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Journal articles on the topic 'Fractional Hamilton-Jacobi equation'

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Author: Grafiati

Published: 11 March 2023

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1

Jarabah, Ola A. "Quantization of Damped Systems Using Fractional WKB Approximation." Applied Physics Research 10, no. 5 (September 27, 2018): 34. http://dx.doi.org/10.5539/apr.v10n5p34.

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The Hamilton Jacobi theory is used to obtain the fractional Hamilton-Jacobi function for fractional damped systems. The technique of separation of variables is applied here to solve the Hamilton Jacobi partial differential equation for fractional damped systems. The fractional Hamilton-Jacobi function is used to construct the wave function and then to quantize these systems using fractional WKB approximation. The solution of the illustrative example is found to be in exact agreement with the usual classical mechanics for regular Lagrangian when fractional derivatives are replaced with the integer order derivatives and r-0 .

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2

Veretennikova, M., and V. Kolokoltsov. "The Fractional Hamilton-Jacobi-Bellman Equation." Journal of Applied Nonlinear Dynamics 1, no. 1 (March 2017): 45–56. http://dx.doi.org/10.5890/jand.2017.03.004.

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3

Dlotko, Tomasz, and Maria B. Kania. "Subcritical Hamilton-Jacobi fractional equation in RN." Mathematical Methods in the Applied Sciences 38, no. 12 (August 18, 2014): 2547–60. http://dx.doi.org/10.1002/mma.3241.

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4

Gomoyunov, Mikhail Igorevich. "Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives." ESAIM: Control, Optimisation and Calculus of Variations 28 (2022): 23. http://dx.doi.org/10.1051/cocv/2022017.

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We consider a Cauchy problem for a Hamilton–Jacobi equation with coinvariant derivatives of an order α ∈ (0, 1). Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives of the order α. We propose a notion of a generalized in the minimax sense solution of the considered problem. We prove that a minimax solution exists, is unique, and is consistent with a classical solution of this problem. In particular, we give a special attention to the proof of a comparison principle, which requires construction of a suitable Lyapunov–Krasovskii functional.

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5

Hoang Luc, Nguyen, Donal O’Regan, and Anh Tuan Nguyen. "Solutions of a Nonlinear Diffusion Equation with a Regularized Hyper-Bessel Operator." Fractal and Fractional 6, no. 9 (September 19, 2022): 530. http://dx.doi.org/10.3390/fractalfract6090530.

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We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our current work is to show the existence and uniqueness of mild solutions. Our desired goal is achieved using the Picard iteration method, and our analysis is based on properties of Mittag–Leffler functions and embeddings between Hilbert scales spaces and Lebesgue spaces.

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6

Jumarie, Guy. "Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function." Journal of Applied Mathematics and Computing 23, no. 1-2 (January 2007): 215–28. http://dx.doi.org/10.1007/bf02831970.

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7

Rakhshan, Seyed Ali, Sohrab Effati, and Ali Vahidian Kamyad. "Solving a class of fractional optimal control problems by the Hamilton–Jacobi–Bellman equation." Journal of Vibration and Control 24, no. 9 (September 14, 2016): 1741–56. http://dx.doi.org/10.1177/1077546316668467.

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The performance index of both the state and control variables with a constrained dynamic optimization problem of a fractional order system with fixed final Time have been considered here. This paper presents a general formulation and solution scheme of a class of fractional optimal control problems. The method is based upon finding the numerical solution of the Hamilton–Jacobi–Bellman equation, corresponding to this problem, by the Legendre–Gauss collocation method. The main reason for using this technique is its efficiency and simple application. Also, in this work, we use the fractional derivative in the Riemann–Liouville sense and explain our method for a fractional derivative of order of [Formula: see text]. Numerical examples are provided to show the effectiveness of the formulation and solution scheme.

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8

Gomoyunov, Mikhail I. "Differential Games for Fractional-Order Systems: Hamilton–Jacobi–Bellman–Isaacs Equation and Optimal Feedback Strategies." Mathematics 9, no. 14 (July 15, 2021): 1667. http://dx.doi.org/10.3390/math9141667.

