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We extend the virtual element method to the two-dimensional time-fractional parabolic PDE, characterized by a fractional derivative of order α ∈ (0 , 1) in time. To illustrate the working of this fractional virtual element scheme, a numerical investigation of the following time-fractional problem over distorted polygonal meshes is conducted. c D t α u (z , t) − Δ u = f (z , t) in z ∈ Ω , t ∈ (0 , T ] , where Ω is a spatial domain, α is fractional order, and t is time variable. Our methodology is based on the fundamental technical component, fractional version of the Grunwald–Letnikov approximation. We prove the method's well-posedness, that is the approximate solution's existence and uniqueness. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L 2 -norm and H 1 -norm over distorted mesh configuration is validated by numerical results, underlining the practical effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We extend the virtual element method to the two-dimensional time-fractional parabolic PDE, characterized by a fractional derivative of order α∈(0,1) in time. To illustrate the working of this fractional virtual element scheme, a numerical investigation of the following time-fractional problem over distorted polygonal meshes is conducted. cDtαu(z,t)−Δu=f(z,t)inz∈Ω,t∈(0,T],where Ω is a spatial domain, α is fractional order, and t is time variable. Our methodology is based on the fundamental technical component, fractional version of the Grunwald–Letnikov approximation. We prove the method’s well-posedness, that is the approximate solution’s existence and uniqueness. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L2-norm and H1-norm over distorted mesh configuration is validated by numerical results, underlining the practical effectiveness of the proposed method.

In this study, we establish a new inertial generalized viscosity approximation method and prove that the resulting sequence strongly converges to a common solution of a split generalized mixed equilibrium problem, fixed point problem for a finite family of nonexpansive mappings and hierarchical fixed point problem in real Hilbert spaces. As an application, we demonstrate the use of our main finding in compressed sensing in signal processing. Additionally, we include numerical examples to evaluate the efficiency of the suggested method and then conduct a comparative analysis of its efficiency with different methods. Our findings can be used in a variety of contexts to improve results.

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element f ∈ X . Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence { u n } ⊂ X , which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws. [ABSTRACT FROM AUTHOR]

In this study, we establish a new inertial generalized viscosity approximation method and prove that the resulting sequence strongly converges to a common solution of a split generalized mixed equilibrium problem, fixed point problem for a finite family of nonexpansive mappings and hierarchical fixed point problem in real Hilbert spaces. As an application, we demonstrate the use of our main finding in compressed sensing in signal processing. Additionally, we include numerical examples to evaluate the efficiency of the suggested method and then conduct a comparative analysis of its efficiency with different methods. Our findings can be used in a variety of contexts to improve results. [ABSTRACT FROM AUTHOR]

This article, published in Mathematical Notes, explores the minimal norm of a projection operator for linear interpolation on an n-dimensional ball. The author proves that the interpolation projection operator, with nodes coinciding with the vertices of a regular simplex inscribed in the boundary sphere, has the least norm. The article provides mathematical formulas and proofs to support this claim. The author also examines the relationship between the simplex and the minimal ellipsoid containing it. The article is written in a technical and mathematical language, aimed at researchers and mathematicians. The author acknowledges A. V. Akopyan for their assistance in proving Theorem 1. [Extracted from the article]

This paper presents a rapidly direct algorithm (RDA) to calculate explicit Fréchet derivatives (EFD) for marine controlled-source electromagnetic measurement (MCSEM) in a three-dimensional (1-D) transversely isotropic (TI) formation. By discretizing the Helmholtz equations about coupled potentials on Yee’s staggered grids by the 3-D finite volume method (FVM), we obtain a complex linear system about the unknown potentials excited by a mass of moving electric current sources. To efficiently determine electromagnetic (EM) fields, we introduce an interpolation operator and projection operator per receiver by using the direct solver PARDISO and 3-D Newtown interpolation. Based on this, the perturbation in goal conductivity is expressed as a piece-wise constant function according to block or pixel model. The spatial scattered electric currents will be decomposed into a series of electric current elements distributed on Yee’s grids due to the conductivity perturbation. We then discretize the scattered electric currents by 3-D FVM and obtain the new right-hand terms about the unknown scattered potentials. This allows for fast production of the linear relationship between the changes in the EM fields and the relative conductivity perturbation per block or pixel, ultimately resulting in the EFD of MCSEM responses. Numerical results demonstrate the efficiency and accuracy of this method. The 3-D pixel sensitivities are presented to further investigate the response characteristics of MCSEM in several cases.

