Stabilization and Analytic Approximate Solutions of an Optimal Control Problem
-
Camelia Pop
Abstract
This paper analyses a dynamical system derived from a left-invariant, drift-free optimal control problem on the Lie group SO(3) × ℝ3 × ℝ3 in deep connection with the important role of the Lie groups in tackling the various problems occurring in physics, mathematics, engineering and economic areas [1, 2, 3, 4, 5]. The stability results for the initial dynamics were inconclusive for a lot of equilibrium points (see [6]), so a linear control has been considered in order to stabilize the dynamics. The analytic approximate solutions of the resulting nonlinear system are established and a comparison with the numerical results obtained via the fourth-order Runge-Kutta method is achieved.
1 Introduction
In the last of years, the optimal control problems on the Lie groups has drawn a lot of attention. The opportunity to study a mechanical system using geometric tools have had a big success in many areas of physics (see for instance [7, 8, 9, 10, 11]), engineering [12, 13, 14, 15]) and biology [16]. The left-invariant, drift-free control systems on the Lie groups belong to this class; having a Lie group as space configuration, the existence of a Hamilton-Poisson formulation is assured and therefore several powerful techniques can be employed such as the reduction of the symmetric system, the determination of the relative equilibria from a global point of view, the energy methods for studying the stability, specific methods for finding the periodical orbits, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems. Using the Lie group framework of the system, the determination of the equilibria and the energy method for determining the nonlinear stability, were performed in [6]. However, the nonlinear stability of some equilibria has remained unsolved, both methods - the energy-Casimir and Arnold’s - being unsuccessful.
The Hamilton-Poisson systems play an important role in mathematics, geometrical mechanics and physics (classical, quantum, or relativistic) due to their deep connections to metriplectic systems and, being useful to achieve their syncronization in information security and cryptography [17, 18, 19, 20].
In this paper, a linear control function is proposed in order to stabilize an equilibria and the analytic approximate solutions via the Optimal Homotopy Asymptotic Method (OHAM) of the controlled system are presented.
2 The Geometrical Framework of the Problem and the Stabilization by one Linear Control
To begin with let us introduce very briefly the left-invariant, drift-free control system on the Lie group G = SO(3) × ℝ3 × ℝ3 considered in [6]
where X ∈ G, ui, i = 1, 9 are the controls and {Ai}i = 1,9 is the standard basis of the corresponding Lie algebra g. Then, the following results hold:
Proposition 2.1
[6]
The system(1)is controllable.
The controls that minimize the function
and steer the system(1)fromX = X0att = 0 toX = Xfatt = tfare given by the solutions of
The main goal of this section is to establish some stability results of the equilibrium points
were already obtained in [6], but the stability problem for the other equilibrium states remains unsolved.
In order to stabilize the equilibrium states
so the controlled system (2)-(3) can be explicitly written:
Proposition 2.2
The controlled system (4) has the Hamilton-Poisson realization
whereΠ−is the minus Lie-Poisson structure on the dualg*of the Lie algebra, given by
and the Hamiltonian function
Proof
Indeed, it is not hard to see that the dynamics (4) can be put into the equivalent form:
as required. More details about Poisson structures can be found in [21]. □
Proposition 2.3
[6] The functionsC1, C2andC3given by:
are Casimirs of our configuration.
Let B be the matrix of linear part of our controlled system (4), that is
At the equilibrium of interest, its characteristic polynomial has the following expression
If MP < 0, we have five zero eigenvalues and four purely imaginary eigenvalues, so we can conclude:
Proposition 2.4
IfMP < 0, M, N, P ∈ ℝ*, then the controlled system (4) may be spectral stabilized about the equilibrium states
Moreover we can prove:
Proposition 2.5
IfMP < 0, M, N, P ∈ ℝ*, then the controlled system (4) may be nonlinear stabilized about the equilibrium states
Proof
Let us consider the function (see [22] for details):
Then we have successively:
∇Fλ,μ(
Considering now
then, for all v ∈ W, i.e. v = (a, b, c, 0, d, 0, e, f, g), a, b, c, d, e, f, g ∈ ℝ, we have
which is positive definite under the restriction λ > 0 and MP < 0, and so
is positive definite.
