
-
Previous Article
A new methodology for solving bi-criterion fractional stochastic programming
- NACO Home
- This Issue
-
Next Article
Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach
Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable
Department of Mathematics, Mechanics and Informatics, University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Ha Noi, Vietnam |
The direct scheme method is applied to construct an asymptotic approximation of any order to a solution of a singularly perturbed optimal problem with scalar state, controlled via a second-order linear ODE and two fixed end points. The error estimates for state and control variables and for the functional are obtained. An illustrative example is given.
References:
[1] |
R. K. Bawa,
Spline based computational technique for linear singularly perturbed boundary value problem, Appl. Math, Comput., 167 (2005), 225-236.
doi: 10.1016/j.amc.2004.06.112. |
[2] |
R. K. Bawa and S. Natesan,
A computational method for self adjoint singular perturbation problems using quintic spline, Int. J. Computers Math. Appl., 50 (2005), 1371-1382.
doi: 10.1016/j.camwa.2005.04.017. |
[3] |
S. V. Belokopytov and M. G. Dmitriev,
Direct scheme in optimal control problems with fast and slow motions, Systems Control Lett., 8 (1986), 129-135.
doi: 10.1016/0167-6911(86)90071-X. |
[4] |
I. P. Boglaev,
A variational difference scheme for a boundary value problem with a small parameter in the highest derivative, USSR Comput. Maths. Math. Phys., 21 (1981), 71-81.
|
[5] |
Y. Boglaev,
On numerical methods for solving singularly perturbed problems, Differ. Uravn., 21 (1985), 1804-1806.
|
[6] |
A. R. Danilin and N. S. Korobitsyna,
Asymptotic estimate for a solution of a singular perturbation optimal control problem on a closed interval under geometric constraints, Proc. Steklov Inst. Math., 285 (2014), 58-67.
doi: 10.1134/s008154381405006x. |
[7] |
A. R. Danilin,
Asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with integral constraint, Proc. Steklov Inst. Math., 291 (2015), 66-76.
doi: 10.1134/s0081543815090059. |
[8] |
W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, 1979. |
[9] |
M. V. Fedoryuk, Asymptotic Analysis for Linear Ordinary Differential Equations, Springer-Verlag Berlin Heidelberg, New York, 1993.
doi: 10.1007/978-3-642-58016-1. |
[10] |
M. G. Gasparo and M. Macconi,
Initial-value methods for second order singularly-perturbed boundary value problems, J. Optim. Theory Appl., 6 (1990), 197-210.
doi: 10.1007/BF00939534. |
[11] |
V. Y. Glizer,
Asymptotic solution of a singularly perturbed set of functional-differential equations of Riccati type encountered in the optimal control theory, Nonlinear Diff. Eq. Appl., 5 (1998), 491-515.
doi: 10.1007/s000300050059. |
[12] |
N. T. Hoai,
Asymptotic solution of a singularly perturbed linear-quadratic problem in critical case with cheap control, J. Optim. Theory Appl., 175 (2017), 324-340.
doi: 10.1007/s10957-017-1156-6. |
[13] |
M. K. Kadalbajoo and K. C. Patidar,
A survey of numerical techniques for solving singularly perturbed ordinary differential equations, Appl. Math. Comput., 130 (2002), 457-510.
doi: 10.1016/S0096-3003(01)00112-6. |
[14] |
P. V. Kokotovic, R. E. Jr. O'Malley and P. Sannuti,
Singular perturbations and order reduction in control theory: An overview, Automatica, 12 (1976), 123-132.
doi: 10.1016/0005-1098(76)90076-5. |
[15] |
P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbations Methods in Control: Analysis and Design, SIAM, Philadelphia, 1999.
doi: 10.1137/1.9781611971118. |
[16] |
M. Kopteva and M. Stynes,
Numerical analysis of a singlarly perturbed non-linear reaction-diffusion problem with multiple solution, Appl. Num. Math., 51 (2004), 273-288.
doi: 10.1016/j.apnum.2004.07.001. |
[17] |
M. Kudu and G. M. Amiraliyev,
Finite difference method for a singularly perturbed differential equations with integral boundary condition, Int. J. Math. Comput., 26 (2015), 72-79.
|
[18] |
G. A. Kurina and M. G. Dmitriev,
Singular perturbations in control problems, Autom. Remote Control, 67 (2006), 1-43.
doi: 10.1134/S0005117906010012. |
[19] |
G. A. Kurina, M. G. Dmitriev and D. S. Naidu,
Discrete singularly perturbed control problems (A Survey), Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 24 (2017), 335-370.
