The estimation of solutions sets of~linear systems of~ordinary differential equations with perturbations based on~the Cauchy operator

A. A. Rogalev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 27 (2023) no. 2, pp. 357-374.

Résumé

The paper outlines a method for numerical analysis of sets of solutions for linear systems of ordinary differential equations that contain perturbations in the right-hand side. The method determines extreme values of the solutions, which comprise the sets of solutions along the coordinate axes or in a specified direction. The estimations are based on using the Cauchy operator, written with symbolic formulas for variations of arbitrary constants. Additionally, control is implemented over the deviation of solutions when calculating a bundle of trajectories. The paper also is devoted to examples of estimating reachability sets of systems under the influence of control and disturbance effects.

Détail

Mots-clés : perturbations, solutions sets, symbolic formulae.
Keywords: trajectories tube, symbolic algorithms

Bibliographie

[1] Abgaryan K. A., “On the stability of motion within a finite interval of time”, Sov. Math., Dokl., 9:3 (1968), 1110–1112 | MR | Zbl

[2] Abgaryan K. A., Vvedenie v teoriiu ustoichivosti dvizheniia na konechnom intervale vremeni [Introduction to the Theory of Stability of Motion over a Finite Time Interval], Nauka, Moscow, 1991, 160 pp. (In Russian) | Zbl

[3] Polyak B. T., Khlebnikov M. V., Shcherbakov P. S., “Linear matrix inequalities in control systems with uncertainty”, Autom. Remote Control, 82:1 (2021), 1–40 | DOI | DOI | Zbl

[4] Kurzhansky A. B., Upravlenie i nabliudenie v usloviiakh neopredelennosti [Management and Supervision in Conditions of Uncertainty], Nauka, Moscow, 1977, 392 pp. (In Russian)

[5] Kurzhansky A. B., Filippova T. F., “On a description of the bundle of surviving trajectories of a control system”, Differ. Uravn., 23:8 (1987), 1303–1315 (In Russian) | MR | Zbl

[6] Chernous'ko F. L., Otsenivanie fazovogo sostoianiia dinamicheskikh sistem. Metod ellipsoidov [Estimation of the Phase State of Dynamic Systems. Ellipsoid Method], Nauka, Moscow, 1988, 320 pp. (In Russian)

[7] Chernous'ko F. L., “Ellipsoidal approximation of the attainability sets of a linear system with an uncertain matrix”, J. Appl. Math. Mech., 60:6 (1996), 921–931 | DOI | Zbl

[8] Kurzhanski A. B., Mesyats A. I., “Control of ellipsoidal trajectories: Theory and numerical results”, Comput. Math. Math. Phys., 54:3 (2014), 418–428 | DOI | DOI | Zbl

[9] Chernous'ko F. L., “Optimal ellipsoidal estimates of control and uncertain systems (survey)”, Applied and Computational Mathematics, 8:2 (2009), 135–151

[10] Ushakov V. N., Ershov A. A., “Reachable sets and integral funnels of differential inclusions depending on a parameter”, Dokl. Math., 104:1 (2021), 200–204 | DOI | DOI | Zbl

[11] Ushakov V. N., Ershov A. A., Ushakov A. V., “Control systems depending on a parameter: Reachable sets and integral funnels”, Mech. Solids, 57:7 (2022), 1672–1688 | DOI | DOI | Zbl

[12] Aubin J.-P., Frankowska H., “The value does not exist! A motivation for extremal analysis”, Probab. Uncertain. Quant. Risk, 7:3 (2022), 195–214 | DOI

[13] Althoff M., Frehse G., Girard A., “Set propagation techniques for reachability analysis”, Annu. Rev. Control Robot. Auton. Syst., 4 (2021), 369–395 | DOI

[14] Villegas Pico H. N., Alipantis D. C., “Reachability analysis of linear dynamic systems with constant, arbitrary, and Lipschitz continuous inputs”, Automatica, 95 (2018), 293–305 | DOI | Zbl

[15] Morozov A. Yu., Reviznikov D. L., “Adaptive interpolation algorithm on sparse meshes for numerical integration of systems of ordinary differential equations with interval uncertainties”, Differ. Equ., 57:7 (2021), 947–958 | DOI | DOI | Zbl

[16] Gidaspov V. Yu., Morozov A. Yu., Reviznikov D. L., “Adaptive interpolation algorithm using TT-decomposition for modeling dynamical systems with interval parameters”, Comput. Math. Math. Phys., 61:9 (2021), 1387–1400 | DOI | DOI | Zbl

[17] Morozov A. Yu., Reviznikov D. L., “Interval approach to solving parametric identification problems for dynamical systems”, Differ. Equ., 58:7 (2022), 952–965 | DOI | DOI | Zbl

[18] Novikov V. A., Rogalev A. N., “Construction of convergent upper and lower bounds of solutions of systems of ordinary differential equations with interval initial data”, Comput. Math. Math. Phys., 33:2 (1993), 193–203 | MR | Zbl

[19] Rogalev A. N., Rogalev A. A., “Numerical estimates of the maximum deviations of aircraft in the atmosphere”, Vestn. SibGAU, 16:1 (2016), 104–112 (In Russian)

[20] Rogalev A. A., “Algorithms for symbolic calculations based on root trees for assessing management capabilities”, Siberian Journal of Science and Technology, 18:4 (2017), 810–819 (In Russian)

[21] Rogalev A. N., Rogalev A. A., Feodorova N. A., “Numerical computations of the safe boundaries of complex technical systems and practical stability”, J. Phys.: Conf. Ser., 1399 (2019), 033112 | DOI

[22] Rogalev A. N., Rogalev A. A., Feodorova N. A., “Malfunction analysis and safety of mathematical models of technical systems”, J. Phys.: Conf. Ser., 1515 (2020), 022064 | DOI

[23] Rogalev A. N., “Regularization of inclusions of differential equations solutions based on the kinematics of a vector field in stability problems”, J. Phys.: Conf. Ser., 2099 (2021), 012045 | DOI

[24] Smirnov A. V., “The bilinear complexity and practical algorithms for matrix multiplication”, Comput. Math. Math. Phys., 53:12 (2013), 1781–1795 | DOI | DOI | MR | Zbl

[25] Abramov S. A., Ryabenko A. A., Khmelnov D. E., “Regular solutions of linear ordinary differential equations and truncated series”, Comput. Math. Math. Phys., 60:1 (2020), 1–14 | DOI | DOI | Zbl

[26] Galeev E. M., Optimizatsiia: teoriia, primery, zadachi [Optimization: Theory, Examples, Tasks], URSS, Moscow, 2002, 304 pp. (In Russian)

[27] Ahlberg J. H., Nilson E. N., Walsh J. L., The Theory of Splines and their Applications, Mathematics in Science and Engineering, Academic Press, New York, 1967, xi+284 pp. | DOI | Zbl

[28] Tolpegin I. G., “Differential-game methods for targeting missiles at high-speed maneuvering targets”, Izv. RARAN, 2003, no. 1, 80–86 (In Russian)