Yu.F. Dolgii, A.N. Sesekin. A study of regularization for a degenerate problem of impulsive stabilization in a system with aftereffect ... P. 80-99

A degenerate problem of stabilization of a linear autonomous system of differential equations with aftereffect and impulse controls is considered. For its regularization, a non-degenerate criterion for the quality of transient processes is used, which is close to a degenerate one. The regularized stabilization problem for impulse controls is replaced by an auxiliary non-degenerate optimal stabilization problem for non-impulse controls containing aftereffect. Bellman’s dynamic programming principle is used to solve the auxiliary problem. When finding the governing system of equations for the coefficients of the quadratic Bellman functional, the formulation of the optimal stabilization problem in the functional spaces of states and controls is used. A representation is obtained for the pulse of the optimal stabilizing control. The difficult problem of finding a solution to the governing system of equations for the Bellman functional is replaced by the problem of finding a solution to the governing system of equations for the coefficients of the representation of the optimal stabilizing control. The latter problem has lower dimension. The asymptotic dependence of the optimal stabilizing control on the regularization parameter is found when the dimension of the control vector coincides with the dimension of the state vector.

Keywords: linear autonomous system, aftereffect, optimal stabilization, impulse control

Received August 24, 2023

Revised October 17, 2023

Accepted October 30, 2023

Yuriy Filippovich Dolgii, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekateriburg, 620000, Russia, e-mail: yurii.dolgii@imm.uran.ru

Alexander Nikolaevich Sesekin, Dr. Phys.-Math. Sci., Prof., Ural Federal University, Yekaterinburg, 620000 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: a.n.sesekin@urfu.ru


1.   Krasovskii N.N. On the analytic construction of an optimal control in a system with time lags. J. Appl. Math. Mech., 1962, vol. 26, no. 1, pp. 50–67. doi: 10.1016/0021-8928(62)90101-6

2.   Krasovskii N.N. Problems of stabilization of controlled motions. In: Malkin I.G., Teoriya ustoichivosti dvizhenii [Theory of stability of motions], Moscow, Nauka Publ., 1966, 532 p.

3.   Hale J.K. Theory of functional differential equations. NY: Springer, 1977, 366 p. doi: 10.1007/978-1-4612-9892-2 . Translated to Russian under the title Teoriya funktsional’no-differentsial’nykh uravnenii. Moscow: Mir Publ., 1984, 421 p.

4.   Dolgii Yu.F., Surkov P.G. Matematicheskie modeli dinamicheskikh sistem s zapazdyvaniem [Matematical models of dynamic systems with delay]. Yekaterinburg, Ural State Univ. Publ., 2012, 122 p. ISBN: 978-5-7996-0772-2 .

5.   Delfour M.C., McCalla C., Mitter S.K. Stability and the infinite-time quadratic cost problem for linear hereditary differential systems. SIAM J. Control, 1975, vol. 13, no. 1, pp. 48–88. doi: 10.1137/0313004

6.   Gibson J.S. Linear-quadratic optimal control of hereditary differential systems: infinite dimensional Riccati equations and numerical approximations. SIAM J. Control Optim., 1983, vol. 21, no. 5, pp. 95–135. doi: 10.1137/0321006

7.   Fiagbedzi Y.A., Pearson A.E. Feedback stabilization of linear autonomous time lag system. IEEE Trans. Automat. Control., 1986, vol. 31, pp. 847–855. doi: 10.1109/TAC.1986.1104417

8.   Khartovskii V.E. A generalization of the problem of complete controllability for differential systems with commensurable delays. J. Computer and Systems Sciences International, 2009, vol. 48, no.  6, pp. 847–855. doi: 10.1134/S106423070906001X

9.   Bensoussan A., Da Prato G., Delfour M.C., Mitter S.K. Representation and control of infinite dimensional systems. Boston; Basel; Berlin: Bikhauser, 2007, 575 p.

10.   Wang G., Xu Y. Periodic feedback stabilization for linear periodic evolution equations. NY; Heidelberg; Dordrecht; London: Springer, 2016. 127 p.

11.   Pandolfi L. Stabilization of neutral functional differential equations. J. Optimization Theory Appl., 1976, vol. 20, no. 2, pp. 191–204. doi: 10.1007/BF01767451

12.   Yanushevsky R.T. Optimal control of linear differential-difference systems of neutral type. Int. J. Control., 1989, vol. 49, no. 6, pp. 1835–1850.

13.   Rabah R., Sklyar G.M., Rezounenko A.V. On strong regular stabilizability of linear neutral type systems. J. Diff. Eq., 2008, vol. 245, no. 3, pp. 569–593. doi: 10.1016/j.jde.2008.02.041

14.   Andreeva E.A., Kolmanovskii V.B., Shaikhet L.E. Upravlenie sistemami s posledeistviem [Control of systems with aftereffect], Moscow, Nauka Publ., 1992, 336 p. ISBN: 5-02-014875-Х .

15.   Krasovskii N.N. The approximation of a problem of analytic design of controls in a system with time-lag. J. Appl. Math. Mech., 1964, vol. 28, no. 4, pp. 876–885. doi: 10.1016/0021-8928(64)90073-5

16.   Krasovskii N.N., Osipov Yu.S. On the stabilization of motions of a plant with delay in a control system. Izv. Akad. Nauk SSSR, Tekh. Kibern., 1963, no. 6, pp. 3–15 (in Russian).

17.   Osipov Yu.S. Stabilization of control systems with delays. Differ. Uravn., 1965, vol. 1, no. 5, pp. 605–618 (in Russian).

18.   Markushin Ye.M., Shimanov S.N. Approximate solution of problem of designing regulator for time-lag systems. Avtomatika i Telemekhanika, 1968, no.  3, pp. 13–20 (in Russian).

19.   Dolgii Yu., Sesekin A. Optimal pulse stabilization of autonomous linear systems of differential equations with aftereffect. In: Proc. Int. Conf. 15th International Conference of stability and oscillations of nonlinear control systems (Pyatnitskiy’s Conference) (STAB 2020). IEEE, 2020. 4 p. doi: 10.1109/STAB49150.2020.9140479

20.   Andreeva I.Yu., Sesekin A.N. An impulse linear-quadratic optimization problem in systems with aftereffect. Russian Math. (Iz. VUZ), 1995, vol. 39, no. 10, pp. 8–12.

21.   Zhelonkina N.I., Lozhnikov A.B., Sesekin A.N. On pulse optimal control of linear systems with aftereffect. Autom. Remote Control, 2013, vol. 74, no. 11, pp. 1802–1809. doi: 10.1134/S0005117913110039

22.   Dmitriev M.G., Kurina G.A. Singular perturbations in control problems. Autom. Remote Control, 2006, vol. 67, no. 1, pp. 1–43. doi: 10.1134/S0005117906010012

23.   Dolgii Yu.F., Sesekin A.N. Regularization analysis of a degenerate problem of impulsive stabilization for a system with time delay. Trudy Inst. Math. Mekh. UrO RAN, 2022, vol. 28, no. 1, pp. 74–95 (in Russian). doi: 10.21538/0134-4889-2022-28-1-74-95

24.   Zavalishchin S.T., Sesekin A.N. Pulse-gliding regimes in nonlinear dynamical systems. Differ. Uravn., 1983, vol. 19, no. 5, pp. 790–799 (in Russian).

Cite this article as: Yu.F. Dolgii, A.N. Sesekin. A study of regularization for a degenerate problem of impulsive stabilization in a system with aftereffect. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 80–99.