Backstepping-based boundary control design for a fractional reaction diffusion system with a space-dependent diffusion coefficient

Chen, Juan; Cui, Baotong; Chen, YangQuan

doi:10.1016/j.isatra.2018.04.013

Cited by 29 publications

(31 citation statements)

References 24 publications

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“…Inter alia, many other systems, including the heterogeneity induced by macromolecular crowding in the cytoplasm and nucleoplasm or the accumulation of large cellular organelles in a perinuclear region [36], the intermittent motion of lipids in a protein-crowded membrane [37,38], and the scaling exponent of a virtual movement [39] also lead to similar effects. Meanwhile, it has been verified that such a position-dependence can result in the violation of classic laws [40][41][42][43][44][45][46], such as substantial deviations from the Stokes-Einstein law for protein diffusion in an Escherichia coli cell. Following these observations, Cherstvy et al put forward alternative theories and various non-ergodic and anomalously diffusive regimes induced by different position-dependent diffusion coefficients [36,44].…”

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confidence: 99%

“…Following these observations, Cherstvy et al put forward alternative theories and various non-ergodic and anomalously diffusive regimes induced by different position-dependent diffusion coefficients [36,44]. Smyshlyaev and Chen applied a position-dependent diffusion coefficient to the boundary feedback control of a diffusive system and proved the Mittag-Leffler stability of the system [45,46]. Several other works have since focussed on position-dependent diffusive systems [47][48][49][50], to name but a few.Differing from a constant diffusion coefficient, the presence of a position-dependent diffusivity involves the problem of how to interpret multiplicative noise in a stochastic equation, particularly, a noise-induced drift, which varies by choosing different integral forms, such as Itô, Stratonovich and isothermal integrals [44,51].…”

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confidence: 99%

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Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

Mei

et al. 2020

New J. Phys.

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This work focuses on the dynamics of particles in a confined geometry with position-dependent diffusivity, where the confinement is modelled by a periodic channel consisting of unit cells connected by narrow passage ways. We consider three functional forms for the diffusivity, corresponding to the scenarios of a constant (D 0 ), as well as a low (D m ) and a high (D d ) mobility diffusion in cell centre of the longitudinally symmetric cells. Due to the interaction among the diffusivity, channel shape and external force, the system exhibits complex and interesting phenomena. By calculating the probability density function, mean velocity and mean first exit time with the Itô calculus form, we find that in the absence of external forces the diffusivity D d will redistribute particles near the channel wall, while the diffusivity D m will trap them near the cell centre. The superposition of external forces will break their static distributions. Besides, our results demonstrate that for the diffusivity D d , a high dependence on the x coordinate (parallel with the central channel line) will improve the mean velocity of the particles. In contrast, for the diffusivity D m , a weak dependence on the x coordinate will dramatically accelerate the moving speed. In addition, it shows that a large external force can weaken the influences of different diffusivities; inversely, for a small external force, the types of diffusivity affect significantly the particle dynamics. In practice, one can apply these results to achieve a prominent enhancement of the particle transport in two-or three-dimensional channels by modulating the local tracer diffusivity via an engineered gel of varying porosity or by adding a cold tube to cool down the diffusivity along the central line, which may be a relevant effect in engineering applications. Effects of different stochastic calculi in the evaluation of the underlying multiplicative stochastic equation for different physical scenarios are discussed.New J. Phys. 22 (2020) 053016 Y Li et al confined spaces, transport of particles in nanopores, zeolites and gel networks [8-10], or protein diffusion in the mammalian cell cytoplasm [11]. In many cases, such structured environments can be viewed as confined channels with different boundaries and properties [12][13][14]. The behaviours of systems in these confined channels have been studied by virtue of the analytical Fick-Jacobs (FJ) equation and numerical simulations of the dynamic equations [15][16][17][18][19][20]. Results for symmetric periodic channels indicate that the mean transport velocity may be proportional to large external forces in the confined environment [19][20][21][22]. For asymmetric channels, it was found that the shape of the confinement contributes to the moving direction and mean displacement [20,23], which can be applied to fine separation of mixtures and also the design of many devices [24,25]. Time-dependent and deformable boundaries were discussed to explore the activity in living cells [26,27]. In addition, the interaction...

