Relevant bibliographies by topics / Singularly perturbed optimal control problems

Academic literature 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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Journal articles on the topic "Singularly perturbed optimal control problems"

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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Singularly perturbed optimal control problems"

1

Howe, Sei. "Upper and lower bounds for singularly perturbed linear quadratic optimal control problems." Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/54758.

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The question of how to optimally control a large scale system is widely considered to be difficult to solve due to the size of the problem. This difficulty is further compounded when a system exhibits a two time-scale structure where some components evolve slowly and others evolve quickly. When this occurs, the optimal control problem is regarded as singularly perturbed with a perturbation parameter epsilon representing the ratio of the slow time-scale to the fast time-scale. As epsilon goes to zero, the system becomes stiff resulting in a computationally intractable problem. In this thesis, we propose an analytic method for constructing bounds on the minimum cost of a singularly perturbed, linear-quadratic optimal control problem that hold for any arbitrary value of epsilon.
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2

Reibiger, Christian. "Optimal Control Problems with Singularly Perturbed Differential Equations as Side Constraints: Analysis and Numerics." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-162862.

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It is well-known that the solution of a so-called singularly perturbed differential equation exhibits layers. These are small regions in the domain where the solution changes drastically. These layers deteriorate the convergence of standard numerical algorithms, such as the finite element method on a uniform mesh. In the past many approaches were developed to overcome this difficulty. In this context it was very helpful to understand the structure of the solution - especially to know where the layers can occur. Therefore, we have a lot of analysis in the literature concerning the properties of solutions of such problems. Nevertheless, this field is far from being understood conclusively. More recently, there is an increasing interest in the numerics of optimal control problems subject to a singularly perturbed convection-diffusion equation and box constraints for the control. However, it is not much known about the solutions of such optimal control problems. The proposed solution methods are based on the experience one has from scalar singularly perturbed differential equations, but so far, the analysis presented does not use the structure of the solution and in fact, the provided bounds are rather meaningless for solutions which exhibit boundary layers, since these bounds scale like epsilon^(-1.5) as epsilon converges to 0. In this thesis we strive to prove bounds for the solution and its derivatives of the optimal control problem. These bounds show that there is an additional layer that is weaker than the layers one expects knowing the results for scalar differential equation problems, but that weak layer deteriorates the convergence of the proposed methods. In Chapter 1 and 2 we discuss the optimal control problem for the one-dimensional case. We consider the case without control constraints and the case with control constraints separately. For the case without control constraints we develop a method to prove bounds for arbitrary derivatives of the solution, given the data is smooth enough. For the latter case we prove bounds for the derivatives up to the second order. Subsequently, we discuss several discretization methods. In this context we use special Shishkin meshes. These meshes are piecewise equidistant, but have a very fine subdivision in the region of the layers. Additionally, we consider different ways of discretizing the control constraints. The first one enforces the compliance of the constraints everywhere and the other one enforces it only in the mesh nodes. For each proposed algorithm we prove convergence estimates that are independent of the parameter epsilon. Hence, they are meaningful even for small values of epsilon. As a next step we turn to the two-dimensional case. To be able to adapt the proofs of Chapter 2 to this case we require bounds for the solution of the scalar differential equation problem for a right hand side f only in W^(1,infty). Although, a lot of results for this problem can be found in the literature but we can not apply any of them, because they require a smooth right hand side f in C^(2,alpha) for some alpha in (0,1). Therefore, we dedicate Chapter 3 to the analysis of the scalar differential equations problem only using a right hand side f that is not very smooth. In Chapter 4 we strive to prove bounds for the solution of the optimal control problem in the two dimensional case. The analysis for this problem is not complete. Especially, the characteristic layers induce subproblems that are not understood completely. Hence, we can not prove sharp bounds for all terms in the solution decomposition we construct. Nevertheless, we propose a solution method. Numerical results indicate an epsilon-independent convergence for the considered examples - although we are not able to prove this.
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3

Reibiger, Christian [Verfasser], Hans-Görg [Akademischer Betreuer] Roos, and Gert [Akademischer Betreuer] Lube. "Optimal Control Problems with Singularly Perturbed Differential Equations as Side Constraints: Analysis and Numerics / Christian Reibiger. Gutachter: Hans-Görg Roos ; Gert Lube. Betreuer: Hans-Görg Roos." Dresden : Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://d-nb.info/106909658X/34.

