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Journal articles on the topic 'Singularly perturbed optimal control problems'

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Published: 10 December 2022

Last updated: 29 January 2023

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1

Cots, Olivier, Joseph Gergaud, and Boris Wembe. "Homotopic approach for turnpike and singularly perturbed optimal control problems." ESAIM: Proceedings and Surveys 71 (August 2021): 43–53. http://dx.doi.org/10.1051/proc/202171105.

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The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one. Then, thanks to this link, we set up a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems. We consider first the turnpike case, then, we generalize the approach to general control problems with singular perturbations (that is with fast but also slow variables). We illustrate each step with an example.

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2

LUBE, GERT, and BENJAMIN TEWS. "OPTIMAL CONTROL OF SINGULARLY PERTURBED ADVECTION-DIFFUSION-REACTION PROBLEMS." Mathematical Models and Methods in Applied Sciences 20, no. 03 (March 2010): 375–95. http://dx.doi.org/10.1142/s0218202510004271.

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In this paper, we consider the numerical analysis of quadratic optimal control problems governed by a linear advection-diffusion-reaction equation without control constraints. In the case of dominating advection, the Galerkin discretization is stabilized via the one- or two-level variant of the local projection approach which leads to a symmetric optimality system at the discrete level. The optimal control problem simultaneously covers distributed and Robin boundary control. In the singularly perturbed case, the boundary control at inflow and/or characteristic parts of the boundary can be seen as regularization of a Dirichlet boundary control. Some numerical tests illustrate the analytical results.

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3

Kostyukova, O. I. "Parametric optimal control problems with singularly perturbed mixed constraints." Journal of Computer and Systems Sciences International 45, no. 1 (January 2006): 44–55. http://dx.doi.org/10.1134/s1064230706010059.

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4

Gaitsgory, Vladimir, and Matthias Gerdts. "On Numerical Solution of Singularly Perturbed Optimal Control Problems." Journal of Optimization Theory and Applications 174, no. 3 (July 14, 2017): 762–84. http://dx.doi.org/10.1007/s10957-017-1138-8.

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5

Salikhov, Z. G., E. N. Ishmet’ev, A. L. Rutkovskii, V. I. Alekhin, and M. Z. Salikhov. "Asymptotic regularization methods in singularly perturbed stochastic optimal-control problems." Steel in Translation 38, no. 1 (January 2008): 17–19. http://dx.doi.org/10.3103/s0967091208010063.

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6

Liu, Dan, Lei Liu, and Ying Yang. "Control of Discrete-Time Singularly Perturbed Systems via Static Output Feedback." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/528695.

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This paper concentrates on control problems of discrete-time singularly perturbed systems via static output feedback. Two methods of designing an controller, which ensures that the resulting closed-loop system is asymptotically stable and meets a prescribed norm bound, are presented in terms of LMIs. Though based on the same matrix transformation, the two approaches are turned into different optimal problems. The first result is given by an -independent LMI, while the second result is related to . Furthermore, a stability upper bound of the singular perturbation parameter is obtained. The validity of the proposed two results is demonstrated by a numerical example.

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7

Kim, Beom-Soo, Young-Joong Kim, and Myo-Taeg Lim. "LQG Control for Nonstandard Singularly Perturbed Discrete-Time Systems." Journal of Dynamic Systems, Measurement, and Control 126, no. 4 (December 1, 2004): 860–64. http://dx.doi.org/10.1115/1.1850537.

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In this paper we present a control method and a high accuracy solution technique in solving the linear quadratic Gaussian problems for nonstandard singularly perturbed discrete time systems. The methodology that exists in the literature for the solution of the standard singularly perturbed discrete time linear quadratic Gaussian optimal control problem cannot be extended to the corresponding nonstandard counterpart. The solution of the linear quadratic Gaussian optimal control problem is obtained by solving the pure-slow and pure-fast reduced-order continuous-time algebraic Riccati equations and by implementing the pure-slow and pure-fast reduced-order Kalman filters. In order to show the effectiveness of the proposed method, we present the numerical result for a one-link flexible robot arm.

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8

Veliov, V. M. "Convergence rate of the solutions of singularly perturbed time-optimal control problems." Banach Center Publications 14, no. 1 (1985): 555–67. http://dx.doi.org/10.4064/-14-1-555-567.

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9

Tuan, H. D. "On reachable set of singularly perturbed differential inclusions and optimal control problems." Optimization 26, no. 3-4 (January 1992): 325–38. http://dx.doi.org/10.1080/02331939208843861.