Therefore, if MP < 0, then the equilibrium states
3 Basic Ideas of the Optimal Homotopy Asymptotic Method
In order to find the approximate analytic solutions of the controlled system (2.4), we shall start with a brief description of this method. For an equation of the form:
subject to the initial conditions of the type:
where 𝓛 is a linear operator, 𝓝 is a nonlinear one and x(t) is the unknown smooth function of the Eq. (5), the approximate analytic solutions can be obtained using the third modified version of Optimal Homotopy Asymptotic Method described hereinafter (see [23] for details).
Following [23, 24] we construct the homotopy given by:
where p ∈ [0, 1] is the embedding parameter, H(t, Cj), (H ≠ 0) is an auxiliary convergence-control function, depending on the variable t and on the parameters C1, C2, …, Cs and the function X(t, p) has the expression:
The following properties hold:
and
The governing equations of x0(t) and x1(t, Cj) can be obtained by equating the coefficients of p0 and p1, respectively:
The expression of x0(t) can be found by solving the linear Eq. (11). Also, to compute x1(t, Cj) we solve the Eq. (12), by taking into consideration that the nonlinear operator N presents the general form:
where m is a positive integer and hk(t) and gk(t) are known functions depending both on x0(t) and N.
Although the Eq. (12) is a nonhomogeneous linear one in the most cases its solution can not be found.
In order to compute the function x1(t, Cj), we do not solve Eq. (12) but from the theory of differential equation, taking into consideration the method of variation of parameters, the method of influence function, Cauchy method, the operator method [25], it is more convenient to consider the unknown function x1(t, Cj) using the following steps:
First we consider one of the following expressions for x1(t, Cj)
or
These expressions of Hk(t, hj(t), Cj) contain both linear combinations of the functions hj and the parameters Cj, j = 1, s. The summation limit m is an arbitrary positive integer number.
Finally, the convergence-control parameters C1, C2, …, Cs, which determine the first-order approximate solution (16), can be optimally computed by means of various methods, such as: the least square method, the Galerkin method, the collocation method, the Kantorowich method or the weighted residual method. ▪
In what follows, we will show that the approximate analytic solution x(t, Cj) given by Eq. (16) is a weak ϵ-approximate solution of the problem Eq. (5) with the initial condition Eq. (6).
Definition 3.1
[26] We call an ϵ-approximate solution of the problem (5) on the domain (0, ∞) a smooth functionx(t, Cj) of the form (16) which satisfies the following condition:
together with the initial condition from Eq. (6), where the residual functionR(t, x(t, Cj)) is obtained by substituting the Eq. (16) into Eq. (5), i.e.
4 Analytic Approximate Solutions of the System (4) using Optimal Homotopy Asymptotic Method
We now apply the procedure described above to the nonlinear differential system given by the Eqs. (4) with the initial conditions
Considering the linear operators 𝓛 [xi(t)]:
and 𝓝[xi(t)], i = 1, 9:
where K > 0 and K1 > 0 are unknown parameters, then the solutions of the Eq. (11) for the initial approximations xi0, i = 1, 9 are given by:
By substituting Eqs. (20) into Eqs. (19), we reach the following results:
Remark 4.1
It is easy to see that the nonlinear operators 𝓝[xi0(t)], i = 1, 9 can be written as linear combinations of the functions
Although the Eq. (12) is a nonhomogeneous linear one, finding its solution could be a difficult problem.
In order to compute the functions xi1(t, Bs, Cs), i = 1, 9, by the Remark 4.1 and the previous Section, the first approximation xi1, i = 1, 9 can be written as:
Now, using the Eqs. (20) and Eq. (22), the first-order analytic approximate solution given by Eq. (16) of the Eqs. (5) - (17) is well-determined and its becomes:
5 Numerical Examples and Discussions
The goal of this section is to prove the accuracy and the validity of the OHAM technique. We make use of the comparison of our approximate solutions with numerical results obtained via the fourth-order Runge-Kutta method in the following case, as follows: we consider the initial value problem given by the system (4) with initial conditions Ai = 0.001, i = 1, 9, M = 15 and N = 20; then, the convergence-control parameters K, K1, ω, Bi, Ci, i = 1, 8 are optimally determined by means of the least-square method using the Wolfram Mathematica 9.0 software.
