On approximation theorems for controllability of non-linear parabolic problems

Kumar, Anil; Joshi, M. C.; Pani, Amiya K.

doi:10.1093/imamci/dnl012

Cited by 5 publications

(9 citation statements)

References 13 publications

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“…Definition 3.2. Let the unconstrained problem (16) has a solution u n ∈ Y with z n ∈ Y as the corresponding mild solution of the fractional system (3), then (u n , z n ) is called the optimal pair of the problem (16).…”

Section: Definition 23 [3]mentioning

confidence: 99%

“…where Θ is a bounded domain in R k with smooth boundary ∂Θ and B = (0, b) × Θ, ∆ = (0, b) × ∂Θ, b > 0, A is the elliptic operator of second order. Kumar et al [16] studied the approximation theorems for controllability of parabolic differential equation which is of the form…”

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confidence: 99%

“…Let X and U be two separable Hilbert spaces, Z = L 2 [0, b ; X] and Y = L 2 [0, b ; U ] be the function spaces, defined on J = [0, b], 0 ≤ b < ∞. Motivated by the abovementioned facts and works described in the papers [15,16], in this manuscript, we consider a control system governed by nonlinear fractional differential equation which is of the form…”

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confidence: 99%

“…Therefore it is natural to extend the concept of optimal control problem for a system governed by the fractional differential equation. For such type of control problems, we obtained the results by using the technique of the paper [16]. However, the analysis is different as we have used the η-resolvent family, fractional calculus and first-order approximation method for the computation of Caputo's fractional derivative.…”

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confidence: 99%

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Approximation theorems for controllability problem governed by fractional differential equation

Dhayal¹,

Malik²,

Abbas³

et al. 2021

Evolution Equations &Amp; Control Theory

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In this manuscript, we discuss the optimal control problem for a nonlinear system governed by the fractional differential equation in a separable Hilbert space X. We utilize the fixed point technique and η-resolvent family to present the existence of control for the fractional system. The optimal pair is obtained as the limit of the optimal pair sequence of the unconstrained problem. Further, we derive some approximation results, which guarantee the convergence of the numerical method to optimal pair sequence. Finally, the main results are validated with the aid of an example.

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Section: Definition 23 [3]mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Approximation theorems for controllability problem governed by fractional differential equation

Dhayal¹,

Malik²,

Abbas³

et al. 2021

Evolution Equations &Amp; Control Theory

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“…The concept of controllability plays an important role in the analysis and design of control systems. Controllability of the deterministic and stochastic dynamical control system in infinite-dimensional spaces is well developed using different kinds of approaches, and the details can be found in various papers, see for example [1,3,6,9,10,14,25,26,27,31], etc and the references therein. From the mathematical point of view, in infinite dimensions, the problems of exact and approximate controllability are to be distinguished.…”

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Approximate controllability of a non-autonomous evolution equation in Banach spaces

Ravikumar¹,

Mohan²,

Anguraj³

2021

NACO

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In this paper, we consider a class of non-autonomous nonlinear evolution equations in separable reflexive Banach spaces. First, we consider a linear problem and establish the approximate controllability results by finding a feedback control with the help of an optimal control problem. We then establish the approximate controllability results for a semilinear differential equation in Banach spaces using the theory of linear evolution systems, properties of resolvent operator and Schauder's fixed point theorem. Finally, we provide an example of a non-autonomous, nonlinear diffusion equation in Banach spaces to validate the results we obtained.

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