|
[20] |
G. A. Kurina and E. V. Smirnova,
Asymptotics of solutions of optimal control problems with intermediate points in quality criterion and small parameters, J. Math. Sci., 170 (2010), 192-228.
doi: 10.1007/s10958-010-0080-1. |
[21] |
G. A. Kurina and T. H. Nguyen,
Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients, Comput. Math. Math. Phys., 52 (2012), 524-547.
doi: 10.1134/S0965542512040100. |
[22] |
G. A. Kurina and T. H. Nguyen,
Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients, Comput. Math. Math. Phys., 52 (2012), 628-652.
doi: 10.1134/S0965542512040100. |
[23] |
R. K. Mohanty and U. Arora,
A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problem with significant first derivatives, Appl. Math. Comput., 172 (2006), 531-544.
doi: 10.1016/j.amc.2005.02.023. |
[24] |
J. Mohapatra and N. R. Reddy,
Exponentially fitted finite difference scheme for singularly perturbed two point boundary value problems, Int. J. Appl. Comput. Math., 1 (2015), 267-278.
doi: 10.1007/s40819-014-0008-4. |
[25] |
N. N. Moiseev and F. L. Chernousko,
Asymptotic methods in the theory of optimal control, IEEE Trans. Automat. Control, 26 (1981), 993-1000.
doi: 10.1109/TAC.1981.1102773. |
[26] |
N. N. Moiseev, Asymptotic Methods for Nonlinear Mechanics, Nauka, Moscow, 1983 (in Russian). |
[27] |
M. Kumar, P. Singh and H. K. Mishra,
A recent survey on computational techniques for solving singularly perturbed boundary value problem, Int. J. Computer Math., 84 (2007), 1439-1463.
doi: 10.1080/00207160701295712. |
[28] |
S. Natesan and N. Ramanujam,
'Shooting method' for the solution of singularly perturbed two-point boundary value problems having less severe boundary layer, Appl. Math. Comput., 133 (2020), 623-641.
doi: 10.1016/S0096-3003(01)00263-6. |
[29] |
R. E. O'Malley, Jr., Singular Perturbations Methods for Ordinary Differential Equations, Springer-Verlag Berlin Heidelberg, New York, 1991.
doi: 10.1007/978-1-4612-0977-5. |
[30] |
S. M. Roberts,
A boundary value technique for singular perturbation problems, J. Math. Anal. Appl., 87 (1982), 489-508.
doi: 10.1016/0022-247X(82)90139-1. |
[31] |
H. G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion and Flow Problems, Springer-Verlag Berlin Heidelberg, New York, 1994.
doi: 10.1007/978-3-662-03206-0. |
[32] |
V. R. Saksena, J. O'Reilly and P. V. Kokotovic,
Singular perturbations and time-scale methods in control theory: A Survey 1976-1983, Automatica, 20 (1984), 273-293.
doi: 10.1016/0005-1098(84)90044-X. |
[33] |
M. Stojanovic,
Global convergence method for singularly perturbed boundary value problem, J. Comput. Appl. Math., 181 (2005), 326-335.
doi: 10.1016/j.cam.2004.12.006. |
[34] |
K. Surla and Z. Uzelac,
A unformly accurate spline collocation method for a normalized flux, J. Comput. Appl. Math., 166 (2004), 291-305.
doi: 10.1016/j.cam.2003.09.021. |
[35] |
T. Valanarasu and N. Ramanujam,
Asymptotic initial-value method for singularly-perturbed boundary problems for second-order ordinary differential equations, J. Optim. Theory Appl., 116 (2003), 167-182.
doi: 10.1023/A:1022118420907. |
[36] |
T. Valanarasu and N. Ramanujam,
An asymptotic initial value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term, J. Appl. Math. Computing, 23 (2007), 141-152.
doi: 10.1007/BF02831964. |
[37] |
A. B. Vasil'eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973 (in Russian). |
[38] |
A. B. Vasil'eva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM. Studies in Applied Mathematics, Philadelphia, 1995.
doi: 10.1137/1.9781611970784. |
[39] |
J. Vigo-Aguiar, S. Natesan and N. Ramanujam,
A numerical algorithm for singular perturbation problems exhibiting weak boundary layers, Int. J. Computers Math. Appl., 45 (2003), 469-479.
doi: 10.1016/S0898-1221(03)80031-7. |
[40] |
L. Wang,
A novel method for a class of non-linear singular perturbation problems, Appl. Math. Comput., 156 (2004), 847-856.
doi: 10.1016/j.amc.2003.06.010. |
show all references
References:
[1] |
R. K. Bawa,
Spline based computational technique for linear singularly perturbed boundary value problem, Appl. Math, Comput., 167 (2005), 225-236.