One can show that these approximate solutions given by Eq. (23) are weak ϵ-approximate solutions by computing the numerical values of the integral of square residual function (to see the Table 4 from Appendix 6), i.e.
with xi(t), i = 1, … 9, given by Eq. (23).
Forx1 the convergence-control parameters are respectively:
The first-order approximation of the solution given by the Eq. (16) becomes:
Finally, Table 1 and Tables 2-3 (see Appendix 6), respectively, emphasizes the accuracy of the OHAM technique by comparing the approximate analytic solutions x3, x5 and x8 respectively presented above with the corresponding numerical integration values.
The comparison between the approximate solutions x3 given by Eq. (A.2) and the corresponding numerical solutions for M = 15 and N = 20 (relative errors: ϵx3 = |x3numerical – x3OHAM|)
| t | x3numerical | x3OHAM given by Eq. (A.2) | ϵx3 |
|---|---|---|---|
| 0 | 0.001 | 0.001 | 0 |
| 1/10 | 0.0025974531 | 0.0026022393 | 4.7862 ⋅ 10–6 |
| 1/5 | 0.0002348415 | 0.0002348153 | 2.6224 ⋅ 10–8 |
| 3/10 | –0.0007638090 | –0.0007667133 | 2.9043 ⋅ 10–6 |
| 2/5 | 0.0065863087 | 0.0065892747 | 2.9660 ⋅ 10–6 |
| 1/2 | –0.0064337350 | –0.0064354063 | 1.6713 ⋅ 10–6 |
| 3/5 | 0.0072796814 | 0.0072795202 | 1.6129 ⋅ 10–7 |
| 7/10 | 0.0002519075 | 0.0002537876 | 1.8801 ⋅ 10–6 |
| 4/5 | –0.0054946676 | –0.0054968279 | 2.1603 ⋅ 10–6 |
| 9/10 | 0.0140979129 | 0.0140967988 | 1.1141 ⋅ 10–6 |
| 1 | –0.0136322814 | –0.0136370940 | 4.8125 ⋅ 10–6 |
The comparison between the approximate solutions x5 given by Eq. (A.4) and the corresponding numerical solutions for M = 15 and N = 20 (relative errors: ϵx5 = |x5numerical – x5OHAM|)
| t | x5numerical | x5OHAM given by Eq. (A.4) | ϵx5 |
|---|---|---|---|
| 0 | 0.001 | 0.001 | 0 |
| 1/10 | –0.0006815591 | –0.0006815657 | 6.6566 ⋅ 10–9 |
| 1/5 | 0.0000919487 | 0.0000919509 | 2.1944 ⋅ 10–9 |
| 3/10 | 0.0005350379 | 0.0005350411 | 3.2094 ⋅ 10–9 |
| 2/5 | –0.0009504616 | –0.0009504672 | 5.6096 ⋅ 10–9 |
| 1/2 | 0.0009887716 | 0.0009887778 | 6.2250 ⋅ 10–9 |
| 3/5 | –0.0006335602 | –0.0006335632 | 2.9127 ⋅ 10–9 |
| 7/10 | 0.0000245359 | 0.0000245330 | 2.9506 ⋅ 10–9 |
| 4/5 | 0.0005972232 | 0.0005972309 | 7.6578 ⋅ 10–9 |
| 9/10 | –0.0009844000 | –0.0009843996 | 3.3406 ⋅ 10–10 |
| 1 | 0.0009815058 | 0.0009816680 | 1.6213 ⋅ 10–7 |
The comparison between the approximate solutions x8 given by Eq. (A.7) and the corresponding numerical solutions for M = 15 and N = 20 (relative errors: ϵx8 = |x8numerical – x8OHAM|)
| t | x8numerical | x8OHAM given by Eq. (A.7) | ϵx8 |
|---|---|---|---|
| 1/10 | –0.0006815591 | –0.0006815657 | 6.6566 ⋅ 10–9 |
| 1/5 | 0.0000919487 | 0.0000919509 | 2.1944 ⋅ 10–9 |
| 3/10 | 0.0005350379 | 0.0005350411 | 3.2094 ⋅ 10–9 |
| 2/5 | –0.0009504616 | –0.0009504672 | 5.6096 ⋅ 10–9 |
| 1/2 | 0.0009887716 | 0.0009887778 | 6.2250 ⋅ 10–9 |
| 3/5 | –0.0006335602 | –0.0006335632 | 2.9127 ⋅ 10–9 |
| 7/10 | 0.0000245359 | 0.0000245330 | 2.9506 ⋅ 10–9 |
| 4/5 | 0.0005972232 | 0.0005972309 | 7.6578 ⋅ 10–9 |
| 9/10 | –0.0009844000 | –0.0009843996 | 3.3406 ⋅ 10–10 |
| 1 | 0.0009815058 | 0.0009816680 | 1.6213 ⋅ 10–7 |