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The paper deals with a two-person zero-sum differential game for a dynamical system described by differential equations with the Caputo fractional derivatives of an order α∈(0,1) and a Bolza-type cost functional. A relationship between the differential game and the Cauchy problem for the corresponding Hamilton–Jacobi–Bellman–Isaacs equation with fractional coinvariant derivatives of the order α and the natural boundary condition is established. An emphasis is given to construction of optimal positional (feedback) strategies of the players. First, a smooth case is studied when the considered Cauchy problem is assumed to have a sufficiently smooth solution. After that, to cope with a general non-smooth case, a generalized minimax solution of this problem is involved.

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9

YAN, Li. "An Optimal Portfolio Problem Presented by Fractional Brownian Motion and Its Applications." Wuhan University Journal of Natural Sciences 27, no. 1 (March 2022): 53–56. http://dx.doi.org/10.1051/wujns/2022271053.

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We use the dynamic programming principle method to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function, and solve the optimal portfolio problem explicitly in a Black-Scholes type of market driven by fractional Brownian motion with Hurst parameter [see formula in PDF]. The results are compared with the corresponding well-known results in the standard Black-Scholes market [see formula in PDF]. As an application of our proposed model, two optimal problems are discussed and solved, analytically.

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10

Silvestre, Luis. "On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion." Advances in Mathematics 226, no. 2 (January 2011): 2020–39. http://dx.doi.org/10.1016/j.aim.2010.09.007.

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11

Iwabuchi, Tsukasa, and Tatsuki Kawakami. "Existence of mild solutions for a Hamilton–Jacobi equation with critical fractional viscosity in the Besov spaces." Journal de Mathématiques Pures et Appliquées 107, no. 4 (April 2017): 464–89. http://dx.doi.org/10.1016/j.matpur.2016.07.007.

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12

Rabei, Eqab M., and Mohammed Al Horani. "Quantization of fractional singular Lagrangian systems using WKB approximation." International Journal of Modern Physics A 33, no. 36 (December 30, 2018): 1850222. http://dx.doi.org/10.1142/s0217751x18502226.

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In this paper, the fractional singular Lagrangian system is studied. The Hamilton–Jacobi treatment is developed to be applicable for fractional singular Lagrangian system. The equations of motion are obtained for the fractional singular systems and the Hamilton–Jacobi partial differential equations are obtained and solved to determine the action integral. Then the wave function for fractional singular system is obtained. Besides, to demonstrate this theory, the fractional Christ-Lee model is discussed and quantized using the WKB approximation.

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13

Kolokoltsov, V. N., and M. S. Troeva. "Fractional McKean–Vlasov and Hamilton–Jacobi–Bellman–Isaacs Equations." Proceedings of the Steklov Institute of Mathematics 315, S1 (December 2021): S165—S177. http://dx.doi.org/10.1134/s0081543821060134.

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14

Giga, Yoshikazu, and Tokinaga Namba. "Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative." Communications in Partial Differential Equations 42, no. 7 (May 8, 2017): 1088–120. http://dx.doi.org/10.1080/03605302.2017.1324880.

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15

Gomoyunov, M. I. "Minimax Solutions of Homogeneous Hamilton–Jacobi Equations with Fractional-Order Coinvariant Derivatives." Proceedings of the Steklov Institute of Mathematics 315, S1 (December 2021): S97—S116. http://dx.doi.org/10.1134/s0081543821060092.

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16

Gomoyunov, Mikhail I. "Dynamic Programming Principle and Hamilton--Jacobi--Bellman Equations for Fractional-Order Systems." SIAM Journal on Control and Optimization 58, no. 6 (January 2020): 3185–211. http://dx.doi.org/10.1137/19m1279368.

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17

Camilli, Fabio, Raul De Maio, and Elisa Iacomini. "A Hopf-Lax formula for Hamilton-Jacobi equations with Caputo time-fractional derivative." Journal of Mathematical Analysis and Applications 477, no. 2 (September 2019): 1019–32. http://dx.doi.org/10.1016/j.jmaa.2019.04.069.

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18

Kolokoltsov, Vassili. "CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control." Fractional Calculus and Applied Analysis 25, no. 1 (February 2022): 128–65. http://dx.doi.org/10.1007/s13540-021-00002-2.

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AbstractInitially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.

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19

Kolokoltsov, V. N., and M. S. Troeva. "Abstract McKean–Vlasov and Hamilton–Jacobi–Bellman Equations, Their Fractional Versions and Related Forward–Backward Systems on Riemannian Manifolds." Proceedings of the Steklov Institute of Mathematics 315, no. 1 (December 2021): 118–39. http://dx.doi.org/10.1134/s0081543821050096.

