Abstract
Approximations of the image and integral funnel of a closed ball of the space \(L_p\), \(p>1\), under a Urysohn-type integral operator are considered. A closed ball of the space \(L_p\), \(p>1\), is replaced by a set consisting of a finite number of piecewise constant functions, and it is proved that, for appropriate discretization parameters, the images of these piecewise constant functions form an internal approximation of the image of the closed ball. This result is applied to approximate the integral funnel of a closed ball of the space \(L_p\), \(p>1\), under a Urysohn-type integral operator by a set consisting of a finite number of points.
References
I. Gohberg and S. Goldberg, Basic Operator Theory, Birkhäuser, Boston, 1981.
M. A. Krasnosel’skii and S. G. Krein, On the principle of averaging in nonlinear mechanics, vol. 10, 1955.
M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustylnik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff International Publishing, Leyden, 1976.
L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
V. I. Blagodatskikh and A. F. Filippov, “Differential inclusions and optimal control. Topology, ordinary differential equations, dynamical systems”, Trudy Mat. Inst. Steklov, 169 (1985), 194–252; English transl.: Proc. Steklov Inst. Math., 169 (1986), 199–259.
N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, Springer, New York, 1988.
A. I. Panasyuk and V. I. Panasyuk, “An equation generated by a differential inclusion”, Mat. Zametki, 27:3 (1980), 429–437; English transl.: Math. Notes, 27:3 (1980), 213–218.
A. A. Ershov, A. V. Ushakov, and V. N. Ushakov, “An approach problem for a control system with a compactum in the phase space in the presence of phase constraints”, Sb. Mat., 210:8 (2019), 1092–1128; English transl.: Sb. Math., 210:8 (2019), 1092–1128.
M. I. Gusev, “On the method of penalty functions for control systems with state constraints under integral constraints on the control”, Tr. Inst. Mat. Mekh. UrO RAN, 27:3 (2021), 59–70.
Kh. G. Guseinov and A. S. Nazlipinar, “An algorithm for approximate calculation of the attainable sets of the nonlinear control systems with integral constraint on controls”, Comput. Math. Appl., 62:4 (2011), 1887–1895.
A. B. Kurzhanskii and P. Varaiya, Dynamics and Control of Trajectory Tubes. Theory and Computation, Birkhäuser, Cham, 2014.
B. T. Polyak, “Convexity of the reachable set of nonlinear systems under \(L_2\) bounded controls”, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 11:2–3 (2004), 255–267.
V. V. Beletskii, Notes on the Motion of Celestial Bodies, Nauka, Moscow, 1972 (Russian).
N. N. Krasovskii, Theory of Control of Motion: Linear Systems, Nauka, Moscow, 1968 (Russian).
N. N. Subbotina and A. I. Subbotin, “Alternative for the encounter-evasion differential game with constraints on the momenta of the players controls”, Prikl. Mat. Mekh., 39:3 (1975), 397–406; English transl.: J. Appl. Math. Mech., 39:3 (1975), 376–385.
R. L. Wheeden and A. Zygmund, Measure and Integral. An Introduction to Real Analysis, M. Dekker, New York, 1977.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 43–58 https://doi.org/10.4213/faa3974.
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Huseyin, A., Huseyin, N. & Guseinov, K.G. Approximations of the Images and Integral Funnels of the \(L_p\) Balls under a Urysohn-Type Integral Operator. Funct Anal Its Appl 56, 269–281 (2022). https://doi.org/10.1134/S0016266322040050
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DOI: https://doi.org/10.1134/S0016266322040050