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Controllability of Differential-Algebraic Equations in the Class of Impulse Effects

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Abstract

Considering a control linear system of differential-algebraic equations with infinitely differentiable coefficients we establish the existence of solutions in the class of Sobolev–Schwartz distributions. The solution is expressed as the sum of a regular generalized function and a singular generalized function. We study controllability with a jump of a regular component and a singular component of the solution.

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References

  1. Cobb D., “On the solution of linear differential equations with singular coefficients,” J. Differ. Equ., vol. 46, 310–323 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  2. Verghese G. C., Levy B., and Kailath T., “A generalized state-space for singular systems,” IEEE Trans. Autom. Control, vol. AC-26, no. 4, 811–831 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  3. Geerts T., “Solvability condition, consistency and weak consistency for linear differential-algebraic equations and time-invariant systems: the general case,” Linear Algebra Appl., vol. 181, 111–130 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  4. Gantmakher F. R., The Theory of Matrices, Amer. Math. Soc., AMS Chelsea Publishing (2000).

    MATH  Google Scholar 

  5. Campbell S. L. and Petzold L. R., “Canonical forms and solvable singular systems of differential equations,” SIAM J. Alg. Disc. Meth., no. 4, 517–521 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  6. Cobb D., “State feedback impulse elimination for singular systems over a Hermite domain,” SIAM J. Control Optim., vol. 44, no. 6, 2189–2209 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  7. Shcheglova A. A., “On generalized solutions of linear algebraic-differential systems,” Russian Math. (Iz. VUZ), no. 4, 64–76 (2006).

    MATH  Google Scholar 

  8. Cobb D., “Controllability, observability, and duality in singular systems,” IEEE Trans. Autom. Control, vol. AC-29, no. 12, 1076–1082 (1984).

    Article  MathSciNet  Google Scholar 

  9. Dai L., Singular Control Systems, Springer-Verlag, Berlin, Heidelberg, and New York (1989) (Lect. Notes Control Inform. Sci.; vol. 118).

  10. Armentano V. A., “The pencil (sE-A) and controllability-observability for generalized linear systems: a geometric approach,” SIAM J. Control Optim., vol. 24, 616–638 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  11. Wang C.-J., “State feedback impulse elimination of linear time-varying singular systems,” Automatica, vol. 32, no. 1, 133–136 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  12. Gařshun I. V., Introduction to the Theory of Nonstationary Linear Systems [Russian], Izdat. Inst. Mat. NAN Belarus, Minsk (1999).

    Google Scholar 

  13. Shcheglova A. A., “The solvability of the initial problem for a degenerate linear hybrid system with variable coefficients,” Russian Math. (Iz. VUZ), vol. 54, no. 9, 49–62 (2010).

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to A. A. Shcheglova.

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Original Russian Text Copyright © 2018 Shcheglova A.A.

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 59, No. 1, pp. 210–224, January–February, 2018; DOI: 10.17377/smzh.2018.59.118

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Shcheglova, A.A. Controllability of Differential-Algebraic Equations in the Class of Impulse Effects. Sib Math J 59, 166–178 (2018). https://doi.org/10.1134/S0037446618010184

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  • DOI: https://doi.org/10.1134/S0037446618010184

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