Optimal Control and Sensitivity Analysis of a Fractional Order TB Model

Rosa, Silvério; Torres, Delfim F. M.

doi:10.19139/soic.v7i3.836

Cited by 20 publications

(25 citation statements)

References 30 publications

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“…When α = 1, the conditions (9) and (12) are verified, and we retrieve from our Theorem 1 the results established in [17,18] about the exponential stability of system (8) on Ω, which is equivalent to +∞ 0 S(t)z 2 dt < ∞, ∀z ∈ L 2 (Ω). In our next theorem, we provide sufficient conditions that guaranty the strong stability of the fractional order differential system (8). The result generalizes the asymptotic result established by Matignon for finite dimensional state spaces, where the dynamics of the system A is considered to be a matrix with constant coefficients in R n [23].…”

Section: Stability Of Time Fractional Differential Systemsmentioning

confidence: 78%

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The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

2020

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We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics is symmetric and uniformly elliptic and by using the properties of the Mittag-Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.

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Section: Stability Of Time Fractional Differential Systemsmentioning

confidence: 78%

“…The next theorem provides necessary and sufficient conditions for exponential stability of the abstract fractional order differential system (8).…”

Section: Stability Of Time Fractional Differential Systemsmentioning

confidence: 99%

The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

2020

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“…, u n ] T , such that the elements of A and U are unknown. By utilizing (5), (9), (13), and the initial conditions given in (3), for s = 1, 2, . .…”

Section: Numerical Solution Of Problem (1)-(3)mentioning

confidence: 99%

“…where u n (t) has been given in (14). For obtaining an error estimate for the state function x(·), we suppose that D α x(·) ∈ C 3 [0, t f ] and is approximated using (13). Also, suppose that D T Ψ(t)…”

Section: Error Estimatementioning

confidence: 99%

“…Besides providing nontrivial generalizations of the classical theory, it opens doors to new and interesting modern scientific problems [10,11]. Given the complexity of the problems, numerical methods are indispensable [12,13,14]. f (t, x(t), u(t))dt (1) subject to the fractional dynamic system D α x(t) = g (t, x(t), D α1 x(t), D α2 x(t), .…”

Section: Introductionmentioning

confidence: 99%

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A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati

Lima

Torres

2019

Communications in Nonlinear Science and Numerical Simulation

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We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann-Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann-Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.Recently, numerical methods based on modified hat functions have shown to provide an effective way for solving fractional differential equations [15,16]. Here, for the first time in the literature, we introduce such a method for solving a general class of fractional optimal control problems. More precisely, in this work we begin by considering the following fractional optimal control problem (FOCP):

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Lyapunov functions for fractional-order systems in biology: Methods and applications

Boukhouima

Hattaf

Lotfi

et al. 2020

Chaos, Solitons & Fractals

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We prove new estimates of the Caputo derivative of order α ∈ (0, 1] for some specific functions. The estimations are shown useful to construct Lyapunov functions for systems of fractional differential equations in biology, based on those known for ordinary differential equations, and therefore useful to determine the global stability of the equilibrium points for fractional systems. To illustrate the usefulness of our theoretical results, a fractional HIV population model and a fractional cellular model are studied. More precisely, we construct suitable Lyapunov functionals to demonstrate the global stability of the free and endemic equilibriums, for both fractional models, and we also perform some numerical simulations that confirm our choices.

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Optimal Control and Sensitivity Analysis of a Fractional Order TB Model

Cited by 20 publications

References 30 publications

The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

A numerical approach for solving fractional optimal control problems using modified hat functions

Lyapunov functions for fractional-order systems in biology: Methods and applications

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