Agnieszka B. Malinowska, Delfim F. M. Torres
This paper presents necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrangian depending on the free end-points. The fractional derivatives are defined in the sense of Caputo.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We generalize the fractional Caputo derivative to the fractional derivative ${^CD^{α,β}_γ}$, which is a convex combination of the left Caputo fractional derivative of order $α$ and the right Caputo fractional derivative of order $β$. The fractional variational problems under our consideration are formulated in terms of ${^CD^{α,β}_γ}$. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.
We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler-Lagrange equation for functionals where the lower and upper bounds of the integral are distinct of the bounds of the Caputo derivative is also proved. Then, the fractional isoperimetric problem is formulated with an integral constraint also containing Caputo derivatives. Normal and abnormal extremals are considered.
We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler-Lagrange equations to the basic and isoperimetric problems are proved, as well as transversality conditions.
We study dynamic minimization problems of the calculus of variations with Lagrangian functionals containing Riemann-Liouville fractional integrals, classical and Caputo fractional derivatives. Under assumptions of regularity, coercivity and convexity, we prove existence of solutions.
Fractional control systems with the Caputo derivative are considered. The modified controllability Gramian and the minimum energy optimal control problem are investigated. Construction of minimizing steering controls for the modified energy functional are proposed.
The study of fractional variational problems in terms of a combined fractional Caputo derivative is introduced. Necessary optimality conditions of Euler-Lagrange type for the basic, isoperimetric, and Lagrange variational problems are proved, as well as transversality and sufficient optimality conditions. This allows to obtain necessary and sufficient Pareto optimality conditions for multiobjective fractional variational problems.
We investigate a class of fractional neutral evolution equations on Banach spaces involving Caputo derivatives. Main results establish conditions for the controllability of the fractional-order system and conditions for existence of a solution to an optimal control problem of minimum energy. The results are proved with the help of fixed-point and semigroup theories.
This paper investigates the regional gradient controllability for ultra-slow diffusion processes governed by the time fractional diffusion systems with a Hadamard-Caputo time fractional derivative. Some necessary and sufficient conditions on regional gradient exact and approximate controllability are first given and proved in detail. Secondly, we propose an approach on how to calculate the minimum number of $ω-$strategic actuators. Moreover, the existence, uniqueness and the concrete form of the optimal controller for the system under consideration are presented by employing the Hilbert Unique…
We consider the regional enlarged observability problem for fractional evolution differential equations involving Caputo derivatives. Using the Hilbert Uniqueness Method, we show that it is possible to rebuild the initial state between two prescribed functions only in an internal subregion of the whole domain. Finally, an example is provided to illustrate the theory.
A Conjugate Gradient Method for Nonlinear Programming Problems using Caputo Fractional Gradients
本文提出了一种基于Caputo分数梯度的共轭梯度方法，用于解决非线性规划问题，通过改进的线搜索技术和正则化方法，提升了优化效率和稳定性。
An Adaptive Order Caputo Fractional Gradient Descent Method for Multi-objective Optimization Problems
本文提出MOAOCFGD算法求解无约束多目标优化问题，通过求解包含自适应阶Caputo分数阶梯度的子问题确定下降方向并结合Armijo型线搜索，在温和假设下证明收敛性，适用于光滑与非光滑问题且无需先验参数。
An Operator-Theoretic Framework for the Optimal Control Problem of Nonlinear Caputo Fractional Systems
本文针对含Caputo导数和非局部初始条件的非线性分数阶系统最优控制问题，提出基于Hammerstein型算子方程的理论框架，建立温和解与最优控制-状态对的存在性条件，并导出二次代价泛函的最优性系统。
We study the regional controllability problem for delayed fractional control systems through the use of the standard Caputo derivative. First, we recall several fundamental results and introduce the family of fractional-order systems under consideration. Afterwards, we formulate the notion of regional controllability for fractional systems with control delays and give some of their important properties. Our main method consists in defining an attainable set, which allow us to prove exact and weak controllability. Moreover, main results include not only those of controllability but also a power…
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 stabi…
We generalize the fractional Caputo derivative to the fractional derivative ${{^CD}^{α,β}_γ}$, which is a convex combination of the left Caputo fractional derivative of order $α$ and the right Caputo fractional derivative of order $β$. The fractional variational problems under our consideration are formulated in terms of ${{^CD}^{α,β}_γ}$. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.
Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem
Caputo意义下的分数阶导数变分问题的研究是一个新的课题，主要结果是Agrawal关于Euler-Lagrange的最优性条件和相应的横截性条件。本文利用Agrawal的Euler-Lagrange方程和Lagrange乘子技巧，得到了Caputo意义下分数阶最优控制问题的一个Noether类定理.
