Journal articles: 'Differential Algebraic Inequality'

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Reis, Timo, and Matthias Voigt. "The Dissipation Inequality for Differential-Algebraic Systems." PAMM 14, no. 1 (December 2014): 11–14. http://dx.doi.org/10.1002/pamm.201410004.

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Reis, Timo, Olaf Rendel, and Matthias Voigt. "The Kalman–Yakubovich–Popov inequality for differential-algebraic systems." Linear Algebra and its Applications 485 (November 2015): 153–93. http://dx.doi.org/10.1016/j.laa.2015.06.021.

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Allouche, Benyamine, Antoine Dequidt, Laurent Vermeiren, and Michel Dambrine. "Modeling and PDC fuzzy control of planar parallel robot." International Journal of Advanced Robotic Systems 14, no. 1 (January 1, 2017): 172988141668711. http://dx.doi.org/10.1177/1729881416687112.

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Many works in the literature have studied the kinematical and dynamical issues of parallel robots. But it is still difficult to extend the vast control strategies to parallel mechanisms due to the complexity of the model-based control. This complexity is mainly caused by the presence of multiple closed kinematic chains, making the system naturally described by a set of differential–algebraic equations. The aim of this work is to control a two-degree-of-freedom parallel manipulator. A mechanical model based on differential–algebraic equations is given. The goal is to use the structural characteristics of the mechanical system to reduce the complexity of the nonlinear model. Therefore, a trajectory tracking control is achieved using the Takagi-Sugeno fuzzy model derived from the differential–algebraic equation forms and its linear matrix inequality constraints formulation. Simulation results show that the proposed approach based on differential–algebraic equations and Takagi-Sugeno fuzzy modeling leads to a better robustness against the structural uncertainties.

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Uppal, Tarun, Soumyendu Raha, and Suresh Srivastava. "Inverse Simulation for Gas Turbine Engine Control through Differential Algebraic Inequality Formulation." International Journal of Turbo & Jet-Engines 35, no. 4 (December 19, 2018): 373–83. http://dx.doi.org/10.1515/tjj-2016-0057.

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Abstract Modern day gas turbines are prime movers in land, air and sea. They have stringent performance requirements to meet the complex mission objectives. Optimal control strategies can help them meet their performance objectives more efficiently. A novel inverse simulation method for optimal control and system analysis studies using Differential Algebraic Equality/Inequality (DAE/DAI) technique is brought out in this paper with a case study. The gas turbine model together with safety constraints and performance specifications is represented as a high index DAI/DAE system. The solution for this DAE/DAI system is obtained using a new numerical approach that is capable of handling both equality and inequality constraints on system dynamics. The algorithm involves direct numerical integration of a DAI formulation in a time stepping manner using Sequential Quadratic Programming (SQP) solver that detects and satisfy active constraints at each time step (mesh point). In this unique approach the model and the constraints are always solved together. The method ensures stable solution at each time step, local minimum at each iteration of simulation and provides a regularised basis to the solver. Compared to other existing computationally intensive techniques in usage, this approach is easy, ensures continuous constraint satisfaction and provides a viable option for Model Predictive Control (MPC) of gas turbine engines.

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Pop, Nicolae. "Generalized Newton’s method for solving nonlinear and nondifferentiable algebraic systems." Journal of Numerical Analysis and Approximation Theory 44, no. 1 (December 18, 2015): 93–99. http://dx.doi.org/10.33993/jnaat441-1058.

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In this paper a model based on non-smooth equations is proposed for solving a non-linear and non-differential equation obtained by discretization of a quasi-variational inequality that models the frictional contact problem. The main aim of this paper is to show that the Newton method based on the plenary hull of the Clarke generalized Jacobians (the non-smooth damped Newton method) can be implemented for solving Lipschitz non-smooth equation.

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Reis, Timo, and Matthias Voigt. "The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions." Systems & Control Letters 86 (December 2015): 1–8. http://dx.doi.org/10.1016/j.sysconle.2015.09.003.

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Qingfei, Pan, Zhang Zifang, and Huang Jingchang. "Stability of the Stochastic Reaction-Diffusion Neural Network with Time-Varying Delays andp-Laplacian." Journal of Applied Mathematics 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/405939.

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The main aim of this paper is to discuss moment exponential stability for a stochastic reaction-diffusion neural network with time-varying delays andp-Laplacian. Using the Itô formula, a delay differential inequality and the characteristics of the neural network, the algebraic conditions for the moment exponential stability of the nonconstant equilibrium solution are derived. An example is also given for illustration.

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Lê, Công-Trình, and Tien-Son Phạm. "On tangent cones at infinity of algebraic varieties." Journal of Algebra and Its Applications 17, no. 08 (July 8, 2018): 1850143. http://dx.doi.org/10.1142/s0219498818501438.