doi: 10.1016/j.amc.2004.06.112. |
[2] |
R. K. Bawa and S. Natesan,
A computational method for self adjoint singular perturbation problems using quintic spline, Int. J. Computers Math. Appl., 50 (2005), 1371-1382.
doi: 10.1016/j.camwa.2005.04.017. |
[3] |
S. V. Belokopytov and M. G. Dmitriev,
Direct scheme in optimal control problems with fast and slow motions, Systems Control Lett., 8 (1986), 129-135.
doi: 10.1016/0167-6911(86)90071-X. |
[4] |
I. P. Boglaev,
A variational difference scheme for a boundary value problem with a small parameter in the highest derivative, USSR Comput. Maths. Math. Phys., 21 (1981), 71-81.
|
[5] |
Y. Boglaev,
On numerical methods for solving singularly perturbed problems, Differ. Uravn., 21 (1985), 1804-1806.
|
[6] |
A. R. Danilin and N. S. Korobitsyna,
Asymptotic estimate for a solution of a singular perturbation optimal control problem on a closed interval under geometric constraints, Proc. Steklov Inst. Math., 285 (2014), 58-67.
doi: 10.1134/s008154381405006x. |
[7] |
A. R. Danilin,
Asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with integral constraint, Proc. Steklov Inst. Math., 291 (2015), 66-76.
doi: 10.1134/s0081543815090059. |
[8] |
W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, 1979. |
[9] |
M. V. Fedoryuk, Asymptotic Analysis for Linear Ordinary Differential Equations, Springer-Verlag Berlin Heidelberg, New York, 1993.
doi: 10.1007/978-3-642-58016-1. |
[10] |
M. G. Gasparo and M. Macconi,
Initial-value methods for second order singularly-perturbed boundary value problems, J. Optim. Theory Appl., 6 (1990), 197-210.
doi: 10.1007/BF00939534. |
[11] |
V. Y. Glizer,
Asymptotic solution of a singularly perturbed set of functional-differential equations of Riccati type encountered in the optimal control theory, Nonlinear Diff. Eq. Appl., 5 (1998), 491-515.
doi: 10.1007/s000300050059. |
[12] |
N. T. Hoai,
Asymptotic solution of a singularly perturbed linear-quadratic problem in critical case with cheap control, J. Optim. Theory Appl., 175 (2017), 324-340.
doi: 10.1007/s10957-017-1156-6. |
[13] |
M. K. Kadalbajoo and K. C. Patidar,
A survey of numerical techniques for solving singularly perturbed ordinary differential equations, Appl. Math. Comput., 130 (2002), 457-510.
doi: 10.1016/S0096-3003(01)00112-6. |
[14] |
P. V. Kokotovic, R. E. Jr. O'Malley and P. Sannuti,
Singular perturbations and order reduction in control theory: An overview, Automatica, 12 (1976), 123-132.
doi: 10.1016/0005-1098(76)90076-5. |
[15] |
P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbations Methods in Control: Analysis and Design, SIAM, Philadelphia, 1999.
doi: 10.1137/1.9781611971118. |
[16] |
M. Kopteva and M. Stynes,
Numerical analysis of a singlarly perturbed non-linear reaction-diffusion problem with multiple solution, Appl. Num. Math., 51 (2004), 273-288.
doi: 10.1016/j.apnum.2004.07.001. |
[17] |
M. Kudu and G. M. Amiraliyev,
Finite difference method for a singularly perturbed differential equations with integral boundary condition, Int. J. Math. Comput., 26 (2015), 72-79.
|
[18] |
G. A. Kurina and M. G. Dmitriev,
Singular perturbations in control problems, Autom. Remote Control, 67 (2006), 1-43.
doi: 10.1134/S0005117906010012. |
[19] |
G. A. Kurina, M. G. Dmitriev and D. S. Naidu,
Discrete singularly perturbed control problems (A Survey), Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 24 (2017), 335-370.
|
[20] |
G. A. Kurina and E. V. Smirnova,
Asymptotics of solutions of optimal control problems with intermediate points in quality criterion and small parameters, J. Math. Sci., 170 (2010), 192-228.
doi: 10.1007/s10958-010-0080-1. |
[21] |
G. A. Kurina and T. H. Nguyen,
Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients, Comput. Math. Math. Phys., 52 (2012), 524-547.
doi: 10.1134/S0965542512040100. |
[22] |
G. A. Kurina and T. H. Nguyen,
Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients, Comput. Math. Math. Phys., 52 (2012), 628-652.
doi: 10.1134/S0965542512040100. |
[23] |
R. K. Mohanty and U. Arora,
A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problem with significant first derivatives, Appl. Math. Comput., 172 (2006), 531-544.
