Dynamic boundary systems with boundary feedback and population system with unbounded birth process

Mei, Zhan‐Dong; Peng, Jigen

doi:10.1002/mma.3175

Cited by 8 publications

(17 citation statements)

References 23 publications

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“…Remark 6.9 We observe that the conditions of the above corollary are the same as Corollary 14 in [30] for i = 1 and Corollary 4.8 in [26] for i = 2. This means that if the conditions hold, the population systems are uniformly exponentially stable both with and without death caused by pregnancies.…”

Section: Application To Population Dynamical Systemsmentioning

confidence: 76%

“…By assumption, it is obtained that (A, B, K) generates a regular linear system and (A, B) generates an abstract linear control system. It follows from the proof of Theorem 3.4 in [26] that boundary control system…”

Section: Feedbackmentioning

confidence: 95%

“…The main difficulty of population dynamical systems with unbounded birth process lies in that the unboundedness makes Desch-Schappacher perturbation theorem invalid. In the recent paper [26], we solved such open problem by using feedback theory of the regular linear system developed by Weiss [40]. In [27], we proved the well-posedness of system with L being unbounded and K being bounded by our admissible invariable theorem developed in [24].…”

Section: Introductionmentioning

confidence: 93%

“…Theorem 6.1 [26] The pair (A, B) is an abstract linear control system. The triple (A, B, K 1 ) and (A, B, K 2 ) generate regular linear systems.…”

mentioning

confidence: 99%

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A Class of Linear Boundary Systems with Delays in State, Input and Boundary Output

Mei¹,

Peng²

2015

Preprint

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In this paper, we consider a class of linear boundary systems with delays in state, input and boundary output. We prove the well-posedness and derive some spectral properties of linear system with delayed boundary feedback under some regularity conditions. Moreover, we show the regularity of linear boundary systems with delays in state and boundary output. With the above results, the regularity of linear boundary systems with delays in state, input and boundary output is verified. As applications, we prove the well-posedness and the asymptotic behavior of population systems with bounded and unbounded birth processes "B 1 (t) = ∞ 0 0 −r β 1 (σ, a)u(t − τ, a)dσda" and "B 2 (t) = ∞ 0 β 2 (a)u(t − τ, a)da", and the well-posedness of population systems with death caused by harvesting.

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Section: Application To Population Dynamical Systemsmentioning

confidence: 76%

Section: Feedbackmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 93%

“…Theorem 6.1 [26] The pair (A, B) is an abstract linear control system. The triple (A, B, K 1 ) and (A, B, K 2 ) generate regular linear systems.…”

mentioning

confidence: 99%

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A Class of Linear Boundary Systems with Delays in State, Input and Boundary Output

Mei¹,

Peng²

2015

Preprint

Self Cite

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“…There is enormous literature dealing with population equation with delay birth process; see e.g. [14,15,16,24]. Recently, Liu et al [13] have studied a spatially and size structured-population model as following…”

mentioning

confidence: 99%

A spatially and size-structured population model with unbounded birth process

Boulouz

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we consider a spatially and size structured population model with unbounded birth process. Firstly, the model is transformed into a closed-loop system, and hence the well-posedness is established by using the feedback theory of regular linear systems. Moreover, the solution to the resulting closed-loop system is given by a perturbed semigroup. Secondly, we give a condition on birth and death rates in such a way that the solution decays exponentially. To do this, we show that the semigroup solution is positive and hence we derive a characterization of exponential stability due to the technique tools of positive semigroups. We mention that our results extend a previous work in [D. Yan and X. Fu, Comm. Pure Appl. Anal. 15 (2016), 637–655] to the unbounded situation.</p>

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Controllability under positivity constraints of a size‐structured population model with delayed birth process

El Gantouh,

Fkirine,

Simpore

2023

Math Methods in App Sciences

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In this paper, we study the approximate controllability, under positivity constraints on the control and state, of a size‐structured population system with time‐delayed behavior. In this model, the birth term is non‐local in size and time, which describes the time‐delay in the recruitment process in newborn individuals population. Our control function is localized at birth and corresponds to a supply of newborn individuals. First, we establish the well‐posedness and positivity property of the model. Second, and more importantly, we derive necessary and sufficient conditions for approximate controllability under positivity constraints. The method relies on the feedback theory of infinite‐dimensional positive systems.

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Dynamic boundary systems with boundary feedback and population system with unbounded birth process

Cited by 8 publications

References 23 publications

A Class of Linear Boundary Systems with Delays in State, Input and Boundary Output

A Class of Linear Boundary Systems with Delays in State, Input and Boundary Output

A spatially and size-structured population model with unbounded birth process

Controllability under positivity constraints of a size‐structured population model with delayed birth process

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