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In this paper, we define the geometric and algebraic tangent cones at infinity of algebraic varieties and establish the following version at infinity of Whitney’s theorem [Local properties of analytic varieties, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton University Press, Princeton, N. J., 1965), pp. 205–244; Tangents to an analytic variety, Ann. of Math. 81 (1965) 496–549]: The geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of the geometric tangent cone at infinity using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. Moreover, we show that the tangent cone at infinity of a complex algebraic variety is actually the part at infinity of this variety [G.-M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, 2nd extended edn. (Springer, Berlin, 2008)]. We also show that the tangent cone at infinity of a complex algebraic variety can be computed using Gröbner bases.

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Wu, Dongli, Hao Zhang, Yunping Liu, Weihua Fang, and Yan Wang. "Real-Time Trajectory Planning and Control for Constrained UAV Based on Differential Flatness." International Journal of Aerospace Engineering 2022 (June 20, 2022): 1–17. http://dx.doi.org/10.1155/2022/8004478.

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The trajectory planning of UAV with nonholonomic constraints is usually taken as differential algebraic equation to solve the optimal control problem of functional extremum under the condition of inequality constraints. However, it can be challenging to meet the requirements of real-time for the high complexity. A differential flat theory based on B-spline trajectory planning can replace the optimal control problem with nonlinear programming and be a good means to achieve the efficient trajectory planning of an UAV under multiple dynamic constraints. This research verifies the feasibility of this theory with actual flight experiments.

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Wu, Qiong, Zhimin Yao, Zhouping Yin, and Hai Zhang. "Fin-TS and Fix-TS on fractional quaternion delayed neural networks with uncertainty via establishing a new Caputo derivative inequality approach." Mathematical Biosciences and Engineering 19, no. 9 (2022): 9220–43. http://dx.doi.org/10.3934/mbe.2022428.

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<abstract><p>This paper investigates the finite time synchronization (Fin-TS) and fixed time synchronization (Fix-TS) issues on Caputo quaternion delayed neural networks (QDNNs) with uncertainty. A new Caputo fractional differential inequality is constructed, then Fix-TS settling time of the positive definite function is estimated, which is very convenient to derive Fix-TS condition to Caputo QDNNs. By designing the appropriate self feedback and adaptive controllers, the algebraic discriminant conditions to achieve Fin-TS and Fix-TS on Caputo QDNNs are proposed based on quaternion direct method, Lyapunov stability theory, extended Cauchy Schwartz inequality, Jensen inequality. Finally, the correctness and validity of the presented results under the different orders are verified by two numerical examples.</p></abstract>

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Siddiqui, Aliya Naaz, Ali Hussain Alkhaldi, and Lamia Saeed Alqahtani. "Generalized Wintgen Inequality for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature." Mathematics 10, no. 10 (May 18, 2022): 1727. http://dx.doi.org/10.3390/math10101727.

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The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for δ(2,2)-invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end.

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Reis, Timo, and Matthias Voigt. "Reprint of “The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions”." Systems & Control Letters 95 (September 2016): 3–10. http://dx.doi.org/10.1016/j.sysconle.2016.05.010.

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Li, Xiaofan, Yuan Ge, Hongjian Liu, Huiyuan Li, and Jian-an Fang. "New Results on Synchronization of Fractional-Order Memristor‐Based Neural Networks via State Feedback Control." Complexity 2020 (September 9, 2020): 1–11. http://dx.doi.org/10.1155/2020/2470972.

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This paper addresses the synchronization issue for the drive-response fractional-order memristor‐based neural networks (FOMNNs) via state feedback control. To achieve the synchronization for considered drive-response FOMNNs, two feedback controllers are introduced. Then, by adopting nonsmooth analysis, fractional Lyapunov’s direct method, Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization of the drive-response FOMNNs. Lastly, for illustrating the effectiveness of the obtained theoretical results, an example is given.

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Zhao, Shuyue, Kelin Li, and Weiyi Hu. "Synchronization in Finite Time of Fuzzy Neural Networks with Hybrid Delays and Uncertain Nonlinear Perturbations." Advances in Fuzzy Systems 2022 (July 5, 2022): 1–22. http://dx.doi.org/10.1155/2022/1448819.

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This paper deals with the finite-time synchronization problem of a class of fuzzy neural networks with hybrid delays and uncertain nonlinear perturbations. By applying the famous finite-time stability theory, combining differential inequality techniques, and the analysis approach, several new algebraic sufficient criteria are obtained to realize finite-time synchronization between the drive system and the response system by designing a state feedback controller and an adaptive controller. Taking discrete delays, distributed delays, and uncertain nonlinear perturbations into account in fuzzy cellular neural networks makes the neural system more general than most existing cellular neural networks. Two different novel types of controllers designed to achieve finite-time synchronization can not only effectively overcome the influence of time delays and perturbations but also change their form according to the change of system state or perturbation to achieve a better control effect. Meanwhile, some algebraic sufficient criteria obtained in this paper can be proved by the parameters of the system itself, and the complex calculation of matrix inequality is avoided. Finally, the validity of our proposed results is confirmed by several examples and simulations. Furthermore, a secure communication problem is presented to further illustrate the fact of the obtained results.