A high-order Kalman filter for full-dimensional variables is proposed for a class of dynamic systems whose state model and measurement model are both nonlinear. The filter requires Taylor expansion of the system equations, and then performs Kronecker product operation on the linear part in the Taylor expansion. Finally, a linear dynamic model is achieved based on the full-dimensional vector formed by the state variables and the high-order dimension expansion variables. After designing the filter, the Kalman filter for the original state variables estimation was selected through the projection operator. The excellent performance of the novel filter is analyzed from the aspects of the information utilization of the state estimation value and the size of the state estimation error covariance matrix. The numerical verification is carried out by computer simulation. [ABSTRACT FROM AUTHOR]

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element f∈X. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence {un}⊂X, which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws.

In this paper, neurodynamic approaches are proposed for solving nonsmooth distributed optimization problems under inequality and set constraints, that is to find the solution that minimizes the sum of local cost functions. A continuous-time neurodynamic approach is designed and its state solution exists globally and converges to an optimal solution of the corresponding distributed optimization problem. Then, a neurodynamic approach with event-triggered mechanism is considered for the purpose of saving communication costs, and then, the convergence and its Zeno-free property are proved. Moreover, to realize the practical application of the neurodynamic approach, a discrete-time neurodynamic approach is proposed to solve nonsmooth distributed optimization problems under inequality and set constraints. It is rigorously proved that the iterative sequence generated by the discrete-time neurodynamic approach converges to the optimal solution set of the distributed optimization problem. Finally, numerical examples are solved to demonstrate the effectiveness of the proposed neurodynamic approaches, and the neurodynamic approach is further applied to solve the ill-conditioned Least Absolute Deviation problem and the load sharing optimization problem. [ABSTRACT FROM AUTHOR]

The Dai-Kou method Dai and Kou (2013), [12] is efficient for solving unconstrained optimization problems. However, its modified variants are quite rare for constrained nonlinear monotone equations. In an attempt to address this, two adaptive versions of the scheme with new and efficient parameter choices are presented in this paper. The schemes are obtained by analyzing eigenvalues of a modified Dai-Kou iteration matrix and constructing two new directions, which are used in the scheme's algorithms. The new methods are derivative-free, which is an attribute required for handling problems with very large dimensions. Both methods also satisfy the required condition for analyzing global convergence in the literature. By applying mild conditions, it is shown that the schemes are globally convergent and description of their effectiveness is achieved through experiments with four effective schemes for solving constrained nonlinear monotone equations. Furthermore, the methods are applied to recover images that are contaminated by impulse noise in compressed sensing.

In this paper, we consider a new class of quasi-variational inequalities, which is called the general quasi-variational inequality. Using the projection operator technique, we establish the equivalence between the general quasi-variational inequalities and the fixed point problems. We use this alternate formulation to propose some new inertial iterative schemes for solving the general quasi-variational inequalities. The convergence criteria of the new inertial projection methods under some appropriate conditions is investigated. Since the general quasi-variational inequalities include the quasi-variational inequalities, variational inequalities, complementarity problems and the related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare the efficiency of the proposed methods with other known methods.

Nonlinear Dynamic Inversion(NDI) control has excellent rapidity and decoupling ability, unfortunately it lacks the essential robustness to disturbance. From the perspective of enhancing the robustness, an adaptive NDI method based on L1 adaptive structure is proposed. The L1 adaptive structure is introduced into the NDI control to enhance its robustness, which also guarantees the stability and expected dynamic performance of the system suffering from the disturbance influence. Secondly, the flight control law of the advanced aircraft is designed based on the present method to improve the robustness and fault tolerance of the flight control system. Finally, the effectiveness of the flight control law based on the present approach is verified under the fault disturbance. The results showed that the flight control law based on L1 adaptive NDI has excellent dynamic performance and strong robustness to parameter uncertainties and disturbances.

In this paper, we introduce and consider some new classes of general quasi variational inequalities, which provide us with unified, natural, novel and simple framework to consider a wide class of unrelated problems arising in pure and applied sciences. We propose some new inertial projection methods for solving the general quasi variational inequalities and related problems. Convergence analysis is investigated under certain mild conditions. Since the general quasi variational inequalities include quasi variational inequalities, variational inequalities, and related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare these methods with other technique for solving quasi variational inequalities for further research activities.