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4

Reibiger, Christian. "Optimal Control Problems with Singularly Perturbed Differential Equations as Side Constraints: Analysis and Numerics." Doctoral thesis, 2014. https://tud.qucosa.de/id/qucosa%3A28578.

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Abstract:
It is well-known that the solution of a so-called singularly perturbed differential equation exhibits layers. These are small regions in the domain where the solution changes drastically. These layers deteriorate the convergence of standard numerical algorithms, such as the finite element method on a uniform mesh. In the past many approaches were developed to overcome this difficulty. In this context it was very helpful to understand the structure of the solution - especially to know where the layers can occur. Therefore, we have a lot of analysis in the literature concerning the properties of solutions of such problems. Nevertheless, this field is far from being understood conclusively. More recently, there is an increasing interest in the numerics of optimal control problems subject to a singularly perturbed convection-diffusion equation and box constraints for the control. However, it is not much known about the solutions of such optimal control problems. The proposed solution methods are based on the experience one has from scalar singularly perturbed differential equations, but so far, the analysis presented does not use the structure of the solution and in fact, the provided bounds are rather meaningless for solutions which exhibit boundary layers, since these bounds scale like epsilon^(-1.5) as epsilon converges to 0. In this thesis we strive to prove bounds for the solution and its derivatives of the optimal control problem. These bounds show that there is an additional layer that is weaker than the layers one expects knowing the results for scalar differential equation problems, but that weak layer deteriorates the convergence of the proposed methods. In Chapter 1 and 2 we discuss the optimal control problem for the one-dimensional case. We consider the case without control constraints and the case with control constraints separately. For the case without control constraints we develop a method to prove bounds for arbitrary derivatives of the solution, given the data is smooth enough. For the latter case we prove bounds for the derivatives up to the second order. Subsequently, we discuss several discretization methods. In this context we use special Shishkin meshes. These meshes are piecewise equidistant, but have a very fine subdivision in the region of the layers. Additionally, we consider different ways of discretizing the control constraints. The first one enforces the compliance of the constraints everywhere and the other one enforces it only in the mesh nodes. For each proposed algorithm we prove convergence estimates that are independent of the parameter epsilon. Hence, they are meaningful even for small values of epsilon. As a next step we turn to the two-dimensional case. To be able to adapt the proofs of Chapter 2 to this case we require bounds for the solution of the scalar differential equation problem for a right hand side f only in W^(1,infty). Although, a lot of results for this problem can be found in the literature but we can not apply any of them, because they require a smooth right hand side f in C^(2,alpha) for some alpha in (0,1). Therefore, we dedicate Chapter 3 to the analysis of the scalar differential equations problem only using a right hand side f that is not very smooth. In Chapter 4 we strive to prove bounds for the solution of the optimal control problem in the two dimensional case. The analysis for this problem is not complete. Especially, the characteristic layers induce subproblems that are not understood completely. Hence, we can not prove sharp bounds for all terms in the solution decomposition we construct. Nevertheless, we propose a solution method. Numerical results indicate an epsilon-independent convergence for the considered examples - although we are not able to prove this.
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5

Lo, Kuang Lin, and 羅光淋. "Optimal Control of Singularly Perturbed Stochastic Hybrid Systems." Thesis, 1993. http://ndltd.ncl.edu.tw/handle/04652626089263255311.

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Books on the topic "Singularly perturbed optimal control problems"

1

Myo-Taeg, Lim, ed. Optimal control of singularly perturbed linear systems and applications: High-accuracy techniques. New York: Marcel Dekker, 2001.

