Solving a class of Hamilton-Jacobi-Bellman equations using pseudospectral methods

Mehrali-Varjani, Mohsen; Shamsi, M.; Malek, Alaeddin

doi:10.14736/kyb-2018-4-0629

Cited by 6 publications

(6 citation statements)

References 33 publications

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“…As an optimal control problem Mehrali-Varjani et al 18 solved a class of Hamilton Jacobi-bellman equations using pseudospectral methods in the year 2018. Abbasi and Malek 19 presented hyperthermia cancer therapy by domain decomposition methods using strongly continuous semigroups in the year 2019.…”

Section: Introductionmentioning

confidence: 99%

Optimal controlling of anti-TGF-$$\beta$$ and anti-PDGF medicines for preventing pulmonary fibrosis

Bahram Yazdroudi,

Malek

2023

Sci Rep

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In the repair of injury, some transforming growth factor-$$\beta$$ β s (TGF-$$\beta$$ β s) and platelet-derived growth factors (PDGFs) bind to fibroblast receptors as ligands and cause the differentiation of fibroblasts into myofibroblasts. When the injury repair is repeated, the myofibroblasts proliferate excessively, forming fibrotic tissue. We goal to control myofibroblasts proliferation and apoptosis with anti-transforming growth factor-$$\beta$$ β (anti-TGF-$$\beta$$ β ) and anti-platelet-derived growth factor (anti-PDGF) medicines. The novel optimal regulator control problem with two controls (medicines) is proposed to simulate how to the preventing pulmonary fibrosis. Idiopathic pulmonary fibrosis (IPF) consists of restoring a system of cells, protein, and tissue networks with injury and scar. Myofibroblasts proliferation back to its equilibrium position after it has been disturbed by abnormal repair. Thus, the optimal regulator control problem with a parabolic partial differential equation as a constraint, zero flux boundary, and given specific initial conditions, is considered. The myofibroblast diffusion equation stands as a governing dynamic system while the objective function is the summation of myofibroblast, anti-TGF-$$\beta$$ β and anti-PDGF medicines for the fixed final time. Here, myofibroblast is a nonlinear state of time while anti-TGF-$$\beta$$ β and anti-PDGF are two unknown control functions. In order to solve the corresponding problem a weighted Galerkin method is used. Firstly, we convert the myofibroblast diffusion equation to a system of ordinary differential equations using the Lagrangian interpolation polynomials defined at Gauss-Lobatto integration points. Secondly, by the calculus of variations, the optimal control problem is solved successfully using canonical Hamiltonian and extended Riccati equations. Numerical results are given, and the plots are depicted. Moreover, solutions to the problem in which there is no control are compared. Numerical results show that, over time, the myofibroblast increases and then remains constant when there is no control. In contrast, the current solution decreases and vanishes after 300 days by prescribing controller medicines for anti-TGF-$$\beta$$ β and anti-PDGF. The optimal strategy proposed in this paper helps practitioners to reduce myofibroblasts by controlling both anti-TGF-$$\beta$$ β and anti-PDGF medicines.

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Section: Introductionmentioning

confidence: 99%

Optimal controlling of anti-TGF-$$\beta$$ and anti-PDGF medicines for preventing pulmonary fibrosis

Bahram Yazdroudi,

Malek

2023

Sci Rep

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“…In 2018, Malek and Varjani solved a class of Hamilton Jacobi-bellman equations using pseudospectral methods [14]. In 2019, Malek and Abbasi introduced hyperthermia cancer therapy by domain decomposition methods using strongly continuous semigroups [15].…”

Section: Introductionmentioning

confidence: 99%

“…ðnÀ 1Þ�1: ð16ÞI is identity matrix, b j is a column vector of zeros and known boundary values which are added with a fixed value of c m .2.1.3 Solution of the linear-quadratic regulator.The minimization of the performance index J will be done using Pontryagin's minimum principle[14,29]. The Hamiltonian is HðmðtÞ; UðtÞ; λðtÞ; tÞ ¼ mðtÞ T mðtÞ þ UðtÞ T UðtÞ þλ T ðtÞðAmðtÞ þ BUðtÞ þ bÞ;…”

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confidence: 99%

Optimal control of TGF-β to prevent formation of pulmonary fibrosis

Yazdroudi

Malek

2022

PLoS ONE

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In this paper, three optimal control problems are proposed to prevent forming lung fibrosis while control is transforming growth factor-β (TGF-β) in the myofibroblast diffusion process. Two diffusion equations for fibroblast and myofibroblast are mathematically formulated as the system’s dynamic, while different optimal control model problems are proposed to find the optimal TGF-β. During solving the first optimal control problem with the regulator objection function, it is understood that the control function gets unexpected negative values. Thus, in the second optimal control problem, for the control function, the non-negative constraint is imposed. This problem is solved successfully using the extended canonical Hamiltonian equations with no flux boundary conditions. Pontryagin’s minimum principle is used to solve the related optimal control problems successfully. In the third optimal control problem, the fibroblast equation is added to a dynamic system consisting of the partial differential equation. The two-dimensional diffusion equations for fibroblast and myofibroblast are transferred to a system of ordinary differential equations using the central finite differences explicit method. Three theorems and two propositions are proved using extended Pontryagin’s minimum principle and the extended Hamiltonian equations. Numerical results are given. We believe that this optimal strategy can help practitioners apply some medication to reduce the TGF-β in preventing the formation of pulmonary fibrosis.

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“…Thus, we proposed a hybrid method based on first discretize, and then optimize (direct method). In section ''Numerical method,'' pseudospectral method (see, for example, Betts et al (2016); Ma et al (2017); Malek and Momeni-Masuleh (2008); Mehrali-Varjani et al (2018)) is chosen for discretization of the problem. Using pseudospectral discretization and domain decomposition in nine regions yields a quadratic programming problem.…”

Section: Introductionmentioning

confidence: 99%

Solving two-dimensional bioheat optimal control problem in solid and vessel domain by pseudospectral discretization

Dastyar

Malek

Yousefi

2022

Transactions of the Institute of Measurement and Control

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In this paper, an optimal control problem subject to an energy transfer equation in thermally significant blood vessel coupled with the Pennes governing equation is solved. The proposed dynamical system is modeled by a system of conduction-reaction and convection-conduction equations. Chebyshev–Gauss–Lobatto (CGL) and Legendre–Gauss–Lobatto (LGL) points are applied to discretize optimal control problem. Then, the problem transferred into a quadratic programming problem using domain decomposition and matching the solution and its first derivative across the related interfaces. The Tikhonov regularization method is used to compute the optimal regularization parameter for the objective function. Some theories have been discussed to prove that the discrete problem has a solution and it is also consistent with the continuous problem. Numerical results are compared and numerical convergent is shown. Optimal and non-optimal external heat source is compared and it is shown that the mathematical modeling is effective in saving the external heat energy. Numerical results show that by this process of heating at the final time, the temperature is very close to the desired temperature without damaging the vessel domain. Numerical results and corresponding graphs are depicted.

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Solving a class of Hamilton-Jacobi-Bellman equations using pseudospectral methods

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Cited by 6 publications

References 33 publications

Optimal controlling of anti-TGF-$$\beta$$ and anti-PDGF medicines for preventing pulmonary fibrosis

Optimal controlling of anti-TGF-$$\beta$$ and anti-PDGF medicines for preventing pulmonary fibrosis

Optimal control of TGF-β to prevent formation of pulmonary fibrosis

Solving two-dimensional bioheat optimal control problem in solid and vessel domain by pseudospectral discretization

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