A high-order Kalman filter for full-dimensional variables is proposed for a class of dynamic systems whose state model and measurement model are both nonlinear. The filter requires Taylor expansion of the system equations, and then performs Kronecker product operation on the linear part in the Taylor expansion. Finally, a linear dynamic model is achieved based on the full-dimensional vector formed by the state variables and the high-order dimension expansion variables. After designing the filter, the Kalman filter for the original state variables estimation was selected through the projection operator. The excellent performance of the novel filter is analyzed from the aspects of the information utilization of the state estimation value and the size of the state estimation error covariance matrix. The numerical verification is carried out by computer simulation.

This paper studies the problem of the secure state estimation of cyber-physical systems. The system model we studied is a class of discrete-time systems with state delay and sparse sensor attacks. In the design of the observer, a Projection Operator is utilized to determine the correct attack set, and the gain matrix will be switched accordingly. In this way, the excessive estimation error could be avoided effectively. In addition, based on the generalization of the classical Krasovskii Stability Theorem for stability analysis of time-delay systems, a Lyapunov-Krasovskii functional is chosen and a sufficient condition for the existence of the observer together with its design method is determined. The designed observer overcomes the difficulties caused by state delay and sensor attacks, and has good performance in secure estimation. Finally, two examples are given to verify that the observer can restore the desired estimate in a short time when a sudden attack occurs.

We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full eigenvalue spectrum of the mixed-state presentation of a process's ε-machine causal-state dynamic. Measures include correlation functions, power spectra, past-future mutual information, transient and synchronization informations, and many others. As a result, a direct and complete analysis of intrinsic computation is now available for the temporal organization of finitary hidden Markov models and nonlinear dynamical systems with generating partitions and for the spatial organization in one-dimensional systems, including spin systems, cellular automata, and complex materials via chaotic crystallography.

Orthogonal direct-sum decompositions of finite games into potential, harmonic and nonstrategic components exist in the literature. In this paper we study the issue of decomposing games that are strategically equivalent from a game-theoretical point of view, for instance games obtained via transformations such as duplications of strategies or positive affine mappings of the payoffs. We show the need to define classes of decompositions to achieve commutativity of game transformations and decompositions. [ABSTRACT FROM AUTHOR]

A method for investigating the problem of the existence and construction of a multiperiodic solution of the quasilinear strictly hyperbolic systems is proposed. [ABSTRACT FROM AUTHOR]

This paper develops a neural-network-based model reference adaptive control (MRAC) scheme for a rotorcraft in the presence of input saturation. Such a control scheme provides acceptable tracking performance for the rotorcraft in a wide range of flight conditions. Combined with hyperbolic tangent functions, the MRAC scheme is capable of tracking the reference signals without violating input constraints. A modified projection operator is utilized to prevent system from parameter drift due to the strong nonlinearity and uncertainty of the rotorcraft mathematical model. Stability of the proposed MRAC scheme is proved based on Lyapunov stability theory. The performance of the resulting controller is tested by conducting numerical simulations for an autonomous rotorcraft in various flight missions. • A robust modification of adaptive control is proposed to prevent nonlinear system from parameter drift. • A neural network based MRAC scheme is conducted in the presence of input saturation using hyperbolic tangent function. • The proposed MRAC scheme is applied to a nonlinear rotorcraft model. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze a new approximation schemes (3) to solve the extended general variational inequalities (2), which were introduced by Muhammad Aslam Noor (see[7, 9]). Using the projection operator technique, we establish the equivalence between the extended general variational inequalities and the fixed-point problem. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed. [ABSTRACT FROM AUTHOR]

A robust controller with an adaptive projection-based parameter estimator is presented for the air-breathing hypersonic vehicles. First, a novel parameter-varying model is established wherein the number of variable parameters is significantly reduced. In this way, the problem of overparameterization is avoided, and the controller design procedure is simplified. Then, the framework of the controller is established based on the backstepping approach, in which a second-order filter is introduced to solve the problem of “explosion of terms”. In order to enhance the robustness of the controller, an adaptive parameter estimator is designed. The Lipschitz continuous dead-zone modification is used to solve the problem of parameter draft in the conventional adaptive control and a sufficiently smooth projection operator is introduced to guarantee the uniform boundedness of adaptive parameters. The simulation results show that the proposed robust controller achieves the stable tracking of the reference commands under the parameter perturbation, and the variable parameters are limited within the preset range.