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10

Gaitsgory, Vladimir, and Sergey Rossomakhine. "Averaging and Linear Programming in Some Singularly Perturbed Problems of Optimal Control." Applied Mathematics & Optimization 71, no. 2 (June 11, 2014): 195–276. http://dx.doi.org/10.1007/s00245-014-9257-1.

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11

Kurina, G. A., and T. H. Nguyen. "Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients." Computational Mathematics and Mathematical Physics 52, no. 4 (April 2012): 524–47. http://dx.doi.org/10.1134/s0965542512040100.

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12

Kecman, V., S. Bingulac, and Z. Gajic. "Eigenvector approach for order-reduction of singularly perturbed linear-quadratic optimal control problems." Automatica 35, no. 1 (January 1999): 151–58. http://dx.doi.org/10.1016/s0005-1098(98)00141-1.

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13

Loghmani, G. B., and M. Ahmadinia. "Numerical Solution of Singularly Perturbed Boundary Value Problems Based on Optimal Control Strategy." Acta Applicandae Mathematicae 112, no. 1 (October 31, 2009): 69–78. http://dx.doi.org/10.1007/s10440-009-9553-y.

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14

Gaitsgory, V. "A new reduction technique for a class of singularly perturbed optimal control problems." IEEE Transactions on Automatic Control 40, no. 4 (April 1995): 721–24. http://dx.doi.org/10.1109/9.376101.

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15

Kadalbajoo, Mohan K., and Arindama Singh. "A Boundary value technique for solving singularly perturbed, fixed end-point optimal control problems." Optimal Control Applications and Methods 9, no. 4 (October 29, 2007): 443–48. http://dx.doi.org/10.1002/oca.4660090407.

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16

Gaitsgory, Vladimir, and Sergey Rossomakhine. "Erratum to: Averaging and Linear Programming in Some Singularly Perturbed Problems of Optimal Control." Applied Mathematics & Optimization 71, no. 2 (August 12, 2014): 277–78. http://dx.doi.org/10.1007/s00245-014-9265-1.

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17

Xu, Han, and Yinlai Jin. "The asymptotic expansion for a class of non-linear singularly perturbed problems with optimal control." Journal of Nonlinear Sciences and Applications 09, no. 05 (May 25, 2016): 2718–26. http://dx.doi.org/10.22436/jnsa.009.05.68.

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18

Babu, G. Kishor. "Singular Perturbation Method for Boundary Value and Optimal Problems to Power Factor Correction Converter Application." WSEAS TRANSACTIONS ON ELECTRONICS 11 (May 19, 2020): 42–53. http://dx.doi.org/10.37394/232017.2020.11.6.

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A linear discrete stable control system is considered. The Power Factor Correction (PFC) converter to allow independent control of current and voltage. It converter are fast and slow states to inheres sty present small parameters inductor and capacitor its computes stiffness and to include switching ripple effects. As an alternative a Singular Perturbation Method (SPM) is presented Boundary Value Problem (BVP) and Optimal Problem. It is applied to two state switching power converters to provide rigorous justification of\ the time scale separation. It is modeled as a one parameter singularly perturbed system. SPM consists of an outer series solution and one boundary layer correction (BLC) solution. A boundary layer correction is required to recover the initial conditions lost in the process of degeneration and to improve the solution. SPM is carried out up to second-order approximate solution for the PFC converter model for BVP and optimal control problems. The results are compared with the exact solution (between with and without parameters). The results substantiate the application.

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19

Gaitsgory, Vladimir, Ludmila Manic, and Sergey Rossomakhine. "On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria." Set-Valued and Variational Analysis 23, no. 1 (October 12, 2014): 87–131. http://dx.doi.org/10.1007/s11228-014-0306-3.

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20

Achdou, Yves, and Nicoletta Tchou. "Hamilton-Jacobi Equations on Networks as Limits of Singularly Perturbed Problems in Optimal Control: Dimension Reduction." Communications in Partial Differential Equations 40, no. 4 (January 12, 2015): 652–93. http://dx.doi.org/10.1080/03605302.2014.974764.

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21

Koripaeva, Yu, and D. Kozlov. "Asymptotic solutions of singularly perturbed optimal control problems for systems described by motion of a point." Актуальные направления научных исследований XXI века: теория и практика 2, no. 5 (November 11, 2014): 109–11. http://dx.doi.org/10.12737/6356.

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22

Langer, Ulrich, Olaf Steinbach, and Huidong Yang. "Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization." Computational Methods in Applied Mathematics 22, no. 1 (October 10, 2021): 97–111. http://dx.doi.org/10.1515/cmam-2021-0169.

