On the robustness of the integrable trajectories of the control systems with limited control resources

• Autorzy: Huseyin Nesir , Huseyin Anar , Guseinov Khalik G.

Identyfikatory

DOI

10.24425/acs.2023.146958

• Języki publikacji: EN

Abstrakty

EN

The control system described by Urysohn type integral equation is considered where the system is nonlinear with respect to the phase vector and is affine with respect to the control vector. The control functions are chosen from the closed ball of the space L_q(Ω; ℝ^m), q > 1, with radius r and centered at the origin. The trajectory of the system is defined as p-integrable multivariable function from the space L_p(Ω; ℝⁿ), (1/q) + (1/p) = 1, satisfying the system’s equation almost everywhere. It is shown that the system’s trajectories are robust with respect to the fast consumption of the remaining control resource. Applying this result it is proved that every trajectory can be approximated by the trajectory obtained by full consumption of the total control resource.

Słowa kluczowe

EN

nonlinear control system; integral equation; integral constraint; integrable trajectory; robustness

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2023
• Tom: Vol. 33, No. 3
• Strony: 527--537
• Opis fizyczny: Bibliogr. 16 poz., wzory

Twórcy

autor

Huseyin Nesir

• Department of Mathematics and Science Education, Sivas Cumhuriyet University, 58140 Sivas, Turkey

• https://orcid.org/0000-0001-7652-1505

autor

Huseyin Anar

• Department of Statistics and Computer Sciences, Sivas Cumhuriyet University, 58140 Sivas, Turkey

• https://orcid.org/0000-0002-3911-2304

autor

Guseinov Khalik G.

• Department of Mathematics, Eskisehir Technical University, 26470 Eskisehir, Turkey

• https://orcid.org/0000-0001-7487-8666

Bibliografia

• [1] C. Bjorland, L. Brandolese, D. Iftimie and M.E. Schonbek: 𝐿𝑝-solutions of the steady-state Navier-Stokes equations with rough external forces. Communications in Partial Differential Equations, 36(2), (2011), 216-246. DOI: 10.1080/03605302.2010.485286.
• [2] R. Conti: Problemi di Controllo e di Controllo Ottimale. UTET, Torino, 1974. In Italian.
• [3] C. Corduneanu: Integral Equations and Applications. Cambridge University Press, Cambridge, 1991.
• [4] I. Gohberg and S. Goldberg: Basic Operator Theory. Birkhäuser, Boston, 1981.
• [5] M.I. Gusev and I.V. Zykov: On the geometry of the reachable sets of control systems with isoperimetric constraints. Trudy Instituta Matematiki i Mekhaniki UrO RAN. 24(1), (2018), 63-75. DOI: 10.21538/0134-4889-2018-24-1-63-75.
• [6] D. Hilbert: Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen. Druck und Verlag von B.G. Teubner, Leipzig und Berlin, 1912. In German.
• [7] N. Huseyin: On the properties of the set of 𝑝-integrable trajectories of the control system with limited control resources. International Journal of Control, 93(8), (2020), 1810-1816. DOI: 10.1080/00207179.2018.1533254.
• [8] A. Huseyin, N. Huseyin and Kh.G. Guseinov: Approximations of the image and integral funnel of the 𝐿𝑝 balls under Urysohn type integral operator. Functional Analysis and Its Applications, 56(4), (2022), 269-281. DOI: 10.1134/S0016266322040050.
• [9] N. Huseyin, A. Huseyin and Kh.G. Guseinov: Approximations of the set of trajectories and integral funnel of the non-linear control systems with 𝐿𝑝 norm constraints on the control functions. IMA Journal of Mathematical Control and Information, 39(4), (2022), 1213-1231. DOI: 10.1093/imamci/dnac028.
• [10] L.V. Kantorovich and G.P. Akilov: Functional Analysis. Nauka, Moscow, 1977.
• [11] M.A. Krasnoselskii and S.G. Krein: On the principle of averaging in nonlinear mechanics. Uspekhi Matematicheskikh Nauk, 10(3), (1955), 147-152.
• [12] N.N. Krasovskii: Theory of Control of Motion: Linear Systems. Nauka, Moscow, 1968.
• [13] M. Kwapisz: Weighted norms and existence and uniqueness of 𝐿𝑝 solutions for integral equations in several variables. Journal of Differential Equations, 97, (1992), 246-262. DOI: 10.1016/0022-0396(92)90072-U.
• [14] P. Rousse, P.-L. Garoche and D. Henrion: Parabolic set simulation for reachability analysis of linear time-invariant systems with integral quadratic constraint. European Journal of Control, 58, (2021), 152-167. DOI: 10.1016/j.ejcon.2020.08.002.
• [15] S.V. Shaposhnikov: On the uniqueness of integrable and probabilistic solutions of the Cauchy problem for the Fokker-Planck-Kolmogorov equation. Doklady Mathematics, 84 (2011), 565-570. DOI: 10.1134/S106456241104020X.
• [16] N.N. Subbotina and A.I. Subbotin: Alternative for the encounter-evasion differential game with constraints on the momenta of the players controls. Journal of Applied Mathematics and Mechanics, 39(3), (1975), 376-385. DOI: 10.1016/0021-8928(75)90002-7.