The aim of this brief was to develop a class of state estimator modified with a projection function to reconstruct the non-measurable variables of a state-constraint system. The main characteristic of this observer was that the estimated states remain inside the compact set defined by the given variables constraints, independently of the output noise and internal modeling uncertainties. The application of the projection operator introduced a non-smooth form for the observer design. Hence, integral Lyapunov functions can be used to realize the stability study. The design of a Lyapunov-Krasovskii functional overcome the smoothness problem and justifies the ultimate boundedness of the estimation error. The upper bound for the estimation error is rationally related to the observer’s lag and it is nonlinear associated with the power of output noises and modeling errors. Two numerical examples (the Chua’s circuit and the ozonation of toxic compounds adsorbed in soil) illustrated the application of the suggested observer. [ABSTRACT FROM AUTHOR]

This paper puts forward a projection-based adaptive command filtered fuzzy nonsingular terminal sliding mode backstepping (PACFTB) control method for the speed control of the linear induction motor (LIM) with unknown end effects. Firstly, the technique of fuzzy logic systems (FLS) is investigated to approximate the nonlinear components of the LIM's mathematical model, which reduces the difficulty and cost of controller design. Then, a constrained command-filtered backstepping controller is designed with a filtering compensator compensating for the inherent error of constrained filter. Moreover, the nonsingular terminal sliding mode control method is combined in the controller design for its advantages of finite-time convergence of the system, and the projection-operator-based adaptive laws are established at the same time. Finally, the stability analysis proves that the boundedness and stability of all signals can be ensured with the proposed PACFTB controller, and the simulation results along with experiment results verify that the proposed control strategy has better control performance than the conventional command filter backstepping and PI controller. [ABSTRACT FROM AUTHOR]

Studying spectral properties of operator polynomials is reduced to studying the corresponding spectral properties of operators defined by operator matrices. The results are used to investigate second-order differential operators by associating them with the corresponding first-order differential operators and using their properties related to invertibility. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that under some conditions, wavelet packet basis of L 2 (ℝ) can be used as a tool for the uniform approximation of an M -times (M > 0) continuously differentiable and square integrable function f. Sufficient conditions which establish that the approximations of wavelet packet sequences of square integrable function f at lower levels are uniformly reliable and they uniformly approach zero as j → ∞ are given. Finally, a method based on wavelet packet expansion to find the definite integral of a function in L 2 (ℝ) is given and its error analysis has been discussed. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive chaos control is proposed for a typical vibratory piezomagnetoelastic energy harvesting system to return the chaotic behavior to a periodic one. Piezomagnetoelastic energy harvesting systems show chaotic behaviors in spite of harmonic input. Although, the chaotic behavior of the system gives higher output voltage than the periodic motion, it is preferred to the output voltage as this is periodic for charging a battery or a capacitor efficiently. Therefore, the chaos control is important in this system. The physical model is composed of the upper and lower piezoelectric layers on a cantilever taper beam, one attached tip magnet, and two external magnets (EM). Position of the EM is controlled by inputs. Firstly, chaotic and periodic regions are detected by utilizing the bifurcation diagrams, phase plan portrait, and Poincaré maps. Then an adaptive controller is proposed for controlling of the chaotic behaviors in the presence of uncertainty due to magnetic forces. The control law is derived based on the inverse dynamic method and the uncertainty elements of the controller are estimated using radial basis function (RBF) network. The weights of the RBF network are obtained using an adaptation law. The adaptation laws are derived based on Lyapunov stability theory and a projection operator. The distance of the tip magnet and the EM as well as the gap distance of two EM are used to control the chaotic behavior. Simulation results show that the proposed controller can return the chaotic motion to a periodic one in spite of the uncertainties in the magnetic forces. [ABSTRACT FROM AUTHOR]

The projector onto the Minkowski sum of closed convex sets is generally not equal to the sum of individual projectors. In this work, we provide a complete answer to the question of characterizing the instances where such an equality holds. Our results unify and extend the case of linear subspaces and Zarantonello's results for projectors onto cones. A detailed analysis in the case of convex combinations is carried out, and we also establish the partial sum property for projectors onto convex cones. [ABSTRACT FROM AUTHOR]

We investigate via a conjugate duality approach general nonlinear minmax location problems formulated by means of an extended perturbed minimal time function, necessary and sufficient optimality conditions being delivered together with characterizations of the optimal solutions in some particular instances. A parallel splitting proximal point method is employed in order to numerically solve such problems and their duals. We present the computational results obtained in matlab on concrete examples, successfully comparing these, where possible, with earlier similar methods from the literature. Moreover, the dual employment of the proximal method turns out to deliver the optimal solution to the considered primal problem faster than the direct usage on the latter. Since our technique successfully solves location optimization problems with large data sets in high dimensions, we envision its future usage on big data problems arising in machine learning. [ABSTRACT FROM AUTHOR]

