The following articles are submitted and still “under review” in the appropriate journals:
Arzu Ahmadova, Nazim Mahmudov I; Nieto, Juan J Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle Journal Article Evolution Equations and Control Theory, 11 (6), 2022. @article{Ahmadova2021bb, title = {Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle}, author = {Arzu Ahmadova, Nazim I Mahmudov and Juan J. Nieto }, doi = {doi: 10.3934/eect.2022008}, year = {2022}, date = {2022-12-01}, journal = {Evolution Equations and Control Theory}, volume = {11}, number = {6}, abstract = {In this paper, we obtain a closed-form representation of a mild solution to the fractional stochastic degenerate evolution equation in a Hilbert space using the subordination principle and semigroup theory. We study aforesaid abstract fractional stochastic Cauchy problem with nonlinear state-dependent terms and show that if the Sobolev type resolvent families describing the linear part of the model are exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions for nonlinearity. We also establish conditions for stabilizability and prove that the stochastic nonlinear fractional Cauchy problem is exponentially stabilizable when the stabilizer acts linearly on the control systems. Finally, we provide applications to show the validity of our theory.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper, we obtain a closed-form representation of a mild solution to the fractional stochastic degenerate evolution equation in a Hilbert space using the subordination principle and semigroup theory. We study aforesaid abstract fractional stochastic Cauchy problem with nonlinear state-dependent terms and show that if the Sobolev type resolvent families describing the linear part of the model are exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions for nonlinearity. We also establish conditions for stabilizability and prove that the stochastic nonlinear fractional Cauchy problem is exponentially stabilizable when the stabilizer acts linearly on the control systems. Finally, we provide applications to show the validity of our theory. |
Arzu Ahmadova Ismail T. Huseynov, Nazim Mahmudov I Approximate controllability of fractional stochastic degenerate evolution equations Journal Article Differential Equations and Dynamical Systems, 2022. @article{Ahmadova2021bb, title = {Approximate controllability of fractional stochastic degenerate evolution equations}, author = {Arzu Ahmadova, Ismail T. Huseynov, Nazim I. Mahmudov}, year = {2022}, date = {2022-11-16}, journal = {Differential Equations and Dynamical Systems}, abstract = {We study a class of dynamic control systems described by nonlinear fractional stochastic degenerate evolution equations in Hilbert spaces. Using fixed point technique, fractional calculations, stochastic analysis technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for approximate controllability of fractional stochastic degenerate evolution equations is formulated and proved. In particular, we discuss the approximate controllability of nonlinear fractional stochastic degenerate control system under the assumptions that the corresponding linear system is approximately controllable. The results in this paper are generalization and continuation of the recent results on this issue. As an application of main results we consider fractional partial differential equations and prove approximate controllability results by verifying main assumptions.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We study a class of dynamic control systems described by nonlinear fractional stochastic degenerate evolution equations in Hilbert spaces. Using fixed point technique, fractional calculations, stochastic analysis technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for approximate controllability of fractional stochastic degenerate evolution equations is formulated and proved. In particular, we discuss the approximate controllability of nonlinear fractional stochastic degenerate control system under the assumptions that the corresponding linear system is approximately controllable. The results in this paper are generalization and continuation of the recent results on this issue. As an application of main results we consider fractional partial differential equations and prove approximate controllability results by verifying main assumptions. |
Nazim I Mahmudov, Arzu Ahmadova Some Results on Backward Stochastic Differential Equations of Fractional Order Journal Article Qualitative Theory of Dynamical Systems, 21 (4), pp. 1-23, 2022. @article{Ahmadova2021bb, title = {Some Results on Backward Stochastic Differential Equations of Fractional Order}, author = {Nazim I Mahmudov, Arzu Ahmadova}, url = {https://doi.org/10.1007/s12346-022-00657-z}, doi = {https://doi.org/10.1007/s12346-022-00657-z}, year = {2022}, date = {2022-09-20}, journal = {Qualitative Theory of Dynamical Systems}, volume = {21}, number = {4}, pages = {1-23}, abstract = {In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order $\alpha\in (1/2, 1)$ whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order $alphain (1/2, 1)$ whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution. |