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Abstract We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 {L_{2}} -norm regularization is replaced by the H - 1 {H^{-1}} -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ ⁢ h {u_{\varrho h}} to the state u and the desired state u ¯ {\overline{u}} in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 {\varrho=h^{2}} ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ {\overline{u}} . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

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23

Subbotina, N. N. "The value functions of singularly perturbed time-optimal control problems in the framework of Lyapunov functions method." Mathematical and Computer Modelling 45, no. 11-12 (June 2007): 1284–93. http://dx.doi.org/10.1016/j.mcm.2006.11.004.

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24

Fridman, E. "Exact slow-fast decomposition of a class of non-linear singularly perturbed optimal control problems via invariant manifolds." International Journal of Control 72, no. 17 (January 1999): 1609–18. http://dx.doi.org/10.1080/002071799220074.

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25

Danilin, A. R., and A. A. Shaburov. "Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables." Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta 55 (May 2020): 33–41. http://dx.doi.org/10.35634/2226-3594-2020-55-03.

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The paper deals with the problem of optimal control with a Boltz-type quality index over a finite time interval for a linear steady-state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A.B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady-state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time-optimal problems.

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26

Bobodzhanov, A. A., B. T. Kalimbetov, and V. F. Safonov. "Singularly perturbed control problems in the case of the stability of the spectrum of the matrix of an optimal system." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 96, no. 4 (December 30, 2019): 22–38. http://dx.doi.org/10.31489/2019m4/22-38.

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27

Koripaeva, Yu. "Especially the position of the switching points of the optimal control and their asymptotic expansions in the case of matrix-singularly perturbed linear problems of time optimal control with constraints on the control." Актуальные направления научных исследований XXI века: теория и практика 3, no. 5 (November 6, 2015): 111–15. http://dx.doi.org/10.12737/14468.

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28

Olwal, Thomas Otieno, Karim Djouani, Okuthe P. Kogeda, and Barend Jacobus Van Wyk. "Joint queue-perturbed and weakly coupled power control for wireless backbone networks." International Journal of Applied Mathematics and Computer Science 22, no. 3 (September 1, 2012): 749–64. http://dx.doi.org/10.2478/v10006-012-0056-z.

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Abstract Wireless Backbone Networks (WBNs) equipped with Multi-Radio Multi-Channel (MRMC) configurations do experience power control problems such as the inter-channel and co-channel interference, high energy consumption at multiple queues and unscalable network connectivity. Such network problems can be conveniently modelled using the theory of queue perturbation in the multiple queue systems and also as a weak coupling in a multiple channel wireless network. Consequently, this paper proposes a queue perturbation and weakly coupled based power control approach forWBNs. The ultimate objectives are to increase energy efficiency and the overall network capacity. In order to achieve this objective, a Markov chain model is first presented to describe the behaviour of the steady state probability distribution of the queue energy and buffer states. The singular perturbation parameter is approximated from the coefficients of the Taylor series expansion of the probability distribution. The impact of such queue perturbations on the transmission probability, given some transmission power values, is also analysed. Secondly, the inter-channel interference is modelled as a weakly coupled wireless system. Thirdly, Nash differential games are applied to derive optimal power control signals for each user subject to power constraints at each node. Finally, analytical models and numerical examples show the efficacy of the proposed model in solving power control problems in WBNs.

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29

SAGARA, Muneomi, Hiroaki MUKAIDANI, and Toru YAMAMOTO. "Near-Optimal Control for Singularly Perturbed Stochastic Systems." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E92-A, no. 11 (2009): 2874–82. http://dx.doi.org/10.1587/transfun.e92.a.2874.

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30

Kabanov, Yu M., and S. M. Pergamenshchikov. "Optimal control of singularly perturbed stochastic linear systems." Stochastics and Stochastic Reports 36, no. 2 (August 1991): 109–35. http://dx.doi.org/10.1080/17442509108833713.

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31

Bidani, M., N. E. Radhy, and B. Bensassi. "Optimal control of discrete-time singularly perturbed systems." International Journal of Control 75, no. 13 (January 2002): 955–66. http://dx.doi.org/10.1080/00207170210156152.

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32

Wang, Yue-Yun, Paul M. Frank, and N. Eva Wu. "Near-optimal control of nonstandard singularly perturbed systems." Automatica 30, no. 2 (February 1994): 277–92. http://dx.doi.org/10.1016/0005-1098(94)90030-2.

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33

Bouzaouache, Hajer, Ennaceur Ben Hadj Braiek, and Mohamed Benrejeb. "Reduced Optimal Control of Nonlinear Singularly Perturbed Systems." Systems Analysis Modelling Simulation 43, no. 1 (January 2003): 75–87. http://dx.doi.org/10.1080/0232929031000116353.