The numerical values of the integral of square residual function given by Eq. (24) corresponding to the approximate solutions given by Eqs. (25), (A.1)-(A.8) for M = 15 and N = 20.
| i | |
|---|---|
| 1 | 2.5681235091717695 ⋅ 10–7 |
| 2 | 6.4073772699416544 ⋅ 10–6 |
| 3 | 1.4346282316189278 ⋅ 10–7 |
| 4 | 1.083313166638645 ⋅ 10–12 |
| 5 | 2.4998964657011165 ⋅ 10–12 |
| 6 | 6.764954176305842 ⋅ 10–13 |
| 7 | 1.2075523363812237 ⋅ 10–12 |
| 8 | 2.5292599216354187 ⋅ 10–12 |
| 9 | 6.764881094330037 ⋅ 10–13 |
On the other hand, the approximate analytic solutions x2 ÷ x9 can be found in Appendix 6.
Remark 5.1
The Figures 1-9 present numerical comparisons between the analytical approximate solutions obtained by OHAM Method and numerical results obtained by Runge-Kutta 4-steps integrator. We can see that the analytical approximate solutions offer us the same numerical results as Runge-Kutta 4-steps integrator.
Comparison between the approximate solutions x1 given by Eq. (25) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Comparison between the approximate solutions x2 given by Eq. (A.1) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Comparison between the approximate solutions x3 given by Eq. (A.2) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Comparison between the approximate solutions x4 given by Eq. (A.4) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Comparison between the approximate solutions x5 given by Eq. (A.4) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Comparison between the approximate solutions x6 given by Eq. (A.5) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Comparison between the approximate solutions x7 given by Eq. (A.6) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Comparison between the approximate solutions x8 given by Eq. (A.7) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
Comparison between the approximate solutions x9 given by Eq. (A.8) and the corresponding numerical solutions: ---------- numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ OHAM solution
6 Conclusion
In this paper, the stabilization of a dynamical system arising from an optimal control problem on the Lie group SO(3) × ℝ3 × ℝ3 is investigated. The conditions of the spectral and nonlinear stability of a specific equilibria were obtained using a linear control added to the initial system. Due to the existence of the Poisson formulation, the results were obtained using specific tools, such as Arnold’s method.
Finally, the analytic approximate solutions of the controlled dynamics are obtained via the Optimal Homotopy Asymptotic Method. Numerical simulations and a comparison with Runge-Kutta 4-steps integrator are also presented.
Conflict of Interest
Conflict of Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.
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Appendix
The following conditions K1 = 7.00700137 ⋅ 10–6, ω = 1.6655500501 hold for all unknown functions xi, i = 1, 9, so:
forx2:
forx3:
forx4:
forx5:
forx6:
forx7:
forx8:
forx9:
© 2018 C. Pop et al., published by De Gruyter
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.
