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December  2021, 11(4): 495-512. doi: 10.3934/naco.2020040

Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable

Nguyen Thi Hoai

Department of Mathematics, Mechanics and Informatics, University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Ha Noi, Vietnam

Received  May 2020 Revised  September 2020 Published  December 2021 Early access  September 2020

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.

Keywords: Singular perturbation, asymptotic expansion, optimal problem, second-order ODE.
Mathematics Subject Classification: Primary: 34E10, 34D15; Secondary: 49K15, 41A60.
Citation: Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040
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. Mumar, 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. Mumar, 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.

Figure 1.  The graph of $ x(t,\varepsilon) $ and its approximations
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Figure 2.  The graph of $ u(t,\varepsilon) $ and its approximations
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Table 1.  Table of values of performance index
$ \varepsilon $ $ \;\;J_{\varepsilon}(\overline{u}_{0})\;\; $ $ \;\;J_{\varepsilon}(\widetilde{u}_{0})\;\; $ $ \;\;J_{\varepsilon}(\widetilde{u}_{1})\;\; $ $ \;\;J_{\varepsilon}(u)\;\; $
0.1 0.11594 0.11240 0.11020 0.11013
0.2 0.16038 0.15884 0.15330 0.15266
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$ \varepsilon $ $ \;\;J_{\varepsilon}(\overline{u}_{0})\;\; $ $ \;\;J_{\varepsilon}(\widetilde{u}_{0})\;\; $ $ \;\;J_{\varepsilon}(\widetilde{u}_{1})\;\; $ $ \;\;J_{\varepsilon}(u)\;\; $
0.1 0.11594 0.11240 0.11020 0.11013
0.2 0.16038 0.15884 0.15330 0.15266
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