Yuri F. Dolgii     (Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russian Federation)


For optimal stabilization of an autonomous linear system of differential equations with aftereffect and impulse controls, the formulation of the problem in the functional state space is used. For a system with aftereffect, approximating systems of ordinary differential equations proposed by S.N. Shimanov and J. Hale are used. A method for constructing approximations for optimal stabilizing control of an autonomous linear system with aftereffect and impulse controls is proposed. Matrix Riccati equations are used to find approximating controls.


Differential equation with aftereffect, Canonical approximation, Optimal stabilization, Impulse control

Full Text:



  1. Andreeva I.Yu., Sesekin A.N. An impulse linear-quadratic optimization problem in systems with after effect. Russian Math. (Iz. VUZ), 1995. Vol. 39, No. 10. P. 8–12.
  2. Bykov D.S., Dolgii Yu.F. Error estimate for approximations of an optimal stabilizing control in a delay system. Trudy Inst. Mat. i Mekh. UrO RAN, 2012. Vol. 18, No. 2. P. 38–47. (in Russian)
  3. Bykov D.S., Dolgii Yu.F. Canonical approximations in task of optimal stabilization of autonomous systems with aftereffect. Trudy Inst. Mat. i Mekh. UrO RAN, 2011. Vol. 17, No. 2. P. 20–34. (in Russian)
  4. 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. P. 48–88. DOI: 10.1137/0313004
  5. Dolgii Yu.F. Stabilization of linear autonomous systems of differential equations with distributed delay. Autom. Remote Control, 2007. Vol. 68, No. 1. P. 1813–1825. DOI: 10.1134/S0005117907100098
  6. Dolgii Yu.F., Sesekin A.N. Regularization analysis of a degenerate problem of impulsive stabilization for a system with time delay. Trudy Inst. Mat. i Mekh. UrO RAN, 2022. Vol. 28, No. 1. P. 74–95. DOI: 10.21538/0134-4889-2022-28-1-74-95 (in Russian)
  7. Dolgii Y. Approximation of the problem of optimal impulse stabilization for an autonomous linear system with delay. In: Proc. 2022 16th Int. Conf. ”Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy’s Conference 2022). V.N. Tkhai (ed.). IEEE Xplore, 2022. P. 1–4. DOI: 10.1109/STAB54858.2022.9807487
  8. 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. 1. P. 95–139. DOI: 10.1137/0321006
  9. Hale J.K. Theory of Functional Differential Equations. NY: Springer-Verlag, 1977. 366 p. DOI: 10.1007/978-1-4612-9892-2
  10. Kolmanovskiĭ V.B., Shaĭkhet L.E. Control of Systems with Aftereffect. Ser. Monogr. Math., vol. 157. Providence, R.I.: American Mathematical Society, 1996. 336 p.
  11. Krasovskii N.N. On the analytic construction of an optimal control in a system with time lags. J. Appl. Mat. Mech., 1962. Vol. 26, No. 1. P. 50–67. DOI: 10.1016/0021-8928(62)90101-6
  12. 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. P. 876–885. (in Russian)
  13. Krasovskii N.N., Osipov Yu.S. On stabilization of a controlled object with a delay in a control system. Izv. Akad. Nauk SSSR, Tekh. Kibern., 1963. No. 6. P. 3–15. (in Russian)
  14. Krasovskii N.N. Stability of Motions. Stanford: University Press, 1963. 212 p.
  15. Lukoyanov N.Yu., Plaksin A.R. Finite-dimensional modeling guides in time-delay systems. Trudy Inst. Mat. i Mekh. UrO RAN, 2013. Vol. 19, No. 1. P. 182–195. (in Russian)
  16. Markushin E.M., Shimanov S.N. Approximate solution of the problem of an analytic construction of a regulator for an equation with retardation. Differ. Uravn., 1966. Vol. 2, No. 8. P. 1018–1026. (in Russian)
  17. Osipov Yu.S. On stabilization of control systems with delay. Differ. Uravn., 1965. Vol. 1, No. 5. P. 605–618. (in Russian)
  18. Pandolfi L. Stabilization of neutral functional differential equations. J. Optim. Theory Appl., 1976. Vol. 20, No. 2. P. 191–204. DOI: 10.1007/BF01767451
  19. Pandolfi L. Canonical realizations of systems with delays. SIAM J. Control Optim., 1983. Vol. 21, No. 4. P. 598–613. DOI: 10.1137/0321036
  20. Shimanov S.N. On the theory of linear differential equations wits after-effect. Differ. Uravn., 1965. Vol. 1, No. 1. P. 102–116. (in Russian)
  21. Zhelonkina N.I., Lozhnikov A.B., Sesekin A.N. On pulse optimal control of linear systems with after-effect. Autom. Remote Control, 2013. Vol. 74, No. 11. P. 1802–1809. DOI: 10.1134/S0005117913110039


Article Metrics

Metrics Loading ...


  • There are currently no refbacks.