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- Magnetic transmission gear finite element simulation with iron pole hysteresis
- Pulsed excitation terahertz tomography – multiparametric approach
- Low and high frequency model of three phase transformer by frequency response analysis measurement
- Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion
- Optimal design of a for middle-low-speed maglev trains
- Eddy current modeling in linear and nonlinear multifilamentary composite materials
- The visual attention saliency map for movie retrospection
- AC/DC current ratio in a current superimposition variable flux reluctance machine
- Influence of material uncertainties on the RLC parameters of wound inductors modeled using the finite element method
- Cogging force reduction in linear tubular flux switching permanent-magnet machines
- Modeling hysteresis curves of La(FeCoSi)13 compound near the transition point with the GRUCAD model
- Electro-magneto-hydrodynamic lubrication
- 3-D Electromagnetic field analysis of wireless power transfer system using K computer
- Simplified simulation technique of rotating, induction heated, calender rolls for study of temperature field control
- Design, fabrication and testing of electroadhesive interdigital electrodes
- A method to reduce partial discharges in motor windings fed by PWM inverter
- Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine
- Special Issue Applications of Nonlinear Dynamics
- Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
- Microblog topic evolution computing based on LDA algorithm
- Modeling the creep damage effect on the creep crack growth behavior of rotor steel
- Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
- Chinese open information extraction based on DBMCSS in the field of national information resources
- 10.1515/phys-2018-0079
- CPW-fed circularly-polarized antenna array with high front-to-back ratio and low-profile
- Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
- Temperature detection technology of power equipment based on Fiber Bragg Grating
- Research on a rotational speed control strategy of the mandrel in a rotary steering system
- Dynamic load balancing algorithm for large data flow in distributed complex networks
- Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
- Fractal-based techniques for physiological time series: An updated approach
- Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
- Application of modified culture Kalman filter in bearing fault diagnosis
- Exact solutions and conservation laws for the modified equal width-Burgers equation
- On topological properties of block shift and hierarchical hypercube networks
- Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
- A note on the transmission feasibility problem in networks
- Ontology learning algorithm using weak functions
- Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
- Parametric simulation analysis and reliability of escalator truss
- A new algorithm for real economy benefit evaluation based on big data analysis
- Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
- Multi-level encryption algorithm for user-related information across social networks
- Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
- Fast recognition method of moving video images based on BP neural networks
- Compressed sensing image restoration algorithm based on improved SURF operator
- Design of load optimal control algorithm for smart grid based on demand response in different scenarios
- Face recognition method based on GA-BP neural network algorithm
- Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
- Localization and recognition algorithm for fuzzy anomaly data in big data networks
- Urban road traffic flow control under incidental congestion as a function of accident duration
- Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
- Feasibility simulation of aseismic structure design for long-span bridges
- Construction of renewable energy supply chain model based on LCA
- The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
- A text-Image feature mapping algorithm based on transfer learning
- Fast recognition algorithm for static traffic sign information
- Topical Issue: Clean Energy: Materials, Processes and Energy Generation
- An investigation of the melting process of RT-35 filled circular thermal energy storage system
- Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
- Energy converting layers for thin-film flexible photovoltaic structures
- Effect of convection heat transfer on thermal energy storage unit
Articles in the same Issue
- Regular Articles
- A modified Fermi-Walker derivative for inextensible flows of binormal spherical image
- Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics
- Three-dimensional atom localization via probe absorption in a cascade four-level atomic system
- Determination of the energy transitions and half-lives of Rubidium nuclei
- Three phase heat and mass transfer model for unsaturated soil freezing process: Part 1 - model development
- Three phase heat and mass transfer model for unsaturated soil freezing process: Part 2 - model validation
- Mathematical model for thermal and entropy analysis of thermal solar collectors by using Maxwell nanofluids with slip conditions, thermal radiation and variable thermal conductivity
- Constructing analytic solutions on the Tricomi equation
