scholarly journals Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Moulay Rchid Sidi Ammi ◽  
Mostafa Tahiri ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Derivative Operator ◽  
Sir Epidemic ◽  
Non Local

<p style='text-indent:20px;'>The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.</p>

scholarly journals Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth

2020 ◽  
Author(s):  
Jürgen Sprekels ◽  
Fredi Tröltzsch
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Optimality Conditions ◽  
Phase Field ◽  
Phase Field Model ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Sufficient Optimality Conditions ◽  
Phase Field System

In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the $L^1$-norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.

scholarly journals Parameter identification for nonlocal phase field models for tumor growth via optimal control and asymptotic analysis

2021 ◽  
pp. 1-52
Author(s):  
Elisabetta Rocca ◽  
Luca Scarpa ◽  
Andrea Signori
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Parameter Identification ◽  
Optimality Conditions ◽  
A Priori ◽  
Three Dimensional ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Original System ◽  
Identification Problem

In this paper, we introduce the problem of parameter identification for a coupled nonlocal Cahn–Hilliard-reaction-diffusion PDE system stemming from a recently introduced tumor growth model. The inverse problem of identifying relevant parameters is studied here by relying on techniques from optimal control theory of PDE systems. The parameters to be identified play the role of controls, and a suitable cost functional of tracking-type is introduced to account for the discrepancy between some a priori knowledge of the parameters and the controls themselves. The analysis is carried out for several classes of models, each one depending on a specific relaxation (of parabolic or viscous type) performed on the original system. First-order necessary optimality conditions are obtained on the fully relaxed system, in both the two- and three-dimensional cases. Then, the optimal control problem on the non-relaxed models is tackled by means of asymptotic arguments, by showing convergence of the respective adjoint systems and the minimization problems as each one of the relaxing coefficients vanishes. This allows obtaining the desired necessary optimality conditions, hence to solve the parameter identification problem, for the original PDE system in case of physically relevant double-well potentials.

scholarly journals Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 850-856 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Yun-Yun Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Vibration System ◽  
Derivative Operator ◽  
Vibration Equation ◽  
State Response ◽  
Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

scholarly journals Optimal control of stochastic phase-field models related to tumor growth

2020 ◽  
Vol 26 ◽  
pp. 104
Author(s):  
Carlo Orrieri ◽  
Elisabetta Rocca ◽  
Luca Scarpa
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Phase Field ◽  
Phase Field Model ◽  
Growth Dynamics ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Adjoint System ◽  
Reaction Diffusion Equation ◽  
Cahn Hilliard Equation

We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.

scholarly journals Numerical Algorithms For State-Linear Optimal Impulsive Control Problems Based On Feedback Necessary Optimality Conditions

2020 ◽  
pp. 152-158
Author(s):  
Stepan Sorokin ◽  
Maxim Staritsyn
Keyword(s):  
Optimality Conditions ◽  
Impulsive Control ◽  
Numerical Algorithms ◽  
Necessary Optimality Conditions ◽  
Maximum Principles ◽  
Control Problems ◽  
Impulsive Systems ◽  
Feedback Controls ◽  
Non Local ◽  
Optimal Impulsive Control

We propose and compare three numeric algorithms for optimal control of state-linear impulsive systems. The algorithms rely on the standard transformation of impulsive control problems through the discontinuous time rescaling, and the so-called “feedback”, direct and dual, maximum principles. The feedback maximum principles are variational necessary optimality conditions operating with feedback controls, which are designed through the usual constructions of the Pontryagin’s Maximum Principle (PMP); though these optimality conditions are formulated completely in the formalism of PMP, they essentially strengthen it. All the algorithms are non-local in the sense that they are aimed at improving non-optimal extrema of PMP (local minima), and, therefore, show the potential of global optimization.

scholarly journals Necessary Optimality Conditions for a Class of Impulsive and Switching Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Lihua Li ◽  
Yan Gao ◽  
Gexia Wang
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Necessary Optimality Conditions ◽  
Switching Systems ◽  
Jump Conditions ◽  
Fréchet Subdifferential ◽  
Extended State ◽  
Adjoint Variables

An optimal control problem for a class of hybrid impulsive and switching systems is considered. By defining switching times as part of extended state, we get the necessary optimality conditions for this problem. It is shown that the adjoint variables satisfy certain jump conditions and the Hamiltonian are continuous at switching instants. In addition, necessary optimality conditions of Fréchet subdifferential form are presented in this paper.

scholarly journals Optimal Control with Time Delays via the Penalty Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mohammed Benharrat ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Calculus Of Variations ◽  
Convergence Theorem ◽  
Unknown Function ◽  
Time Delays ◽  
Penalty Method ◽  
Necessary Optimality Conditions

We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general optimal control problem with time delays is considered. Main result gives a convergence theorem, allowing us to obtain a solution to the delayed optimal control problem by considering a sequence of delayed problems of the calculus of variations.

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