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20

Malinowska, Agnieszka, and Delfim Torres. "Towards a combined fractional mechanics and quantization." Fractional Calculus and Applied Analysis 15, no. 3 (January 1, 2012). http://dx.doi.org/10.2478/s13540-012-0029-9.

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AbstractA fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional derivatives. The obtained results provide tools to carry out the quantization of nonconservative problems through combined fractional canonical equations of Hamilton type.

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21

Razminia, Abolhassan, Mehdi Asadizadehshiraz, and Delfim F. M. Torres. "Fractional Order Version of the Hamilton–Jacobi–Bellman Equation." Journal of Computational and Nonlinear Dynamics 14, no. 1 (November 28, 2018). http://dx.doi.org/10.1115/1.4041912.

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We consider an extension of the well-known Hamilton–Jacobi–Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.

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22

Kolokoltsov, Vassili N., and Maria A. Veretennikova. "A fractional Hamilton Jacobi Bellman equation for scaled limits of controlled Continuous Time Random Walks." Communications in Applied and Industrial Mathematics 6, no. 1 (October 30, 2014). http://dx.doi.org/10.1685/journal.caim.484.

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23

Sun, Jiaojiao, and Feng Dong. "Optimal reduction and equilibrium carbon allowance price for the thermal power industry under China’s peak carbon emissions target." Financial Innovation 9, no. 1 (January 8, 2023). http://dx.doi.org/10.1186/s40854-022-00410-0.

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AbstractAs the largest source of carbon emissions in China, the thermal power industry is the only emission-controlled industry in the first national carbon market compliance cycle. Its conversion to clean-energy generation technologies is also an important means of reducing CO2 emissions and achieving the carbon peak and carbon neutral commitments. This study used fractional Brownian motion to describe the energy-switching cost and constructed a stochastic optimization model on carbon allowance (CA) trading volume and emission-reduction strategy during compliance period with the Hurst exponent and volatility coefficient in the model estimated. We defined the optimal compliance cost of thermal power enterprises as the form of the unique solution of the Hamilton–Jacobi–Bellman equation by combining the dynamic optimization principle and the fractional Itô’s formula. In this manner, we obtained the models for optimal emission reduction and equilibrium CA price. Our numerical analysis revealed that, within a compliance period of 2021–2030, the optimal reductions and desired equilibrium prices of CAs changed concurrently, with an increasing trend annually in different peak-year scenarios. Furthermore, sensitivity analysis revealed that the energy price indirectly affected the equilibrium CA price by influencing the Hurst exponent, the depreciation rate positively impacted the CA price, and increasing the initial CA reduced the optimal reduction and the CA price. Our findings can be used to develop optimal emission-reduction strategies for thermal power enterprises and carbon pricing in the carbon market.

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24

Goffi, Alessandro. "Transport equations with nonlocal diffusion and applications to Hamilton–Jacobi equations." Journal of Evolution Equations, June 8, 2021. http://dx.doi.org/10.1007/s00028-021-00720-3.

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AbstractWe investigate regularity and a priori estimates for Fokker–Planck and Hamilton–Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $$s\in (1/2,1)$$ s ∈ ( 1 / 2 , 1 ) . As for Fokker–Planck equations, we establish integrability estimates under a fractional version of the Aronson–Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans’ nonlinear adjoint method we prove new integral, sup-norm and Hölder estimates for weak and strong solutions to fractional Hamilton–Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calderón–Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo–Nirenberg inequalities, we deduce optimal $$L^q$$ L q -regularity for fractional Hamilton–Jacobi equations.

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25

Ciomaga, Adina, Daria Ghilli, and Erwin Topp. "Periodic homogenization for weakly elliptic Hamilton-Jacobi-Bellman equations with critical fractional diffusion." Communications in Partial Differential Equations, July 6, 2021, 1–38. http://dx.doi.org/10.1080/03605302.2021.1941108.

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26

Camilli, Fabio, and Alessandro Goffi. "Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative." Nonlinear Differential Equations and Applications NoDEA 27, no. 2 (March 12, 2020). http://dx.doi.org/10.1007/s00030-020-0624-0.

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