- Feynman diagrams and rooted maps
- New type of chaos synchronization in discrete-time systems: the F-M synchronization
- Unsteady flow of fractional Oldroyd-B fluids through rotating annulus
- A note on the uniqueness of 2D elastostatic problems formulated by different types of potential functions
- On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics
- Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications
- Siewert solutions of transcendental equations, generalized Lambert functions and physical applications
- Numerical solution of mixed convection flow of an MHD Jeffery fluid over an exponentially stretching sheet in the presence of thermal radiation and chemical reaction
- A new three-dimensional chaotic flow with one stable equilibrium: dynamical properties and complexity analysis
- Dynamics of a dry-rebounding drop: observations, simulations, and modeling
- Modeling the initial mechanical response and yielding behavior of gelled crude oil
- Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation
- Solitary wave solutions of two KdV-type equations
- Applying industrial tomography to control and optimization flow systems
- Reconstructing time series into a complex network to assess the evolution dynamics of the correlations among energy prices
- An optimal solution for software testing case generation based on particle swarm optimization
- Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation
- Alternative methods for solving nonlinear two-point boundary value problems
- Global model simulation of OH production in pulsed-DC atmospheric pressure helium-air plasma jets
- Experimental investigation on optical vortex tweezers for microbubble trapping
- Joint measurements of optical parameters by irradiance scintillation and angle-of-arrival fluctuations
- M-polynomials and topological indices of hex-derived networks
- Generalized convergence analysis of the fractional order systems
- Porous flow characteristics of solution-gas drive in tight oil reservoirs
- Complementary wave solutions for the long-short wave resonance model via the extended trial equation method and the generalized Kudryashov method
- A Note on Koide’s Doubly Special Parametrization of Quark Masses
- On right-angled spherical Artin monoid of type Dn
- Gas flow regimes judgement in nanoporous media by digital core analysis
- 4 + n-dimensional water and waves on four and eleven-dimensional manifolds
- Stabilization and Analytic Approximate Solutions of an Optimal Control Problem
- On the equations of electrodynamics in a flat or curved spacetime and a possible interaction energy
- New prediction method for transient productivity of fractured five-spot patterns in low permeability reservoirs at high water cut stages
- The collinear equilibrium points in the restricted three body problem with triaxial primaries
- Detection of the damage threshold of fused silica components and morphologies of repaired damage sites based on the beam deflection method
- On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem
- Ion acoustic quasi-soliton in an electron-positron-ion plasma with superthermal electrons and positrons
- Analysis of projectile motion in view of conformable derivative
- Computing multiple ABC index and multiple GA index of some grid graphs
- Terahertz pulse imaging: A novel denoising method by combing the ant colony algorithm with the compressive sensing
- Characteristics of microscopic pore-throat structure of tight oil reservoirs in Sichuan Basin measured by rate-controlled mercury injection
- An activity window model for social interaction structure on Twitter
- Transient thermal regime trough the constitutive matrix applied to asynchronous electrical machine using the cell method
- On the zagreb polynomials of benzenoid systems
- Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
- The Greek parameters of a continuous arithmetic Asian option pricing model via Laplace Adomian decomposition method
- Quantifying the global solar radiation received in Pietermaritzburg, KwaZulu-Natal to motivate the consumption of solar technologies
- Sturm-Liouville difference equations having Bessel and hydrogen atom potential type
- Study on the response characteristics of oil wells after deep profile control in low permeability fractured reservoirs
- Depiction and analysis of a modified theta shaped double negative metamaterial for satellite application
- An attempt to geometrize electromagnetism
- Structure of traveling wave solutions for some nonlinear models via modified mathematical method
- Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation
- Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations
- Effect of magnetic field and heat source on Upper-convected-maxwell fluid in a porous channel
- Physical cues of biomaterials guide stem cell fate of differentiation: The effect of elasticity of cell culture biomaterials
- Shooting method analysis in wire coating withdrawing from a bath of Oldroyd 8-constant fluid with temperature dependent viscosity
- Rank correlation between centrality metrics in complex networks: an empirical study
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
- Modeling of electric and heat processes in spot resistance welding of cross-wire steel bars
- Dynamic characteristics of triaxial active control magnetic bearing with asymmetric structure