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Yuldashev, T. K., and E. T. Karimov. "Mixed type integro-differential equation with fractional order Caputo operators and spectral parameters." Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta 57 (May 2021): 190–205. http://dx.doi.org/10.35634/2226-3594-2021-57-10.

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The issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters are considered. A mixed type integro-differential equation is a partial integro-differential equation of fractional order in both positive and negative parts of multidimensional rectangular domain under consideration. The fractional Caputo operator's order is less in the positive part of the domain, than the order of Caputo operator in the negative part of the domain. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels are obtained. Further, a method of degenerate kernels is used. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system, regular and irregular values of spectral parameters are calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. To prove the convergence of Fourier series, the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. The continuous dependence of the problem solution on a small parameter for regular values of spectral parameters is also studied. The results are formulated as a theorem.

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Bennett, Jonathan, and Neal Bez. "Generating monotone quantities for the heat equation." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 756 (November 1, 2019): 37–63. http://dx.doi.org/10.1515/crelle-2017-0025.

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AbstractThe purpose of this article is to expose and further develop a simple yet surprisingly far-reaching framework for generating monotone quantities for positive solutions to linear heat equations in euclidean space. This framework is intimately connected to the existence of a rich variety of algebraic closure properties of families of sub/super-solutions, and more generally solutions of systems of differential inequalities capturing log-convexity properties such as the Li–Yau gradient estimate. Various applications are discussed, including connections with the general Brascamp–Lieb inequality and the Ornstein–Uhlenbeck semigroup.

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Yu, Tianhu, Shengqiang Liu, Huamin Wang, Yingjia Cui, and Dengqing Cao. "Robust delay-dependent stability of uncertain inertial neural networks with impulsive effects and distributed-delay." International Journal of Biomathematics 12, no. 01 (January 2019): 1950010. http://dx.doi.org/10.1142/s1793524519500104.

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The robust stability problem of uncertain inertial neural networks with impulsive effects and distributed-delay is considered in the present paper. The average impulsive interval and differential inequality for delay differential equations are used to obtain the global exponential stability of the inertial neural networks. The robust distributed-delay-dependent stability criteria here are proposed in terms of both linear matrix inequalities and algebraic inequalities. Our results can not only be used to obtain the stability of the uncertain inertial neural network with impulsive disturbance, but also be utilized to design the impulsive control for the uncertain inertial neural networks. The novel criteria complement and extend the previous works on uncertain inertial neural network with/without impulsive effects. Typical numerical examples are used to test the validity of the developed stability criteria finally.

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Fan, Yingjie, Zhongliang Wei, and Meixuan Li. "Switching-Jumps-Dependent Quasi-Synchronization Criteria for Fractional-Order Memrisive Neural Networks." Fractal and Fractional 7, no. 1 (December 24, 2022): 12. http://dx.doi.org/10.3390/fractalfract7010012.

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This paper investigates the switching-jumps-dependent quasi-synchronization issue for fractional-order memristive neural networks (FMNNs). First, a simplied linear feedback controller is applied. Then, in terms of several fractional order differential inequalities and two kinds of Lyapunov functions, two quasi-synchronization criteria expressed by linear matrix inequality (LMI)-based form and algebraic form are established, respectively. Meanwhile, the co-designed scheme for error bound and control gain is established. Compared with the previous quasi-synchronization results, a strong assumption that the system states must be bounded is removed. Finally, some simulation examples are carried out to display the feasibility and validity of the proposed analysis methods.

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Bouassem, Karim, Abdellatif El Assoudi, Jalal Soulami, and El Hassane El Yaagoubi. "Unknown Input Observer Design for a Class of Linear Descriptor Systems." E3S Web of Conferences 229 (2021): 01019. http://dx.doi.org/10.1051/e3sconf/202122901019.

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This paper addresses the problem of unknown inputs observer (UIO) design for a class of linear descriptor systems. The unknown inputs affect both state and output of the system. The basic idea of the proposed approach is based on the separation between dynamic and static relations in the descriptor model. Firstly, the method used to separate the differential part from the algebraic part is developed. Secondly, an observer design permitting the simultaneous estimation of the system state and the unknown inputs is proposed. The developed approach for the observer design is based on the synthesis of an augmented model which regroups the differential variables and unknown inputs. The exponential stability of the estimation error is studied using the Lyapunov theory and the stability condition is given in term of linear matrix inequality (LMI). Finally, to illustrate the efficiency of the proposed methodology, a heat exchanger pilot model is considered.

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KOJIC, Vedran. "On a Geometric Programming Approach to Profit Maximization: The Case of CES Technology." Eurasia Proceedings of Science Technology Engineering and Mathematics 18 (October 20, 2022): 7–15. http://dx.doi.org/10.55549/epstem.1182630.

