equations Using One-Sided Lipschitz Constants and Model Reduction

Coent

Fribourg

2020

Cyber Physical Systems. Model-Based Design

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“…Let us mention that the change of variables has a similar structure to the one used for tackling a class of hybrid optimal control problems. For such problems, the time at which a switch in the state equation occurs has to be optimized (see [CKP17] and [CKP18]). The switching time plays an analogous role to the geometric parameter considered in the present article.…”

Section: Introductionmentioning

confidence: 99%

Optimal control problem for viscous systems of conservation laws, with geometric parameter, and application to the Shallow-Water equations

Court

Kunisch

Pfeiffer

2019

Interfaces Free Bound.

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A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters.Notation. Most of the variables introduced in this paper are multi-dimensional. The functional spaces in which they lie is denoted in bold, as for example L p (Ω) = [L p (Ω)] m , W s,p = [W s,p (Ω)] m , H s = [H s (Ω)] m , for some integer m ∈ N, and for p ≥ 1, s ≥ 0.

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“…In future work, we will utilize the FD method to impulsive control [16], sliding mode control [23,36], tracking control [30], a class of Convolutional Broad Network [43] and neural networks [20,34,35], and meanwhile, we will apply the FD method for switched systems [38] and multi-agent systems [19,48] under the condition of considering the impulsive effects [17]. In addition, optimal control technique [4,9,10] will be attempted to solve the parameters.…”

mentioning

confidence: 99%

Event-based fault detection for interval type-2 fuzzy systems with measurement outliers

Li¹,

Han²,

Lu³

2021

Discrete &Amp; Continuous Dynamical Systems - S

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This paper investigates the event-based fault detection (FD) problem for a category of discrete-time interval type-2 fuzzy systems with measurement outliers. For the sake of decreasing the utilization of limited communication bandwidth, an event-based mechanism is introduced. Based on the saturation function technique, a novel event-based FD observer is first designed to reduce the influence of outliers in the dynamic systems. Then, on the basis of Lyapunov stability theory, sufficient conditions are provided to ensure that the error system satisfies the H∞ performance and the H∞ fault performance in different cases, respectively. In contrast to the existing event-based FD results, the false alarm, which is induced by measurement outliers, can be effectively avoided by the designed FD observer with saturation function. Lastly, some simulation results are given to verify the effectiveness of the method presented in this paper.

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Hybrid optimal control problems for a class of semilinear parabolic equations

Cited by 6 publications

References 22 publications

Guaranteed Optimal Reachability Control of Reaction-Diffusion Equations Using One-Sided Lipschitz Constants and Model Reduction

Guaranteed Optimal Reachability Control of Reaction-Diffusion Equations Using One-Sided Lipschitz Constants and Model Reduction

Optimal control problem for viscous systems of conservation laws, with geometric parameter, and application to the Shallow-Water equations

Event-based fault detection for interval type-2 fuzzy systems with measurement outliers

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