Estimates of the phase trajectories of controlled systems with multi-valued impulses

O. V. Filippova

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 143, pp. 326-334

Résumé

We consider a controlled system for the differential equation $$ \dot{x}(t)=f(t,x(t),u(t), \xi), \ \ t \in [a,b] , \ \ x(a)=\mathrm{x},$$ where the parameter $\xi$ is an element of some given metric space, the control $u$ satisfies the constraint $$ u(t)\in U(t,x(t), \xi), \ \ t \in [a,b].$$ It is assumed that at each given moment of time $t_k\in (a,b)$ a solution $x:[a,b]\to \mathbb{R}^n$ (a phase trajectory) suffers discontinuity, the magnitude of which belongs to a non-empty compact set $I_k( x(t_k))\subset \mathbb{R}^n,$ and is an absolutely continuous function on intervals $(t_{k-1},t_k]$. The control function is assumed to be measurable. A theorem on estimating the distance from a given piece-wise absolutely continuous function $y:[a,b]\to \mathbb{R}^n$ to the set of phase trajectories for all initial values from a neighborhood of a vector $x_0$ and for all parameters from a neighborhood of a point $\xi_0$ is proven. It is assumed that for the given initial value $\mathrm{x}=x_0$ of the solution and for the value $\xi=\xi_0$ of the parameter, the set of phase trajectories is a priori limited. The proven theorem allows, by selecting the function $y$, to obtain an approximate solution of the controlled system, as well as an estimate of the error of such solution.

Détail

Keywords: differential inclusion, Cauchy problem, multi-valued impulses, phase trajectory.

Bibliographie

[11] A. F. Filippov, “On some questions of the theory of optimal controll”, Moscow Universities Reports, 1959, no. 2, 25–32 (In Russian) | Zbl

[12] T. Wazewski, “Systemes de commande et equations au contingent”, Bull. Acad. Polon. Sci., Ser. Math. Astr., Phys., 9:3 (1961), 151–155 | MR | Zbl

[13] T. Wazewski, “Sur une generalisation de la notion des solution d'une equations au contingent”, Bull. Acad. Polon. Sci., Ser. Math., Astr., Phys., 10:1 (1962), 11–15 | MR | Zbl

[14] S. V. Emelyanov, A. V. Ilyin, S. K. Korovin, V. V. Fomichev, A. S. Fursov, Mathematical Methods of Control Theory. Problems of Stability, Controllability and Observability, 1st. ed., FIZMATLIT Publ., Moscow, 2014 (In Russian)

[15] O. V. Filippova, “Differential equations with a parameter, with multivalued impulses and with control”, Vestnik rossiyskikh universitetov. Matematika = Russian Universities Reports. Mathematics, 25:132 (2020), 441–447 (In Russian) | Zbl

[16] P. I. Chugunov, “Properties of solutions for differential switching and controlled systems”, Applied Mathematics and Application Packages, 1980, 155–179 (In Russian)

[17] V. I. Blagodatskikh, A. F. Filippov, “Differentsialnye vklyucheniya i optimalnoe upravlenie”, Topologiya, obyknovennye differentsialnye uravneniya, dinamicheskie sistemy, Sbornik obzornykh statei. 2. K 50-letiyu instituta, Tr. MIAN SSSR, 169, 1985, 194–252 ; V. I. Blagodatskikh, A. F. Filippov, “Differential inclusions and optimal control”, Proc. Steklov Inst. Math., 169 (1986), 199–259 | Zbl | MR | Zbl

[18] Yu. G. Borisovich, B. D. Gelman, A. D. Myshkis, V. V. Obukhovsky, Introduction to the Theory of Multivalued Mappings and Differential Inclusions, 2nd ed., Book House “LIBROKOM”, Moscow, 2016 | MR

[19] A. I. Bulgakov, O. V. Filippova, “The functional differential inclusions with impulses and with the right-hand side not necessarily convex-valued with respect to switching”, Izv. IMI UdGU, 2014, no. 1(43), 3–48 (In Russian) | Zbl

[20] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, 5-e izd., Nauka, M., 2007 ; A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, v. I, II, Dover Publications, Mineola, New York, 1957, 1961 | MR | MR

[21] A. V. Arutyunov, Lectures on Convex and Multivalued Analysis, FIZMATLIT Publ., Moscow, 2014 (In Russian)

[22] A. I. Bulgakov, E. V. Korchagina, O. V. Filippova, “Functional-differential inclusions with impulses. Part I–VI”, Vestnik Tambovskogo universiteta. Seriya: estestvennye i tekhnicheskie nauki = Tambov University Reports. Series: Natural and Technical Sciences, 14:6 (2009), 1275–1313 (In Russian) | DOI