Abstract
We formulate sufficient conditions for the technical stability on given bounded and infinite time intervals and for the asymptotic technical stability of continuously controlled linear dynamical processes with distributed parameters. By using the comparison method and the method of Lagrange multipliers in combination with the Lyapunov direct method, we obtain criteria which define a set of controls providing the technical stability of the output process. We select the optimal control that realizes the least value of the norm corresponding to a given process.
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References
K. A. Abgaryan, “Stability of motion on a finitie time interval”, in: Itogi VINITI, Ser. Obshchaya Mekh. [in Russian], Vol. 3, VINITI, Moscow (1976), pp. 43–127.
F. D. Bairamov, “Guaranteeing the technical stability of controlled systems”, in: Problems of Analytical Mechanics, Stability, and Motion Control [in Russian], Nauka, Novosibirsk (1991), pp. 134–139.
A. G. Butkovskii, “Control over systems with distributed parameters”, Avtomat. Telemekh, No. 11, 16–65 (1979).
V. I. Zubov, Dynamics of Controlled Systems [in Russian], Vyshcha Shkola, Kiev (1982).
G. V. Kamenkov, “On stability on a finite time interval”, Prikl. Mat. Mekh., 17, Issue 5, 529–540 (1953).
N. F. Kirichenko, Introduction into the Theory of Stabilization of Motion [in Russian], Vyshcha Shkola, Kiev (1978).
A. M. Letov, Mathematical Theory of Stabilization of Motion [in Russian], Nauka, Moscow (1987).
J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod Gauthier-Villars, Paris (1968).
K. A. Lur'ye, “The Mayer-Boltz problem for multiple integrals and optimization of the behavior of systems with distributed parameters”, Prikl. Mat. Mekh., 27, Issue 5, 842–853 (1963).
V. M. Kuntsevich and M. M. Lychak, Synthesis of Systems of Automatic Control with the Use of Lyapunov Functions [in Russian], Nauka, Moscow (1977).
T. K. Sirazetdinov, Stability of Systems with Distributed Parameters [in Russian], Kazan' University, Kazan' (1971).
K. S. Matviichuk, “On the comparison method for differential equations close to hyperbolic ones”, Differents. Uravn., 20, No. 11, 2009–2011 (1984).
K. S. Matviichuk, “Technical stability of parametrically perturbed distributed processes”, Prikl. Mat. Mekh., 50, Issue 2, 210–218 (1986).
K. S. Matviichuk, “On the technical stability of a system of automatic control with varying structure”, Prikl. Mekh., 30, No. 10, 74–78 (1994).
J. Szarski, Differential Inequalities, PWN, Warszawa (1967).
N. A. Kil'chevskii, A Course of Theoretical Mechanics [in Russian], Vol. 1, Nauka, Moscow (1977).
Additional information
Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10. pp. 1337–1344, October, 1997.
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Matviichuk, K.S. On conditions of technical stability of controlled processes with distributed parameters. Ukr Math J 49, 1507–1515 (1997). https://doi.org/10.1007/BF02487437
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DOI: https://doi.org/10.1007/BF02487437