Geodesic
On flexibility of constraints system under approximation of optimal control problems
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 59 (2022), pp. 114-130.

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For finite-dimensional mathematical programming problems (approximating problems) being obtained by a parametric approximation of control functions in lumped optimal control problems with functional equality constraints, we introduce concepts of rigidity and flexibility for a system of constraints. The rigidity in a given admissible point is treated in the sense that this point is isolated for the admissible set; otherwise, we call a system of constraints as flexible in this point. Under using a parametric approximation for a control function with the help of quadratic exponentials (Gaussian functions) and subject to some natural hypotheses, we establish that in order to guarantee the flexibility of constraints system in a given admissible point it suffices to increase the dimension of parameter space in the approximating problem. A test of our hypotheses is illustrated by an example of the soft lunar landing problem.
Keywords: lumped optimal control problems with functional equality constraints, parametric approximation of control, rigidity and flexibility of constraints system, Gaussian functions, quadratic exponentials. 
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A. V. Chernov. On flexibility of constraints system under approximation of optimal control problems. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 59 (2022), pp. 114-130. http://geodesic.mathdoc.fr/item/IIMI_2022_59_a7/

[1] Conway B. A., “A survey of methods available for the numerical optimization of continuous dynamic systems”, Journal of Optimization Theory and Applications, 152:2 (2012), 271–306 | DOI | MR | Zbl

[2] Teo K. L., Li B., Yu Ch., Rehbock V., Applied and computational optimal control, Springer, Cham, 2021 | DOI | MR | Zbl

[3] Li B., Guo X., Zeng X., Dian S., Guo M., “An optimal PID tuning method for a single-link manipulator based on the control parametrization technique”, Discrete and Continuous Dynamical Systems — S, 13:6 (2020), 1813–1823 | DOI | MR | Zbl

[4] Farooqi H., Fagiano L., Colaneri P., Barlini D., “Shrinking horizon parametrized predictive control with application to energy-efficient train operation”, Automatica, 112 (2020), 108635 | DOI | MR | Zbl

[5] Zhong W., Lin Q., Loxton R., Teo K. L., “Optimal train control via switched system dynamic optimization”, Optimization Methods and Software, 36:2–3 (2021), 602–626 | DOI | MR | Zbl

[6] Liu P., Liu X., Wang P., Li G., Xiao L., Yan J., Ren Zh., “Control variable parameterisation with penalty approach for hypersonic vehicle reentry optimisation”, International Journal of Control, 92:9 (2019), 2015–2024 | DOI | MR | Zbl

[7] Wu D., Bai Y., Yu Ch., “A new computational approach for optimal control problems with multiple time-delay”, Automatica, 101 (2019), 388–395 | DOI | MR | Zbl

[8] Mu P., Wang L., Liu Ch., “A control parameterization method to solve the fractional-order optimal control problem”, Journal of Optimization Theory and Applications, 187:1 (2020), 234–247 | DOI | MR | Zbl

[9] Liu Ch., Gong Zh., Yu Ch., Wang S., Teo K. L., “Optimal control computation for nonlinear fractional time-delay systems with state inequality constraints”, Journal of Optimization Theory and Applications, 191:1 (2021), 83–117 | DOI | MR | Zbl

[10] Aliev F. A., Larin V. B., “A historical perspective on the parametrization of all stabilizing feedback controllers”, Applied and Computational Mathematics, 18:3 (2019), 326–328 | MR | Zbl

[11] Zhang Y., Zhang J.-F., Liu X.-K., “Implicit function based adaptive control of non-canonical form discrete-time nonlinear systems”, Automatica, 129 (2021), 109629 | DOI | MR | Zbl

[12] Volin J. M., Ostrovskii G. M., “A method of successive approximations for calculating optimal modes of some distributed-parameter systems”, Automation and Remote Control, 26 (1966), 1188–1194 | MR | Zbl

[13] Teo K. L., Goh C. J., Wong K. H., A unified computational approach to optimal control problems, John Wiley and Sons, New York, 1991 | MR | Zbl

[14] Teo K. L., Jennings L. S., Lee H. W. J., Rehbock V., “The control parameterization enhancing transform for constrained optimal control problems”, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 40:3 (1999), 314–335 | DOI | MR | Zbl

[15] Li R., Teo K. L., Wong K. H., Duan G. R., “Control parameterization enhancing transform for optimal control of switched systems”, Mathematical and Computer Modelling, 43:11–12 (2006), 1393–1403 | DOI | MR | Zbl

[16] Chernov A. V., “On the application of Gaussian functions for discretization of optimal control problems”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 27:4 (2017), 558–575 (in Russian) | DOI | MR | Zbl

[17] Chernov A. V., “On application of Gaussian functions for numerical solution of optimal control problems”, Automation and Remote Control, 80:6 (2019), 1026–1040 | DOI | DOI | MR | Zbl

[18] Chernov A. V., “On differentiation of functionals of approximating problems in the frame of solution of free time optimal control problems by the sliding nodes method”, Tambov University Reports. Series: Natural and Technical Sciences, 23:124 (2018), 861–876 (in Russian) | DOI

[19] Afanas'ev V. N., Kolmanovskij V. B., Nosov V. R., Mathematical theory of control systems design, Kluwer Academic Publishers, Dordrecht, 1996 | MR | Zbl

[20] Maz'ya V., Schmidt G., Approximate approximations, AMS, Providence, 2007 | MR | Zbl

[21] Luh L.-T., “The shape parameter in the Gaussian function”, Computers and Mathematics with Applications, 63:3 (2012), 687–694 | DOI | MR | Zbl

[22] Chernov A. V., “On using Gaussian functions with varied parameters for approximation of functions of one variable on a finite segment”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 27:2 (2017), 267–282 (in Russian) | DOI | MR | Zbl

[23] Buhmann M. D., Radial basis functions: theory and implementations, Cambridge University Press, Cambridge, 2003 | DOI | MR | Zbl

[24] Laforgia A., Natalini P., “Exponential, gamma and polygamma functions: simple proofs of classical and new inequalities”, Journal of Mathematical Analysis and Applications, 407:2 (2013), 495–504 | DOI | MR | Zbl

[25] Chernov A. V., “On approximate solution of free time optimal control problems”, Vestnik of Lobachevsky University of Nizhni Novgorod, 2012, no. 6(1), 107–114 (in Russian)

[26] Sukharev A. G., Timokhov A. V., Fedorov V. V., A course in optimization methods, Nauka, M., 2008 | MR