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2

Weak convergence methods and singularly perturbed stochastic control and filtering problems. Boston: Birkhäuser, 1990.

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Kushner, Harold J. Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Boston, MA: Birkhäuser Boston, 1990. http://dx.doi.org/10.1007/978-1-4612-4482-0.

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Yoo, Seog Hwan. Asymptotic expansions for a nonlinear singularly perturbed optimal control problem with free final time. 1989.

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Gajic, Zoran. Optimal Control of Singularly Perturbed Linear Systems and Applications. Taylor & Francis Group, 2001.

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Gajic, Zoran. Optimal Control of Singularly Perturbed Linear Systems and Applications. Taylor & Francis Group, 2001.

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Gajic, Zoran. Optimal Control of Singularly Perturbed Linear Systems and Applications. Taylor & Francis Group, 2001.

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Gajic, Zoran. Optimal Control of Singularly Perturbed Linear Systems and Applications. Taylor & Francis Group, 2001.

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Gajic, Zoran. Optimal Control of Singularly Perturbed Linear Systems and Applications. Taylor & Francis Group, 2001.

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Gajic, Zoran. Optimal Control of Singularly Perturbed Linear Systems and Applications (Control Engineering, Number 7). CRC, 2001.

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Book chapters on the topic "Singularly perturbed optimal control problems"

1

Naidu, Desineni S., and Ayalasomayajula K. Rao. "Singularly perturbed nonlinear difference equations and closed-loop discrete optimal control problem." In Lecture Notes in Mathematics, 148–88. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0074764.

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Kushner, Harold J. "Controlled Singularly Perturbed Systems." In Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, 61–91. Boston, MA: Birkhäuser Boston, 1990. http://dx.doi.org/10.1007/978-1-4612-4482-0_4.

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Gajić, Zoran, and Xuemin Shen. "Singularly Perturbed Weakly Coupled Linear Control Systems." In Parallel Algorithms for Optimal Control of Large Scale Linear Systems, 293–312. London: Springer London, 1993. http://dx.doi.org/10.1007/978-1-4471-3219-6_10.

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Sobolev, Vladimir A. "Thrice Critical Case in Singularly Perturbed Control Problems." In Trends in Mathematics, 83–87. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-01153-6_15.

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Kabanov, Yu M., and S. M. Pergamenshchikov. "On Optimal Control of Singularly Perturbed Stochastic Differential Equations." In Modeling, Estimation and Control of Systems with Uncertainty, 200–209. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4612-0443-5_13.

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Kushner, Harold J. "Singularly Perturbed Wide-Band Noise Driven Systems." In Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, 171–90. Boston, MA: Birkhäuser Boston, 1990. http://dx.doi.org/10.1007/978-1-4612-4482-0_8.

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Gajić, Zoran, and Xuemin Shen. "Quasi Singularly Perturbed and Weakly Coupled Linear Control Systems." In Parallel Algorithms for Optimal Control of Large Scale Linear Systems, 255–92. London: Springer London, 1993. http://dx.doi.org/10.1007/978-1-4471-3219-6_9.

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Gajić, Zoran, and Xuemin Shen. "Optimal Control of Singularly Perturbed and Weakly Coupled Bilinear Systems." In Parallel Algorithms for Optimal Control of Large Scale Linear Systems, 399–432. London: Springer London, 1993. http://dx.doi.org/10.1007/978-1-4471-3219-6_14.

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Xu, Hua, and Koichi Mizukami. "Nonstandard Extension of H ∞-Optimal Control for Singularly Perturbed Systems." In Advances in Dynamic Games and Applications, 81–94. Boston, MA: Birkhäuser Boston, 2000. http://dx.doi.org/10.1007/978-1-4612-1336-9_4.

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Pan, Zigang, and Tamer Başar. "H∞-Optimal Control of Singularly Perturbed Systems with Sampled-State Measurement." In Advances in Dynamic Games and Applications, 23–55. Boston, MA: Birkhäuser Boston, 1994. http://dx.doi.org/10.1007/978-1-4612-0245-5_2.