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34

Clarke, F. "Perturbed optimal control problems." IEEE Transactions on Automatic Control 31, no. 6 (June 1986): 535–42. http://dx.doi.org/10.1109/tac.1986.1104318.

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35

Sobolev, Vladimir. "One Critical Case in Singularly Perturbed Control Problems." Journal of Physics: Conference Series 811 (February 2017): 012017. http://dx.doi.org/10.1088/1742-6596/811/1/012017.

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36

Liu, Lei, and Shaofan Feng. "Multi-Objective Optimal Control for Uncertain Singularly Perturbed Systems." IEEE Access 9 (2021): 108340–45. http://dx.doi.org/10.1109/access.2021.3101211.

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37

Mukaidani, H., T. Shimomura, and Hua Xu. "Near-optimal control of linear multiparameter singularly perturbed systems." IEEE Transactions on Automatic Control 47, no. 12 (December 2002): 2051–57. http://dx.doi.org/10.1109/tac.2002.805676.

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38

Berg, J. M. "Optimal control of singularly perturbed linear systems and applications." Automatica 39, no. 2 (February 2003): 369–72. http://dx.doi.org/10.1016/s0005-1098(02)00236-4.

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39

Wu Li-Meng and Ni Ming-Kang. "Internal layer solution of singularly perturbed optimal control problem." Acta Physica Sinica 61, no. 8 (2012): 080203. http://dx.doi.org/10.7498/aps.61.080203.

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40

Salikhov, Z. G. "Modeling of Singularly Perturbed multi-criterion Optimal Control Workflows." IFAC-PapersOnLine 48, no. 3 (2015): 1254–58. http://dx.doi.org/10.1016/j.ifacol.2015.06.256.

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41

Sagara, Muneomi, Hiroaki Mukaidani, and Vasile Dragan. "Near-optimal control for multiparameter singularly perturbed stochastic systems." Optimal Control Applications and Methods 32, no. 1 (January 2011): 113–25. http://dx.doi.org/10.1002/oca.934.

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42

Bielecki, T., and ? Stettner. "On ergodic control problems for singularly perturbed Markov processes." Applied Mathematics & Optimization 20, no. 1 (July 1989): 131–61. http://dx.doi.org/10.1007/bf01447652.

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43

Abbad, M., J. A. Filar, and T. R. Bielecki. "Algorithms for singularly perturbed limiting average Markov control problems." IEEE Transactions on Automatic Control 37, no. 9 (1992): 1421–25. http://dx.doi.org/10.1109/9.159585.

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44

Wu, Mingkang Ni and Limeng. "Step-Like Contrast Structure of Singularly Perturbed Optimal Control Problem." Journal of Computational Mathematics 30, no. 1 (June 2012): 2–13. http://dx.doi.org/10.4208/jcm.1110-m11si07.

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45

Kecman, Vojislav, and Zoran Gajic. "Optimal Control and Filtering for Nonstandard Singularly Perturbed Linear Systems." Journal of Guidance, Control, and Dynamics 22, no. 2 (March 1999): 362–65. http://dx.doi.org/10.2514/2.4388.

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46

Xu, HUA, HIROAKI MUKAIDANI, and KOICHI MIZUKAMI. "New method for composite optimal control of singularly perturbed systems." International Journal of Systems Science 28, no. 2 (February 1997): 161–72. http://dx.doi.org/10.1080/00207729708929375.

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47

Karimi, H. R., P. Jabedar Maralani, B. Moshiri, and B. Lohmann. "APPROXIMATED OPTIMAL CONTROL OF SINGULARLY PERTURBED SYSTEMS VIA HAAR WAVELETS." IFAC Proceedings Volumes 38, no. 1 (2005): 353–58. http://dx.doi.org/10.3182/20050703-6-cz-1902.00920.

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48

Wu, Limeng, Mingkang Ni, and Haibo Lu. "Step-like contrast structure of singularly perturbed optimal control problem." Electronic Journal of Qualitative Theory of Differential Equations, no. 46 (2011): 1–16. http://dx.doi.org/10.14232/ejqtde.2011.1.46.

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49

Danilin, A. R. "Approximation of a singularly perturbed elliptic problem of optimal control." Sbornik: Mathematics 191, no. 10 (October 31, 2000): 1421–31. http://dx.doi.org/10.1070/sm2000v191n10abeh000512.

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50

Fridman, E. "Near-optimal H/sup ∞/ control of linear singularly perturbed systems." IEEE Transactions on Automatic Control 41, no. 2 (1996): 236–40. http://dx.doi.org/10.1109/9.481525.

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