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The profit maximization problem takes a central place in the theory of the firm, especially when conditions for perfect competition hold. In this paper, we solve the profit maximization problem of a perfectly competitive firm when the constant elasticity of substitution (CES) production function with n≥2 inputs describes its technology. Commonly, this problem is solved by using multivariable differential calculus. However, to avoid tedious algebraic manipulations and bypass checking nontrivial necessary and sufficient conditions, we employ geometric programming (GP), and the power mean inequality (PMI) as an elegant complementary tool to multivariable calculus. Since the GP and the PMI are simple optimization techniques without derivatives, they can provide new insights into the given problem to managers, students, and other audiences who may be unfamiliar with multivariable differential calculus. Additionally, by using the properties of limits, we show that the solution to the profit maximization problem with Cobb-Douglas technology is a limiting case of our result.

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Hu, Weiyi, Kelin Li, and Shuyue Zhao. "Periodicity of Nonautonomous Fuzzy Neural Networks with Reaction-Diffusion terms and Distributed Time Delays." Mathematical Problems in Engineering 2022 (October 13, 2022): 1–15. http://dx.doi.org/10.1155/2022/9186393.

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In this paper, the periodicity of a class of nonautonomous fuzzy neural networks with impulses, reaction-diffusion terms, and distributed time delays are investigated. By establishing an integro-differential inequality with impulsive initial conditions and time-varying coefficients, employing the M -matrix theory, Poincar mappings, and fixed point theory, several new sufficient conditions to ensure the periodicity and global exponential stability of the formulated system are obtained. It is worthwhile to mention that our technical methods are practical, in the sense that all new stability conditions are stated in simple algebraic forms, and an optimization method is provided to estimate the exponential convergence rate, so their verification and applications are straightforward and convenient. The validity and generality of our methods are illustrated by two numerical examples.

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Yuldashev, Tursun K., and Erkinjon T. Karimov. "Inverse Problem for a Mixed Type Integro-Differential Equation with Fractional Order Caputo Operators and Spectral Parameters." Axioms 9, no. 4 (October 20, 2020): 121. http://dx.doi.org/10.3390/axioms9040121.

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The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators’ orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag–Leffler function of two variables, the Cauchy–Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem.

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Uppal, T., S. Raha, and S. Srivastava. "Trajectory feasibility evaluation using path prescribed control of unmanned aerial vehicle in differential algebraic equations framework." Aeronautical Journal 121, no. 1240 (May 31, 2017): 770–89. http://dx.doi.org/10.1017/aer.2017.36.

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ABSTRACTMission simulation is a critical activity in the development and operation of Unmanned Aerial Vehicles (UAVs). It is important to ascertain the feasibility of a trajectory in a mission. In this work, an algorithm has been developed for feasibility study of a trajectory of a UAV using prescribed path optimal control through an inverse simulation method. This has been done under a Differential Algebraic Equations (DAE)/Inequalities (DAI) framework. The UAV model together with constraints is represented as a high index DAE system. The trajectory that UAV shall take is prescribed as one of the constraint equations. The solution for the DAE system is obtained using a variation of the alpha method that is capable of handling both equality and inequality constraints on system dynamics. The algorithm involves direct numerical integration of a DAI formulation in a time-stepping manner using a Sequential Quadratic Programming (SQP) solver that detects and satisfy active path constraints at each time step (mesh point). In this unique approach, the model and the constraints are always solved together. The method ensures stable solution at each time step, local minimum at each iteration of simulation and provides a regularised basis to the solver. A typical UAV trajectory has been simulated and demonstrated in this paper. This new approach can be used for path planning of UAVs before the actual control law is designed for flight control computer. Compared to other existing computationally intensive techniques, this approach is computationally simple, ensures continuous constraint satisfaction and provides a viable option for model predictive control of UAVs.

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Tamba, Tua A., Jonathan Chandra, and Bin Hu. "Stability analysis of a hybrid DC-DC buck converter model using dissipation inequality and convex optimization." Journal of Mechatronics, Electrical Power, and Vehicular Technology 14, no. 1 (July 31, 2023): 47–54. http://dx.doi.org/10.14203/j.mev.2023.v14.47-54.

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The stability analysis of a DC-DC buck converter is a challenging problem due to the hybrid systems characteristic of its dynamics. Such a challenge arises from the buck converter operation which depends upon the ON/OFF logical transitions of its electronic switch component to correspondingly activate different continuous vector fields of the converter’s temporal dynamics. This paper presents a sum of squares (SOS) polynomial optimization approach for stability analysis of a hybrid model of buck converter which explicitly takes into account the converter’s electronic switching behavior. The proposed method first transforms the converter’s hybrid dynamics model into an equivalent polynomial differential algebraic equation (DAE) model. An SOS programming algorithm is then proposed to computationally prove the stability of the obtained DAE model using Lyapunov’s stability concept. Based on simulation results, it was found that the proposed method requires only 8.5 seconds for proving the stability of a buck converter model. In contrast, exhaustive simulations based on numerical integration scheme require 15.6 seconds to evaluate the stability of the same model. These results thus show the effectiveness of the proposed method as it can prove the converter stability in shorter computational times without requiring exhaustive simulations using numerical integration.