Nazim I Mahmudov Arzu Ahmadova, Ismail Huseynov T A novel technique for solving Sobolev-type fractional multi-order evolution equations Journal Article Computational and Applied Mathematics, 41 (2), pp. 1-35, 2022. @article{Mahmudov2021b, title = {A novel technique for solving Sobolev-type fractional multi-order evolution equations}, author = {Nazim I Mahmudov, Arzu Ahmadova, Ismail T Huseynov}, doi = {https://doi.org/10.1007/s40314-022-01781-x}, year = {2022}, date = {2022-03-01}, journal = {Computational and Applied Mathematics}, volume = {41}, number = {2}, pages = {1-35}, abstract = {A strong inspiration for studying Sobolev-type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev-type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler-type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representation of solutions for multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices.}, keywords = {}, pubstate = {published}, tppubtype = {article} } A strong inspiration for studying Sobolev-type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev-type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler-type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representation of solutions for multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices. |
Arzu Ahmadova, Nazim Mahmudov I 2021. @online{Ahmadova2021bb, title = {Existence and uniqueness result of an adapted solution of a singular backward stochastic nonlinear Volterra integral equation}, author = {Arzu Ahmadova, Nazim I Mahmudov }, doi = {arXiv:2109.08950}, year = {2021}, date = {2021-05-30}, abstract = {Our main result in this paper is to establish a fundamental lemma to prove the global existence and uniqueness of an adapted solution to a singular backward stochastic nonlinear Volterra integral equation (for short singular BSVIE) of order alpha in ( 1/2 ; 1) under a weaker condition than Lipschitz one in Hilbert space.}, keywords = {}, pubstate = {published}, tppubtype = {online} } Our main result in this paper is to establish a fundamental lemma to prove the global existence and uniqueness of an adapted solution to a singular backward stochastic nonlinear Volterra integral equation (for short singular BSVIE) of order alpha in ( 1/2 ; 1) under a weaker condition than Lipschitz one in Hilbert space. |
Arzu Ahmadova, Nazim Mahmudov I Asymptotic separation of solutions to fractional stochastic multi-term differential equations Journal Article Fractal and Fractional, 5 (4), pp. 256, 2021. @article{Ahmadova2021bb, title = {Asymptotic separation of solutions to fractional stochastic multi-term differential equations}, author = {Arzu Ahmadova, Nazim I Mahmudov}, doi = {https://doi.org/10.3390/fractalfract5040256}, year = {2021}, date = {2021-03-13}, journal = {Fractal and Fractional}, volume = {5}, number = {4}, pages = {256}, abstract = {In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order α∈(12,1) and β∈(0,1) whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on λ. Furthermore, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order α∈(12,1) and β∈(0,1) whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on λ. Furthermore, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general. |
Ismail T. Huseynov Arzu Ahmadova, Nazim Mahmudov I 2020. @online{Huseynov2020b, title = {Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications}, author = {Ismail T. Huseynov, Arzu Ahmadova, Nazim I. Mahmudov }, doi = {arXiv preprint arXiv:2012.11360}, year = {2020}, date = {2020-12-18}, abstract = {In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type differentiation operators with general fractional-order. A rule of fractional differentiation under integral sign with general order is necessary and applicable tool for verification by substitution for candidate solutions of inhomogeneous multi-term fractional differential equations. We derive explicit analytical solutions of generalized Bagley-Torvik equations in terms of recently defined bivariate Mittag-Leffler type functions that based on fractional Green's function method and verified solutions by substitution in accordance by applying the fractional Leibniz integral rule. Furthermore, we study an oscillator equation as a special case of differential equations with multi-orders via the Leibniz integral rule.}, keywords = {}, pubstate = {published}, tppubtype = {online} } In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type differentiation operators with general fractional-order. A rule of fractional differentiation under integral sign with general order is necessary and applicable tool for verification by substitution for candidate solutions of inhomogeneous multi-term fractional differential equations. We derive explicit analytical solutions of generalized Bagley-Torvik equations in terms of recently defined bivariate Mittag-Leffler type functions that based on fractional Green's function method and verified solutions by substitution in accordance by applying the fractional Leibniz integral rule. Furthermore, we study an oscillator equation as a special case of differential equations with multi-orders via the Leibniz integral rule. |