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Optimal stabilization for differential systems with delays – Malkin’s approach. (English) Zbl 1414.93160
Summary: The paper considers a process controlled by a system of delayed differential equations. Under certain assumptions, a control function is determined such that the zero solution of the system is asymptotically stable and, for an arbitrary solution, the integral quality criterion with infinite upper limit exists and attains its minimum value in a given sense. To solve this problem, Malkin’s approach to ordinary differential systems is extended to delayed functional differential equations, and Lyapunov’s second method is applied. The results are illustrated by examples, and applied to some classes of delayed linear differential equations.

##### MSC:
 93D20 Asymptotic stability in control theory 93C23 Control/observation systems governed by functional-differential equations 49K40 Sensitivity, stability, well-posedness
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##### References:
 [1] Athans, M.; Falb, P. L., Optimal Control, An Introduction to the Theory and Its Applications, (2007), Dover Publications, Inc. [2] Banks, H. T.; Rosen, I. G., A spline based technique for computing riccati operators and feedback controls in regulator problems for delay equations, SIAM J. Sci. Stat. Comput., 5, 4, 830-855, (1984) · Zbl 0553.65047 [3] Basin, M.; Perez, J.; Martinez-Zuniga, R., Optimal filtering for nonlinear polynomial systems over linear observations with delay, Int. J. Innov. Comput. Inf. Control, 2, 4, 863-874, (2006) [4] Basin, M.; Rodriguez-Gonzalez, J.; Fridman, L., Optimal and robust control for linear state-delay systems, J. Frankl. Inst., 344, 9, 830-845, (2007) · Zbl 1119.49021 [5] Bekiaris-Liberis, N.; Krstic, M., Nonlinear control under nonconstant delays, Advances in Design and Control, (2013), SIAM · Zbl 1417.93006 [6] Bellman, R., Dynamic Programming, (2003), Dover Publications, Inc. · Zbl 1029.90076 [7] Bokov, G. V., Pontryagin’s maximum principle of optimal control problems with time delay, (russian), J. Math. Sci., 172, 623-634, (2011) · Zbl 1222.49028 [8] Diblík, J., Long-time behaviour of solutions of delayed-type linear differential equations, Electron. J. Qual. Theory Differ. Equ., 47, 1-23, (2018) · Zbl 1413.34210 [9] Diblík, J., Asymptotic representation of solutions of equation $$\dot{y}(t) = \beta(t) [y(t) - y(t - \tau(t))],$$, J. Math. Anal. Appl., 217, 200-215, (1998) · Zbl 0892.34067 [10] Dolgii, Y. F.; Koshkin, E. V., Use of finite-dimensional approximations in a problem of stabilization of periodic systems with aftereffect, (russian), Izv. Vyssh. Uchebn. Zaved. Mat., 1, 29-45, (2015) [11] Driver, R. D., Ordinary and Delay Differential Equations,, (1977), Springer-Verlag New York Inc · Zbl 0374.34001 [12] Eller, D. H.; Aggarwal, J. K.; Banks, H. T., Optimal control of linear time-delay systems, IEEE Trans. Automat. Control, AC14, 6, 678-687, (1969) [13] Fiagbedzi, Y. A.; Peartson, A. E., Feedback stabilization of linear autonomous time-lag systems, IEEE Trans. Automat. Control, 31, 9, 847-855, (1986) · Zbl 0601.93045 [14] Friedman, E., Introduction to time-delay systems, analysis and control, Systems & Control: Foundations & Applications, (2014), Birkhäauser [15] Gabasov, R.; Kirillova, F., The Qualitative Theory of Optimal Processes, (1976), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York, Basel · Zbl 0339.49002 [16] Gabasov, R.; Kirillova, F. M.; Prischepova, S. V., Optimal feedback control, Lectures Notes in Control and Information Sciences 207, (1995), Springer · Zbl 0885.93004 [17] Hajipour, M.; Jajarmi, A., Numerical solution of the state-delayed optimal control problems by a fast and accurate finite difference theta-method, Commun. Nonlinear Sci. Numer. Simul., 55, 265-279, (2017) [18] Hale, J. K.; Lunel, S. V., Introduction to Functional Differential Equations, (1993), Springer-Verlag · Zbl 0787.34002 [19] Jajarmi, A.; Hajipour, M., An efficient recursive shooting method for the optimal control of time-varying systems with state time-delay, Appl. Math. Model., 40, 4, 2756-2769, (2016) [20] Kim, A. V.; Ivanov, A. V., Systems with Delay, Analysis, Control, and Computations, (2015), Scrivener Publishing, Wiley · Zbl 1331.34002 [21] Kirk, D. E., Optimal Control Theory, An Introduction, (2004), Dover Publications, Inc. [22] Krasovskii, N. N., Stability of Motion, Application of Lyapunov’s Second Method to Differential Systems and Equations with Delay, translated by J.L. Brenner, (1963), Standford University Press · Zbl 0109.06001 [23] Kwon, W. H.; Lee, Y. S.; Han, S. H., General receding horizon control for linear time-delay systems, Automatica J. IFAC, 40, 9, 1603-1611, (2004) · Zbl 1055.93032 [24] Macki, J.; Strauss, A., Introduction to Optimal Control Theory, corrected second printing, Undergraduate Texts in Mathematics, (1995), Springer-Verlag [25] Malkin, I. G., Theory of Stability of Motion, Second revised edition (Russian),, (1966), Nauka Publisher: Nauka Publisher Moscow · Zbl 0136.08502 [26] Malkin, I. G., Theory of stability of motion, translated from a publication of the state publishing house of technical-theoretical literature, moscow-leningrad,, United State Atomic Energy Commission, 455, (1952), Office of Technical Information, Translation Series [27] Mazurov, V. M.; Malov, D. I.; Salomykov, V. I., Systems of control of PH value in absorbtional column with recycle, Himich. Promysh., 4, 63-65, (1974) [28] Michiels, W.; Niculescu, S. I., Stability and stabilization of time-delay systems, an eigenvalue-based approach, Advances in Design and Control, (2007), SIAM: SIAM Philadelphia [29] Park, J. H.; Yoo, H. W.; Han, S.; Kwon, W. H., Receding horizon controls for input-delayed systems, IEEE Trans. Automat. Control, 53, 7, 1746-1752, (2008) · Zbl 1367.93282 [30] Pedregal, P., Introduction to optimization, Texts in Applied Mathematics, (2004), Springer · Zbl 1036.90002 [31] Pontryagin, L. S.; Boltyanskii, V. G.; Gamrelidze, R. V.; Mishchenko, E. F., The Mathematical Theory of Optimal Processes, (1962), Interscience · Zbl 0102.32001 [32] Yanushevsky, R. T., Control of objects with lag, (russian), Series in Theoretical Foundations of Engineering Cybernetics, 416, (1978), Nauka: Nauka Moscow [33] Yi, S.; Nelson, P. W.; Ulsoy, A. G., Analysis and Control Using the Lambert W Function, (2010), World Scientific · Zbl 1269.93036 [34] Zabczyk, J., Mathematical control theory, an introduction, Modern Birkhäuser Classic, (2008), Birkhäuser · Zbl 1123.93003
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