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Conference papers on the topic "Singularly perturbed optimal control problems"

1

Safonov, Valerii Fedorovich. "Regularization of integral operators in singularly perturbed problems in S. A. Lomov's Method." In International Conference "Optimal Control and Differential Games" dedicated to the 110th anniversary of L. S. Pontryagin. Moscow: Steklov Mathematical Institute, 2018. http://dx.doi.org/10.4213/proc23040.

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Liu, Lei, Zejin Feng, and Cunwu Han. "Multi-objected optimal control problem for singularly perturbed systems based on passivity." In 2016 35th Chinese Control Conference (CCC). IEEE, 2016. http://dx.doi.org/10.1109/chicc.2016.7553760.

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Kurina, Galina Alekseevna, and Thi Hoai Nguyen. "On zero order asymptotic solution of singularly perturbed linear - quadratic problems in a critical case." In International Conference "Optimal Control and Differential Games" dedicated to the 110th anniversary of L. S. Pontryagin. Moscow: Steklov Mathematical Institute, 2018. http://dx.doi.org/10.4213/proc23000.

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Garcia, G., J. Daafouz, and J. Bernussou. "A LMI solution in the H/sub 2/ optimal problem for singularly perturbed systems." In Proceedings of the 1998 American Control Conference (ACC). IEEE, 1998. http://dx.doi.org/10.1109/acc.1998.694729.

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Gajic, Z., Dj Petkovski, and N. Harkara. "The Recursive Algorithm for the Optimal Static Output Feedback Control Problem of Linear Singularly Perturbed Systems." In 1988 American Control Conference. IEEE, 1988. http://dx.doi.org/10.23919/acc.1988.4789818.

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Radisavljevic-Gajic, Verica, and Milos Milanovic. "Three-Stage Feedback Controller Design With Applications to a Three Time-Scale System." In ASME 2016 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/dscc2016-9806.

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A new technique was presented that facilitates design of independent full-state feedback controllers at the subsystem levels. Different types of local controllers, for example, eigenvalue assignment, robust, optimal (in some sense L1, H2, H∞, ...) may be used to control different subsystems. This feature has not been available for any known linear feedback controller design. In the second part of the paper, we specialize the results obtained to the three time-scale linear systems (singularly perturbed control systems) that have natural decomposition into slow, fast, and very fast subsystems. The proposed technique eliminates numerical ill-condition of the original three-time scale problems.
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Kurina, Galina, and Nguyen Thi Hoai. "Higher Order Asymptotic Approximation to a Solution of Singularly Perturbed Optimal Control Problem with Intersecting Solutions of the Degenerate Problem." In 2019 23rd International Conference on System Theory, Control and Computing (ICSTCC). IEEE, 2019. http://dx.doi.org/10.1109/icstcc.2019.8885991.

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Danilin, Aleksei Rufimovich, and A. A. Shaburov. "Asymptotic expansion of a solution to a singularly perturbed optimal control problem with a convex integral performance index whose terminal part depends on slow and fast variables." In International Conference "Optimal Control and Differential Games" dedicated to the 110th anniversary of L. S. Pontryagin. Moscow: Steklov Mathematical Institute, 2018. http://dx.doi.org/10.4213/proc22969.

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Liu, Lei, Yi He, and Cunwu Han. "Optimal Tracking Control for Uncertain Singularly Perturbed Systems." In 2021 IEEE 10th Data Driven Control and Learning Systems Conference (DDCLS). IEEE, 2021. http://dx.doi.org/10.1109/ddcls52934.2021.9455506.

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Abbad, M., J. A. Filar, and T. R. Bielecki. "Algorithms for singularly perturbed limiting average Markov control problems." In 29th IEEE Conference on Decision and Control. IEEE, 1990. http://dx.doi.org/10.1109/cdc.1990.203841.

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