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Journal of Automation and Information Sciences
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Journal of Automation and Information Sciences

DOI: 10.1615/JAutomatInfScien.v51.i5.10
pages 1-15

On the Optimal Impulse Control in Descriptor Systems

Larisa A. Vlasenko
Kharkov National University of Radio and Electronics, Kharkov, Ukraine
Anatoliy G. Rutkas
Kharkov National University of Radio and Electronics, Kharkov
Valerii V. Semenets
Kharkiv National University of Radio Electronics, 14 Nauka Ave, Kharkiv 61166, Ukraine
Arkadiy A. Chikriy
V.M. Glushkov Institute of Cybernetics National Academy of Sciences of Ukraine 40 Akadem. Glushkov Ave., Kiev, 03187, Ukraine


We study the optimal impulse control problem with the quadratic performance functional for a descriptor system. The system evolution is described by a linear differential-algebraic equation unsolved with respect to the derivative of the state. The system is controlled by changing the measurable control and the pure impulse control. The pure impulse control is characterized by impulse intensities and moments of impulse applications. The main restriction is that the characteristic matrix pencil corresponding to the state equation is regular. In terms of characteristic matrix pencil we establish the conditions for the existence and uniqueness of the optimal control and the corresponding optimal state. The optimal control and the optimal state are constructed by using the adjoint state which is a solution of the adjoint two-point boundary value problem. The results are illustrated by the example of descriptor system that describes transient states in a radio technical filter. For this system we consider the energetic performance functional with impulse intensities characterizing the energy of inertial elements and input voltage of the filter and also intensities and moments of impulse applications. Transient states under impulsive perturbations of currents and voltages are described by using the formula of constants variation for the impulsive descriptor system.


  1. BensusanA., Lions G.L., Impulse control and quasi variational inequalities [Russian translation], Nauka, Moscow, 1987.

  2. Krasovskiy N.N., Motion control theory. Linear systems [in Russian], Nauka, Moscow, 1968.

  3. Butkovskiy A.G., Structural theory of distributed systems [in Russian], Nauka, Moscow, 1977.

  4. Chikrii A.A., Conflict-controlled processes, Springer Science and Business Media, Dordrecht, 2013, DOLIO. 1007/978-94-017-1135-7.

  5. Chikrii A.A., An analytical method in a dynamic pursuit games, Proceedings of the Steklov Institute of Mathematics, 2010, 271, No. 1, 69-85, DOI: 10.1134/S0081543810040073.

  6. Kryvonos Iu.G., Matychyn I.I., Chikrii A.A., Dynamic games with discontinuous trajectories [in Russian], Naukova dumka, Kiev, 2005.

  7. ChikriiA.A., Matichin I.I., Differential games with impulse control of players, Proceedings of the Steklov Institute of Mathematics (Suppl.), 2005, Suppl. 1, S68-S81, mathscinet-getitem?mr=2156250.

  8. ChikriiA.A., MatychynI.I., ChikriiK.A., Differential games with impulse control. Advances in dynamic game theory, Vol. 9. of Annals of the International Society of Dynamic Games, Birkhauser, Boston, 2007, 37-55, DOI: 10.1007/978-0-8176-4553-3 2.

  9. Matychyn I., Chikrii A., Onyshchenko V., Conflict-controlled processes involving fractional differential equations with impulses, Mathematica Balkanica, 2012, 26, No. 1-2, 159-168,

  10. Vlasenko L.A., Rutkas A.G., Stochastic impulse control of parabolic systems of Sobolev type, Differential Equations, 2011,47, No. 10, 1498-1507, DOI: 10.1134/S0012266111100132.

  11. Vlasenko L.A., Rutkas A.G., Optimal control of a class of random distributed Sobolev type systems with aftereffect, Journal of Automation and Information Sciences, 2013, 45, No. 9, 66-76, DOI: 10.1615/JAutomatInfScien.v45.i9.60.

  12. Dai L., Singular control systems, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989, DOI: 10.1007/Bfb0002475.

  13. KurinaG.A., MarzR., On linear quadratic optimal control problems for time-variables descriptor systems, SLAM J. Control Optim., 2004, 42, No. 6, 2062-2077, DOI: 10.1137/S0363012900380991.

  14. Duan G.R., Analysis and design of descriptor linear systems, Springer, New York, Dordrecht, Heidelberg, London, 2010, 10.1007/978-1-4419-6397-0.

  15. Vlasenko L.A., Rutkas A.G., Semenets V.V., Sequential composition and decomposition of descriptor control systems, Journal of Automation and Lnformation Sciences, 2018, 50, No. 9, 60-75, DOI: 10.1615/JAutomatInfScien.v50.i9.50.

  16. Vlasenko L.A., Rutkas A.G., On the optimal control of systems described by implicit evolution equations, Differential Equations, 2009, 45, No. 3, 429-438, DOI: 10.1134/S0012266109030124.

  17. Vlasenko L.A., Chikrii A.A., The method of resolving functional for a dynamic game in a Sobolev system, Journal of Automation and Lnformation Sciences, 2014, 46, No. 7, 1-11, DOI: 10.1615/JAutomatInfScien.v46.i7.10.

  18. Vlasenko L.A., Rutkas A.G., On a differential game in a system described by an implicit differential- operator equation, Differential Equations, 2015,51, No. 6, 798-807, DOI: 10.1134/S0012266115060117.

  19. Rozenfeld A.S., YakhinsonB.I., Transient processes and generalized functions [in Russian], Nauka, Moscow, 1966.

  20. Vlasenko L.A., PerestyukN.A., On the solvability of impulsive differential-algebraic equations, UkrainianMathematical Journal, 2005, 57, No. 4, 551-564, DOI: 10.1007/sll253-005-0209-4.

  21. Vlasenko L.A., Rutkas A.G., Samoilenko A.M., Problem of impulsive regulator for one dynamical system of the Sobolev type, Ukrainian Mathematical Journal, 2008, 60, No. 8, 1200-1209, DOI: 10.1007/sll253-009-0135-y.

  22. Vlasenko L.A., Samoilenko A.M., Optimal control with impulsive component for systems described by operator differential equations, Ukrainian Mathematical Journal, 2009, 61, No. 8, 1250-1263, DOI: 10.1007/sll253-010-0274-l.

  23. Lions J.-L., Controle optimal de systemes gouvernes par des equations aux derrivees partielles [Russian translation], Mir, Moscow, 1972.

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