doi: 10.1016/j.amc.2005.02.023. |
[24] |
J. Mohapatra and N. R. Reddy,
Exponentially fitted finite difference scheme for singularly perturbed two point boundary value problems, Int. J. Appl. Comput. Math., 1 (2015), 267-278.
doi: 10.1007/s40819-014-0008-4. |
[25] |
N. N. Moiseev and F. L. Chernousko,
Asymptotic methods in the theory of optimal control, IEEE Trans. Automat. Control, 26 (1981), 993-1000.
doi: 10.1109/TAC.1981.1102773. |
[26] |
N. N. Moiseev, Asymptotic Methods for Nonlinear Mechanics, Nauka, Moscow, 1983 (in Russian). |
[27] |
M. Kumar, P. Singh and H. K. Mishra,
A recent survey on computational techniques for solving singularly perturbed boundary value problem, Int. J. Computer Math., 84 (2007), 1439-1463.
doi: 10.1080/00207160701295712. |
[28] |
S. Natesan and N. Ramanujam,
'Shooting method' for the solution of singularly perturbed two-point boundary value problems having less severe boundary layer, Appl. Math. Comput., 133 (2020), 623-641.
doi: 10.1016/S0096-3003(01)00263-6. |
[29] |
R. E. O'Malley, Jr., Singular Perturbations Methods for Ordinary Differential Equations, Springer-Verlag Berlin Heidelberg, New York, 1991.
doi: 10.1007/978-1-4612-0977-5. |
[30] |
S. M. Roberts,
A boundary value technique for singular perturbation problems, J. Math. Anal. Appl., 87 (1982), 489-508.
doi: 10.1016/0022-247X(82)90139-1. |
[31] |
H. G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion and Flow Problems, Springer-Verlag Berlin Heidelberg, New York, 1994.
doi: 10.1007/978-3-662-03206-0. |
[32] |
V. R. Saksena, J. O'Reilly and P. V. Kokotovic,
Singular perturbations and time-scale methods in control theory: A Survey 1976-1983, Automatica, 20 (1984), 273-293.
doi: 10.1016/0005-1098(84)90044-X. |
[33] |
M. Stojanovic,
Global convergence method for singularly perturbed boundary value problem, J. Comput. Appl. Math., 181 (2005), 326-335.
doi: 10.1016/j.cam.2004.12.006. |
[34] |
K. Surla and Z. Uzelac,
A unformly accurate spline collocation method for a normalized flux, J. Comput. Appl. Math., 166 (2004), 291-305.
doi: 10.1016/j.cam.2003.09.021. |
[35] |
T. Valanarasu and N. Ramanujam,
Asymptotic initial-value method for singularly-perturbed boundary problems for second-order ordinary differential equations, J. Optim. Theory Appl., 116 (2003), 167-182.
doi: 10.1023/A:1022118420907. |
[36] |
T. Valanarasu and N. Ramanujam,
An asymptotic initial value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term, J. Appl. Math. Computing, 23 (2007), 141-152.
doi: 10.1007/BF02831964. |
[37] |
A. B. Vasil'eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973 (in Russian). |
[38] |
A. B. Vasil'eva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM. Studies in Applied Mathematics, Philadelphia, 1995.
doi: 10.1137/1.9781611970784. |
[39] |
J. Vigo-Aguiar, S. Natesan and N. Ramanujam,
A numerical algorithm for singular perturbation problems exhibiting weak boundary layers, Int. J. Computers Math. Appl., 45 (2003), 469-479.
doi: 10.1016/S0898-1221(03)80031-7. |
[40] |
L. Wang,
A novel method for a class of non-linear singular perturbation problems, Appl. Math. Comput., 156 (2004), 847-856.
doi: 10.1016/j.amc.2003.06.010. |


0.1 | 0.11594 | 0.11240 | 0.11020 | 0.11013 |
0.2 | 0.16038 | 0.15884 | 0.15330 | 0.15266 |
0.1 | 0.11594 | 0.11240 | 0.11020 | 0.11013 |
0.2 | 0.16038 | 0.15884 | 0.15330 | 0.15266 |
[1] |
Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020164 |
[2] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[3] |
Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 |
[4] |
Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279 |
[5] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[6] |
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020340 |
[7] |
Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 |
[8] |
Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020377 |
[9] |
Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124 |
[10] |
Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103 |
[11] |
Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011 |
[12] |
Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008 |
[13] |
Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020176 |
[14] |
Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001 |
[15] |
Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170 |
[16] |
Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 |
[17] |
Makram Hamouda, Ahmed Bchatnia, Mohamed Ali Ayadi. Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021001 |
[18] |
Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020345 |
[19] |
Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291 |
[20] |
Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020108 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]