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confidence: 99%

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confidence: 99%

Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

Mei

et al. 2020

New J. Phys.

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“…□ Remark 2 Contrary to the work [7, 29, 30], herein the Mittag‐Leffler stability of the fractional plant is proved with some constraints of system parameters by using the fractional Lyapunov method [22] and a useful lemma reported in [38]. From Definition 1, we find that this Mittag‐Leffler stable system is also asymptotical stable (see [22, Remark 4.4], [19] for more details).…”

Section: Full‐state Feedback Controlmentioning

confidence: 89%

“…In this case, some special techniques are needed to overcome the difficulties in solving kernels. Moreover, research on fractional reaction diffusion systems governed by time fractional PDEs [16, 17], especially the time fractional PDE systems with space‐dependent coefficients [18], has recently focused on the boundary feedback control problem (see [19–21]) since a breakthrough on Mittag‐Leffler stability (fractional Lyapunov stability) was achieved by Li [22]. For fractional ODE systems, much attention is given to the solutions/numerical solutions of fractional linear ODEs and extending synchronisation on fractional non‐linear ODEs (chaotic fractional systems) in [23, 24].…”

Section: Introductionmentioning

confidence: 99%

State and output feedback boundary control of time fractional PDE–fractional ODE cascades with space‐dependent diffusivity

Chen¹,

Tepljakov²,

Petlenkov³

et al. 2020

IET Control Theory & Appl

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This study considers the boundary control problem of a time fractional partial differential equation (PDE) – fractional ordinary differential equation (ODE) cascaded system with spatially varying diffusivity and Dirichlet connection. Using the backstepping transformation, a full‐state feedback law is obtained. Meanwhile, the well‐posedness of kernel functions in this transformation is established theoretically. Subsequently, a Mittag‐Leffler convergent boundary observer is derived, which composes with the proposed state feedback law to enable stabilisation by output feedback. With the designed state and output feedback controllers, the Mittag‐Leffler stability of the closed‐loop system is then proved by the fractional Lyapunov method. Finally, the results of numerical simulations are provided for fractional cascaded plants when neither kernel ODE nor kernel PDE has an explicit solution.

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“…Dai et al [13] considered the P-type ILC issue for multi-input multi-output DPSs with singular matrix coefficients, which described by coupled partial differential equations with parabolic and hyperbolic type. The distributed parameter systems have been widely used to describe some engineering applications, such as industrial automation [14], building environment control [15], and fractional reaction diffusion processes [16], [17]. Moore and Chen [18] presented an ILC algorithm for a class of periodic distributed parameter systems with the wireless sensor network, which is described the agricultural irrigation process.…”

Section: Introductionmentioning

confidence: 99%

A PD-Type Iterative Learning Algorithm for Semi-Linear Distributed Parameter Systems With Sensors/Actuators

et al. 2019

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In this paper, the control problem of semi-linear distributed parameter system (DPS) with sensors/actuators is considered using iterative learning control (ILC) method. During the learning process, the output signals of the system exist random data dropout which is described as a Bernoulli random variable. Then, a novel intermittent updating PD-type ILC algorithm is proposed on the basis of the available output information. In this kind ILC algorithm, the scheme only updates its control signal when the output signal is successfully transmitted. Hence, the intermittent updating PD-type ILC algorithm is presented for a semi-linear parabolic DPS based on single sensor and single actuator, and the convergence condition of the output error is obtained by using Bellman-Gronwall lemma and semigroup theory under some given assumptions. Secondly, this kind PD-type ILC law is extended to control the semi-linear DPS with multiple sensors and multiple actuators. Lastly, one example is given to demonstrate the effectiveness of the proposed scheme.

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Backstepping-based boundary control design for a fractional reaction diffusion system with a space-dependent diffusion coefficient

Cited by 29 publications

References 24 publications

Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

State and output feedback boundary control of time fractional PDE–fractional ODE cascades with space‐dependent diffusivity

A PD-Type Iterative Learning Algorithm for Semi-Linear Distributed Parameter Systems With Sensors/Actuators

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