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Pan, Jie, and Lianglin Xiong. "Novel Criteria of Stability for Delayed Memristive Quaternionic Neural Networks: Directly Quaternionic Method." Mathematics 9, no. 11 (June 4, 2021): 1291. http://dx.doi.org/10.3390/math9111291.

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In this paper, we fixate on the stability of varying-time delayed memristive quaternionic neural networks (MQNNs). With the help of the closure of the convex hull of a set the theory of differential inclusion, MQNN are transformed into variable coefficient continuous quaternionic neural networks (QNNs). The existence and uniqueness of the equilibrium solution (ES) for MQNN are concluded by exploiting the fixed-point theorem. Then a derivative formula of the quaternionic function’s norm is received. By utilizing the formula, the M-matrix theory, and the inequality techniques, some algebraic standards are gained to affirm the global exponential stability (GES) of the ES for the MQNN. Notably, compared to the existing work on QNN, our direct quaternionic method operates QNN as a whole and markedly reduces computing complexity and the gained results are more apt to be verified. The two numerical simulation instances are provided to evidence the merits of the theoretical results.

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Shao, Xing-Guo, Zhen-Cai Zhu, Qing-Guo Wang, Peter CY Chen, Bin Zi, and Guo-Hua Cao. "Non-smooth dynamical analysis and experimental validation of the cable-suspended parallel manipulator." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 226, no. 10 (January 17, 2012): 2456–66. http://dx.doi.org/10.1177/0954406211435585.

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The cable-suspended parallel manipulator replaces the rigid links of traditional parallel robot. The unilateral property of the cable complicates the dynamic analysis of such manipulator and further induces difficulty in control problem. The set-valued tension law is proposed to model the unilateral constraint of the cable, and the dynamics of cable-suspended parallel manipulator is analyzed in the framework of non-smooth dynamics. The resulting non-smooth dynamics model consists of a set of differential–algebraic equations with inequality constraints. Its solution is found by the Moreau midpoint method. An experimental setup was established to verify and validate the effectiveness and accuracy of non-smooth dynamics. And the simulation results generally agree with the experimental results, which demonstrate that the non-smooth dynamics is effective and reasonable for the dynamic analysis of the cable-suspended parallel manipulator. The results of this article deeply reveal the dynamics of the cable-suspended parallel manipulator, and may be used to design more accurate controller for its trajectory control.

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Ouarid, Kaoutar, Abdellatif El Assoudi, Jalal Soulami, and El Hassane El Yaagoubi. "Observer Design for Simultaneous State and Fault Estimation for a Class of Discrete-time Descriptor Linear Models." E3S Web of Conferences 229 (2021): 01020. http://dx.doi.org/10.1051/e3sconf/202122901020.

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This paper investigates the problem of observer design for simultaneous states and faults estimation for a class of discrete-time descriptor linear models in presence of actuator and sensor faults. The idea of the present result is based on the second equivalent form of implicit model [1] which permits to separate the differential and algebraic equations in the considered singular model, and the use of an explicit augmented model structure. At that stage, an observer is built to estimate simultaneously the unknown states, the actuator faults, and the sensor faults. Next, the explicit structure of the augmented model is established. Then, an observer is built to estimate simultaneously the unknown states, the actuator faults, and the sensor faults. By using the Lyapunov approach, the convergence of the state estimation error of the augmented system is analyzed, and the observer’s gain matrix is achieved by solving only one linear matrix inequality (LMI). At long last, an illustrative model is given to show the performance and capability of the proposed strategy.

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Wohlmuth, Barbara. "Variationally consistent discretization schemes and numerical algorithms for contact problems." Acta Numerica 20 (April 28, 2011): 569–734. http://dx.doi.org/10.1017/s0962492911000079.

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We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-ordera prioriconvergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-baseda posteriorierror estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

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Aldawish, Ibtisam, Mohamed Jleli, and Bessem Samet. "On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities." Axioms 12, no. 5 (April 29, 2023): 443. http://dx.doi.org/10.3390/axioms12050443.

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Hermite–Hadamard inequality is a double inequality that provides an upper and lower bounds of the mean (integral) of a convex function over a certain interval. Moreover, the convexity of a function can be characterized by each of the two sides of this inequality. On the other hand, it is well known that a twice differentiable function is convex, if and only if it admits a nonnegative second-order derivative. In this paper, we obtain a characterization of a class of twice differentiable functions (including the class of convex functions) satisfying second-order differential inequalities. Some special cases are also discussed.

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Tariq, Muhammad, Soubhagya Kumar Sahoo, Sotiris K. Ntouyas, Omar Mutab Alsalami, Asif Ali Shaikh, and Kamsing Nonlaopon. "Some New Refinements of Trapezium-Type Integral Inequalities in Connection with Generalized Fractional Integrals." Axioms 11, no. 10 (September 27, 2022): 508. http://dx.doi.org/10.3390/axioms11100508.