This paper puts forward a projection-based adaptive command filtered fuzzy nonsingular terminal sliding mode backstepping (PACFTB) control method for the speed control of the linear induction motor (LIM) with unknown end effects. Firstly, the technique of fuzzy logic systems (FLS) is investigated to approximate the nonlinear components of the LIM’s mathematical model, which reduces the difficulty and cost of controller design. Then, a constrained command-filtered backstepping controller is designed with a filtering compensator compensating for the inherent error of constrained filter. Moreover, the nonsingular terminal sliding mode control method is combined in the controller design for its advantages of finite-time convergence of the system, and the projection-operator-based adaptive laws are established at the same time. Finally, the stability analysis proves that the boundedness and stability of all signals can be ensured with the proposed PACFTB controller, and the simulation results along with experiment results verify that the proposed control strategy has better control performance than the conventional command filter backstepping and PI controller.

We show that, under some additional assumptions, all projection operators onto latticially closed subsets of the Orlicz-Musielak space generated by \(\Phi\) are isotonic if and only if \(\Phi\) is convex with respect to its second variable. A dual result of this type is also proven for antiprojections. This gives the positive answer to the problem presented in Opuscula Mathematica in 2012.

For linear bounded operators A, B from the Banach algebra of linear bounded operators acting in a Banach space, we prove a number of statements on the coincidence of the properties of the operators IY − AB, IX − BA related to their kernels and images. In particular, we establish the identical dimension of the kernels, their simultaneous complementability property, the coincidence of the codimensions of the images, their simultaneous Fredholm property and the coincidence of their Fredholm indices. We construct projections onto the image and the kernel of these operators. We establish the simultaneous nonquasianalyticity property of the operators AB and BA. [ABSTRACT FROM AUTHOR]

Distributed energy sources are usually interfaced to the grid using power electronic converters, and lack of inertia in inverter dominated microgrids can affect the system stability. This paper presents a new method for virtual synchronous generator (VSG) control in order to solve the low system inertia and support the grid frequency problem. In this paper, the VSG based on electromagnetic transient characteristics is improved and an adaptive command filter back-stepping controller is designed. Firstly, the rotor swing equation and power part are modeled to complete the controller design for achieving system stability in the islanded, grid-connected and transition modes. In addition, a limited-amplitude command filter is used to deal with computational complexity and nonlinear saturation problems in the design process. Secondly, projection operator, and adaptive inertia and damping control are introduced to reduce the modeling error and disturbance caused by changing parameters. This ensures the boundedness of the estimated value and further improves the frequency response, especially in the transition mode. Finally, simulation results show that the proposed controller is more effective than the traditional control method for achieving power stability and frequency improvement.

In this paper the problem of a particle in an array of hexagons with periodic boundary condition is solved. Using the projection operators, we categorize eigenfunctions corresponding to each of the irreducible representations of the symmetry group . Based on these results, the Dirichlet and Neumann boundary conditions are discussed.

A brief history and a mathematical description of the dynamic projection operators technique is presented. An example of the general Cauchy problem for evolution equations in 1+ 1 dimensions is studied in detail. A boundary regime propagation is formulated in terms of operators and illustrated by the simplest one-dimensional diffusion equation. The problem of temperature waves is discussed.

In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some x * ∈ C : = ∩ i = 1 m { x ∈ H | c i ( x ) ≤ 0 } , where m is a positive integer, H is a real Hilbert space, and { c i } i = 1 m are convex functions defined as H . The key of the CPM is that, for the current iterate x k , the CPM firstly constructs a new level set H k through a convex combination of some of { c i } i = 1 m in an appropriate way, and then updates the new iterate x k + 1 only by using the projection P H k . We also introduce the combination relaxation projection methods (CRPM) to project onto half-spaces to make CPM easily implementable. The simplicity and easy implementation are two advantages of our methods since only one projection is used in each iteration and the projections are also easy to calculate. The weak convergence theorems are proved and the numerical results show the advantages of our methods.

Application of CTRW (continuous time random walks) to dipole hopping transport is reviewed. Conditions of applicability of basic kinetic equations to spin systems are indicated. Correct versions of derivation of the CTRW-equations are presented. Existence of different forms of memory kernels is demonstrated. Correction of Scher-Lax memory kernel within geometrical memory approach is fulfilled in accordance with leading terms of concentration expansion. Approximate solution for autocorrelation function is considered. Modern state of numerical simulation and experimental measurements of autocorrelation function in nuclear polarization delocalization are described. It is shown, that application of the CTRW was more successful in description of dipole transport than for hopping conductivity.