- Design optimization of an axial-field eddy-current magnetic coupling based on magneto-thermal analytical model
- Thermal constitutive matrix applied to asynchronous electrical machine using the cell method
- Temperature distribution around thin electroconductive layers created on composite textile substrates
- Model of the multipolar engine with decreased cogging torque by asymmetrical distribution of the magnets
- Analysis of spatial thermal field in a magnetic bearing
- Use of the mathematical model of the ignition system to analyze the spark discharge, including the destruction of spark plug electrodes
- Assessment of short/long term electric field strength measurements for a pilot district
- Simulation study and experimental results for detection and classification of the transient capacitor inrush current using discrete wavelet transform and artificial intelligence
- Magnetic transmission gear finite element simulation with iron pole hysteresis
- Pulsed excitation terahertz tomography – multiparametric approach
- Low and high frequency model of three phase transformer by frequency response analysis measurement
- Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion
- Optimal design of a for middle-low-speed maglev trains
- Eddy current modeling in linear and nonlinear multifilamentary composite materials
- The visual attention saliency map for movie retrospection
- AC/DC current ratio in a current superimposition variable flux reluctance machine
- Influence of material uncertainties on the RLC parameters of wound inductors modeled using the finite element method
- Cogging force reduction in linear tubular flux switching permanent-magnet machines
- Modeling hysteresis curves of La(FeCoSi)13 compound near the transition point with the GRUCAD model
- Electro-magneto-hydrodynamic lubrication
- 3-D Electromagnetic field analysis of wireless power transfer system using K computer
- Simplified simulation technique of rotating, induction heated, calender rolls for study of temperature field control
- Design, fabrication and testing of electroadhesive interdigital electrodes
- A method to reduce partial discharges in motor windings fed by PWM inverter
- Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine
- Special Issue Applications of Nonlinear Dynamics
- Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
- Microblog topic evolution computing based on LDA algorithm
- Modeling the creep damage effect on the creep crack growth behavior of rotor steel
- Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
- Chinese open information extraction based on DBMCSS in the field of national information resources
- 10.1515/phys-2018-0079
- CPW-fed circularly-polarized antenna array with high front-to-back ratio and low-profile
- Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
- Temperature detection technology of power equipment based on Fiber Bragg Grating
- Research on a rotational speed control strategy of the mandrel in a rotary steering system
- Dynamic load balancing algorithm for large data flow in distributed complex networks
- Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
- Fractal-based techniques for physiological time series: An updated approach
- Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
- Application of modified culture Kalman filter in bearing fault diagnosis
- Exact solutions and conservation laws for the modified equal width-Burgers equation
- On topological properties of block shift and hierarchical hypercube networks
- Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
- A note on the transmission feasibility problem in networks
- Ontology learning algorithm using weak functions
- Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
- Parametric simulation analysis and reliability of escalator truss
- A new algorithm for real economy benefit evaluation based on big data analysis
- Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
- Multi-level encryption algorithm for user-related information across social networks
- Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
- Fast recognition method of moving video images based on BP neural networks
- Compressed sensing image restoration algorithm based on improved SURF operator
- Design of load optimal control algorithm for smart grid based on demand response in different scenarios
- Face recognition method based on GA-BP neural network algorithm
- Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
- Localization and recognition algorithm for fuzzy anomaly data in big data networks
- Urban road traffic flow control under incidental congestion as a function of accident duration
- Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
- Feasibility simulation of aseismic structure design for long-span bridges
- Construction of renewable energy supply chain model based on LCA
- The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
- A text-Image feature mapping algorithm based on transfer learning
- Fast recognition algorithm for static traffic sign information
- Topical Issue: Clean Energy: Materials, Processes and Energy Generation
- An investigation of the melting process of RT-35 filled circular thermal energy storage system
- Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
- Energy converting layers for thin-film flexible photovoltaic structures
- Effect of convection heat transfer on thermal energy storage unit