Regional Gradient Observability for Fractional Differential Equations with Caputo Time-Fractional Derivatives
本文研究含Caputo导数的分数阶次扩散方程的区域梯度能观性，给出相关刻画与基于Hilbert Uniqueness Method的初始梯度恢复方法，数值实验验证了算法在重构误差上的有效性。
This work presents a new mathematical model to depict the effect of obesity on cancerous tumor growth when chemotherapy as well as immunotherapy have been administered. We consider an optimal control problem to destroy the tumor population and minimize the drug dose over a finite time interval. The constraint is a model including tumor cells, immune cells, fat cells, chemotherapeutic and immunotherapeutic drug concentrations with Caputo time fractional derivative. We investigate the existence and stability of the equilibrium points namely, tumor free equilibrium and coexisting equilibrium, ana…
In order to solve fractional variational problems, there exist two theorems of necessary conditions: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives, and other Euler-Lagrange equation that involves only Caputo derivatives. In this article, we make a comparison solving a particular fractional variational problem with both methods to obtain some conclusions about which method gives the optimal solution.
Boundary Regional Controllability of Semilinear Systems Involving Caputo Time Fractional Derivatives
针对含Caputo时间分数阶导数的半线性系统，在近似区域边界可控性假设下，利用分数幂和不动点技术证明边界区域可控性的充分条件，并提出算法和数值模拟验证。
We prove a Noether's theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.
Variational Problems Involving a Caputo-Type Fractional Derivative
本文研究了依赖于Caputo型分数导数的变分问题，推导了第一和第二阶的充分必要条件，并考虑了积分约束和哈密顿约束情况。
We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler-Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.
We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.
We investigate the exact enlarged controllability and optimal control of a fractional diffusion equation in Caputo sense. This is done through a new definition of enlarged controllability that allows us to extend available contributions. Moreover, the problem is studied using two approaches: a reverse Hilbert uniqueness method, generalizing the approach introduced by Lions in 1988, and a penalization method, which allow us to characterize the minimum energy control.
Pharmacokinetic/Pharmacodynamic Anesthesia Model Incorporating psi-Caputo Fractional Derivatives
提出一种融合ψ-Caputo分数阶导数的PK/PD麻醉诱导模型，通过Picard迭代求解非齐次系统，数值分析表明ψ函数和分数阶对个体化药物反应建模至关重要，为精准麻醉给药奠定基础。
Derivatives and integrals of non-integer order were introduced more than three centuries ago, but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus f…
In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does not necessarily coincide with the initial point of the fractional differential operator. A detailed analysis of basic properties of the fundamental solution matrix is carried out. In particular, the Hölder continuity of this matrix with respect to both variables is proved, and its dual definition is given. Based on this, two representation formulas for the s…
We obtain Euler-Lagrange equations, transversality conditions and a Noether-like theorem for Herglotz-type variational problems with Lagrangians depending on generalized fractional derivatives. As an application, we consider a damped harmonic oscillator with time-depending mass and elasticity, and arbitrary memory effects.
Fractional differential equations of a reaction-diffusion SIR model involving the Caputo-fractional time-derivative and a nonlinear diffusion operator
本研究分析反应扩散SIR模型的分数阶微分方程，利用Caputo分数阶导数与p-Laplacian算子，通过疫苗接种控制变量，建立解的存在唯一性并求解最优控制，数值实验验证其有效性。
Since the seminal work of Emmy Noether it is well know that all conservations laws in physics, \textrm{e.g.}, conservation of energy or conservation of momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether's theorem can not yields informations about constants of motion for non-conservative systems since it is not possible to formulate physically meaningful Lagragians for this kind of systems in classical calculus of variation. On the other hand, in recent years the fractional calculus of variation within Lagrangians dep…
We propose a novel Caputo fractional derivative-based optimization algorithm. Upon defining the Caputo fractional gradient with respect to the Cartesian coordinate, we present a generic Caputo fractional gradient descent (CFGD) method. We prove that the CFGD yields the steepest descent direction of a locally smoothed objective function. The generic CFGD requires three parameters to be specified, and a choice of the parameters yields a version of CFGD. We propose three versions -- non-adaptive, adaptive terminal and adaptive order. By focusing on quadratic objective functions, we provide a conv…
Fast and Scalable Caputo Fractional Gradient Descent via Perturbation-Preserving Memory Compression
研究如何让基于卡普托的优化计算可行，核心方法是将分数阶下降方向表示为离散卷积，引入指数和近似与二元分层离散卷积降低内存成本，贡献是在特定假设下保证方法有单调下降和线性收敛性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。