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The main objective of this article is to introduce a new notion of convexity, i.e., modified exponential type convex function, and establish related fractional inequalities. To strengthen the argument of the paper, we introduce two new lemmas as auxiliary results and discuss some algebraic properties of the proposed notion. Considering a generalized fractional integral operator and differentiable mappings, whose initial absolute derivative at a given power is a modified exponential type convex, various improvements of the Hermite–Hadamard inequality are presented. Thanks to the main results, some generalizations about the earlier findings in the literature are recovered.

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Srivastava, Hari Mohan, Muhammad Tariq, Pshtiwan Othman Mohammed, Hleil Alrweili, Eman Al-Sarairah, and Manuel De La De La Sen. "On Modified Integral Inequalities for a Generalized Class of Convexity and Applications." Axioms 12, no. 2 (February 5, 2023): 162. http://dx.doi.org/10.3390/axioms12020162.

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In this paper, we concentrate on and investigate the idea of a novel family of modified p-convex functions. We elaborate on some of this newly proposed idea’s attractive algebraic characteristics to support it. This is used to study some novel integral inequalities in the frame of the Hermite–Hadamard type. A unique equality is established for differentiable mappings. The Hermite–Hadamard inequality is extended and estimated in a number of new ways with the help of this equality to strengthen the findings. Finally, we investigate and explore some applications for some special functions. We think the approach examined in this work will further pique the interest of curious researchers.

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MASON, GEOFFREY. "VECTOR-VALUED MODULAR FORMS AND LINEAR DIFFERENTIAL OPERATORS." International Journal of Number Theory 03, no. 03 (September 2007): 377–90. http://dx.doi.org/10.1142/s1793042107000973.

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We consider holomorphic vector-valued modular forms F of integral weight k on the full modular group Γ = SL(2, ℤ) corresponding to representations of Γ of arbitrary finite dimension p. Assuming that the component functions of F are linearly independent, we prove that the inequality k ≥ 1 - p always holds, and that equality holds only in the trivial case when p = 1 and k = 0. For any p ≥ 2, we show how to construct large numbers of representations of Γ for which k = 2 - p. The key idea is to consider representations of Γ on spaces of solutions of certain linear differential equations whose coefficients are modular forms.

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Zhu, Xinyue, Wei Li, and Xueping Luo. "Stability for a Class of Differential Set-Valued Inverse Variational Inequalities in Finite Dimensional Spaces." Axioms 11, no. 9 (September 16, 2022): 475. http://dx.doi.org/10.3390/axioms11090475.

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In this paper, we introduce and study a new class of differential set-valued inverse variational inequalities in finite dimensional spaces. By applying a result on differential inclusions involving an upper semicontinuous set-valued mapping with closed convex values, we first prove the existence of Carathéodory weak solutions for differential set-valued inverse variational inequalities. Then, by the existence result, we establish the stability for the differential set-valued inverse variational inequality problem when the constraint set and the mapping are perturbed by two different parameters. The closedness and continuity of Carathéodory weak solutions with respect to the two different parameters are obtained.

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Raffoul, Youssef N. "Boundedness and stability in nonlinear systems of differential equations using a modified variation of parameters formula." Cubo (Temuco) 25, no. 1 (April 20, 2023): 37–55. http://dx.doi.org/10.56754/0719-0646.2501.037.

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In this research we introduce a new variation of parameters for systems of linear and nonlinear ordinary differential equations. We use known mathematical methods and techniques including Gronwall’s inequality and fixed point theory to obtain boundedness on all solutions and stability results on the zero solution.

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35

Orhan, Halit, and Luminiţa-Ioana Cotîrlă. "Fekete-Szegö Inequalities for Some Certain Subclass of Analytic Functions Defined with Ruscheweyh Derivative Operator." Axioms 11, no. 10 (October 15, 2022): 560. http://dx.doi.org/10.3390/axioms11100560.

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In our present investigation, we introduce and study some new subclasses of analytic functions associated with Ruscheweyh differential operator Dr. We obtain a Fekete–Szegö inequality for certain normalized analytic function defined on the open unit disk for which Drl′(z)ϑzDrl′(z)Drl(z)1−ϑ≺ez(0≤ϑ≤1) lies in a starlike region with respect to 1 and symmetric with respect to the real axis. As a special case of this result, the Fekete–Szegö inequality for a class of functions defined through Poisson distribution series is obtained.

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Agarwal, Ravi P., Snezhana Hristova, and Donal O’Regan. "Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study." Axioms 12, no. 3 (February 21, 2023): 226. http://dx.doi.org/10.3390/axioms12030226.

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Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type stability, and it is a special type of data dependence of solutions of differential equations. For boundary value problems, this type of stability requires some additional understanding, and, in connection with this, we discuss the Ulam-Hyers stability for different types of differential equations, such as ordinary differential equations and generalized proportional Caputo fractional differential equations. To propose an appropriate idea of Ulam-type stability, we consider a boundary condition with a parameter, and the value of the parameter depends on the chosen arbitrary solution of the corresponding differential inequality. Several examples are given to illustrate the theoretical considerations.

