American Institute of Mathematical Sciences

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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  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 & Optimization, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[5]

Y. Boglaev, On numerical methods for solving singularly perturbed problems, Differ. Uravn., 21 (1985), 1804-1806.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[18]

G. A. Kurina and M. G. Dmitriev, Singular perturbations in control problems, Autom. Remote Control, 67 (2006), 1-43.  doi: 10.1134/S0005117906010012.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

N. N. Moiseev, Asymptotic Methods for Nonlinear Mechanics, Nauka, Moscow, 1983 (in Russian).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

A. B. Vasil'eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973 (in Russian).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[5]

Y. Boglaev, On numerical methods for solving singularly perturbed problems, Differ. Uravn., 21 (1985), 1804-1806.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[18]

G. A. Kurina and M. G. Dmitriev, Singular perturbations in control problems, Autom. Remote Control, 67 (2006), 1-43.  doi: 10.1134/S0005117906010012.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

N. N. Moiseev, Asymptotic Methods for Nonlinear Mechanics, Nauka, Moscow, 1983 (in Russian).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

A. B. Vasil'eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973 (in Russian).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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