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Bayraktar, Bahtiyar, Péter Kórus, and Juan Eduardo Nápoles Valdés. "Some New Jensen–Mercer Type Integral Inequalities via Fractional Operators." Axioms 12, no. 6 (May 25, 2023): 517. http://dx.doi.org/10.3390/axioms12060517.

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In this study, we present new variants of the Hermite–Hadamard inequality via non-conformable fractional integrals. These inequalities are proven for convex functions and differentiable functions whose derivatives in absolute value are generally convex. Our main results are established using the classical Jensen–Mercer inequality and its variants for (h,m)-convex modified functions proven in this paper. In addition to showing that our results support previously known results from the literature, we provide examples of their application.

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Agarwal, Ravi P., and Snezhana Hristova. "Ulam-Type Stability for a Boundary-Value Problem for Multi-Term Delay Fractional Differential Equations of Caputo Type." Axioms 11, no. 12 (December 18, 2022): 742. http://dx.doi.org/10.3390/axioms11120742.

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A boundary-value problem for a couple of scalar nonlinear differential equations with a delay and several generalized proportional Caputo fractional derivatives is studied. Ulam-type stability of the given problem is investigated. Sufficient conditions for the existence of the boundary-value problem with an arbitrary parameter are obtained. In the study of Ulam-type stability, this parameter was chosen to depend on the solution of the corresponding fractional differential inequality. We provide sufficient conditions for Ulam–Hyers stability, Ulam–Hyers–Rassias stability and generalized Ulam–Hyers–Rassias stability for the given problem on a finite interval. As a partial case, sufficient conditions for Ulam-type stability for a couple of multi-term delay, Caputo fractional differential equations are obtained. An example is illustrating the results.

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Hao, Jianwei, Jinrong Wang, and Jiangfeng Han. "Solvability of Conformable Type Frictionless Contact Problem via Hemivariational Inequalities." Axioms 11, no. 6 (June 5, 2022): 271. http://dx.doi.org/10.3390/axioms11060271.

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In this paper, we study a class of conformable frictionless contact problems with the surface traction driven by the conformable impulsive differential equation. The existence of a mild solution for conformable impulsive hemivariational inequality is obtained by the Rothe method, subjectivity of multivalued pseudomonotone operators and the property of the conformable derivative. Notice that we imply some new fractional viscoelastic constitutive laws.

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Zhang, Wenlin, Michal Fečkan, and Jinrong Wang. "The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System." Axioms 11, no. 7 (July 20, 2022): 347. http://dx.doi.org/10.3390/axioms11070347.

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In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique.

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Vivas-Cortez, Miguel, Muhammad Uzair Awan, Sadia Talib, Artion Kashuri, and Muhammad Aslam Noor. "Some New Post-Quantum Integral Inequalities Involving Twice (p, q)-Differentiable ψ-Preinvex Functions and Applications." Axioms 10, no. 4 (October 29, 2021): 283. http://dx.doi.org/10.3390/axioms10040283.

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The main motivation of this article is derive a new post-quantum integral identity using twice (p, q)-differentiable functions. Using the identity as an auxiliary result, we will obtain some new variants of Hermite–Hadamard’s inequality essentially via the class of ψ-preinvex functions. To support our results, we offer some applications to a special means of positive real numbers and twice (p, q)-differentiable functions that are in absolute value bounded as well.

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Xu, Lili, Yalong Xue, Qifa Lin, and Chaoquan Lei. "Global Attractivity of Symbiotic Model of Commensalism in Four Populations with Michaelis–Menten Type Harvesting in the First Commensal Populations." Axioms 11, no. 7 (July 12, 2022): 337. http://dx.doi.org/10.3390/axioms11070337.

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This article revisits the stability property of a symbiotic model of commensalism with Michaelis–Menten type harvesting in the first commensal populations. By constructing some suitable Lyapunov functions, we provide a thorough analysis of the dynamic behaviors of the subsystem composed of the second and third species. After that, by applying the stability results of this subsystem and combining with the differential inequality theory, sufficient conditions which ensure the global attractivity of the equilibria are obtained. The results obtained here essentially improve and generalize some known results.

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Ali, Rosihan M., Satwanti Devi, and A. Swaminathan. "Inclusion properties for a class of analytic functions defined by a second-order differential inequality." Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 112, no. 1 (December 30, 2016): 117–33. http://dx.doi.org/10.1007/s13398-016-0368-1.

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Kiskinov, Hristo, Magdalena Veselinova, Ekaterina Madamlieva, and Andrey Zahariev. "A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis." Axioms 10, no. 2 (April 27, 2021): 75. http://dx.doi.org/10.3390/axioms10020075.

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In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made.

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Chasreechai, Saowaluck, Muhammad Aamir Ali, Muhammad Amir Ashraf, Thanin Sitthiwirattham, Sina Etemad, Manuel De la Sen, and Shahram Rezapour. "On New Estimates of q-Hermite–Hadamard Inequalities with Applications in Quantum Calculus." Axioms 12, no. 1 (January 2, 2023): 49. http://dx.doi.org/10.3390/axioms12010049.

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In this paper, we first establish two quantum integral (q-integral) identities with the help of derivatives and integrals of the quantum types. Then, we prove some new q-midpoint and q-trapezoidal estimates for the newly established q-Hermite-Hadamard inequality (involving left and right integrals proved by Bermudo et al.) under q-differentiable convex functions. Finally, we provide some examples to illustrate the validity of newly obtained quantum inequalities.

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Klimov, Vladimir S. "Isoperimetric and Functional Inequalities." Modeling and Analysis of Information Systems 25, no. 3 (June 30, 2018): 331–42. http://dx.doi.org/10.18255/1818-1015-2018-3-331-342.

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We establish lower estimates for an integral functional$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$where $\Omega$ -- a bounded domain in $\mathbb{R}^n \; (n \geqslant 2)$, an integrand $f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)$ -- a function that is $B$-measurable with respect to a variable $t$ and is convex and even in the variable $p$, $\nabla u(x)$ -- a gradient (in the sense of Sobolev) of the function $u \colon \Omega \rightarrow \mathbb{R}$. In the first and the second sections we utilize properties of permutations of differentiable functions and an isoperimetric inequality $H^{n-1}( \partial A) \geqslant \lambda(m_n A)$, that connects $(n-1)$-dimensional Hausdorff measure $H^{n-1}(\partial A )$ of relative boundary $\partial A$ of the set $A \subset \Omega$ with its $n$-dimensional Lebesgue measure $m_n A$. The integrand $f$ is assumed to be isotropic, i.e. $f(t,p) = f(t,q)$ if $|p| = |q|$.Applications of the established results to multidimensional variational problems are outlined. For functions $ u $ that vanish on the boundary of the domain $\Omega$, the assumption of the isotropy of the integrand $ f $ can be omitted. In this case, an important role is played by the Steiner and Schwartz symmetrization operations of the integrand $ f $ and of the function $ u $. The corresponding variants of the lower estimates are discussed in the third section. What is fundamentally new here is that the symmetrization operation is applied not only to the function $u$, but also to the integrand $f$. The geometric basis of the results of the third section is the Brunn-Minkowski inequality, as well as the symmetrization properties of the algebraic sum of sets.

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Karthikeyan, Kulandhivel, and Ozgur Ege. "Boundary value problems of higher order fractional integro-differential equations involving Gronwall's inequality in Banach spaces." Miskolc Mathematical Notes 24, no. 2 (2023): 805. http://dx.doi.org/10.18514/mmn.2023.4049.

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Liu, Haiyan, Siyan Xue, Yuan Cheng, and Shouheng Tuo. "A New Parameterless Filled Function Method for Global Optimization." Axioms 11, no. 12 (December 19, 2022): 746. http://dx.doi.org/10.3390/axioms11120746.

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The filled function method is an effective way to solve global optimization problems. However, its effectiveness is greatly affected by the selection of parameters, and the non-continuous or non-differentiable properties of the constructed filled function. To overcome the above-mentioned drawbacks, in this paper, a new parameterless filled function is proposed that is continuous and differentiable. Theoretical proofs have been made to show the properties of the proposed filled function. Based on the new filled function, a filled function algorithm is proposed to solve unconstrained global optimization problems. Experiments are carried out on widely used test problems and an application of supply chain problems with equality and inequality constraints. The numerical results show that the proposed filled function is effective.

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Almeida, Ricardo, Ravi P. Agarwal, Snezhana Hristova, and Donal O’Regan. "Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks." Axioms 10, no. 4 (November 27, 2021): 322. http://dx.doi.org/10.3390/axioms10040322.

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A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

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Yaghoubi, Hassan, Assef Zare, and Roohallah Alizadehsani. "Analysis and Design of Robust Controller for Polynomial Fractional Differential Systems Using Sum of Squares." Axioms 11, no. 11 (November 7, 2022): 623. http://dx.doi.org/10.3390/axioms11110623.

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This paper discusses the robust stability and stabilization of polynomial fractional differential (PFD) systems with a Caputo derivative using the sum of squares. In addition, it presents a novel method of stability and stabilization for PFD systems. It demonstrates the feasibility of designing problems that cannot be represented in LMIs (linear matrix inequalities). First, sufficient conditions of stability are expressed for the PFD equation system. Based on the results, the fractional differential system is Mittag–Leffler stable when there is a polynomial function to satisfy the inequality conditions. These functions are obtained from the sum of the square (SOS) approach. The result presents a valuable method to select the Lyapunov function for the stability of PFD systems. Then, robust Mittag–Leffler stability conditions were able to demonstrate better convergence performance compared to asymptotic stabilization and a robust controller design for a PFD equation system with unknown system parameters, and design performance based on a polynomial state feedback controller for PFD-controlled systems. Finally, simulation results indicate the effectiveness of the proposed theorems.

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Journal articles on the topic 'Differential Algebraic Inequality'

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