Approximate analytic solution of composed linear descriptor operator system using functional analysis approach

Document Type : Research Paper


Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq


This paper focuses on the solvability and approximate analytic solution of composed linear descriptor operator with constant coefficient, using functional (Variational) approach. This approach is based on finding a suitable functional form whose critical points are the solution of the proposed problem and the solution of a proposed problem is a critical point of the obtained variational functional defined on suitable reflexive Hilbert space since the existence of this approach is based on the symmetry and positivity of the composed linear descriptor operator on Hilbert space. The necessary mathematical requirements are derived and proved. A step-by-step computational algorithm is proposed. Illustration and computation with the giving exact solution are also proposed.


[1] J. Bals, G. Hofer, A. Pfeiffer, and C. Schallert, Virtual Iron Bird: A Multidisciplinary Modelling and Simulation Platform for new Aircraft System Architectures, In Deutscher Luftund Raumfahrtkongress, Friedrichshafen, Germany, 2005.
[2] A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, Springer-Verlag New York Inc., 2003.
[3] S.L. Campbell, P. Kunkel and V. Mehrmann, Regularization of Linear and Nonlinear Descriptor Systems, In L. T. Biegler, S. L. Campbell, and V. Mehrmann, editors, Control and Optimization with Differential-Algebraic Constraints, Advances in Control and Design, SIAM Publications, 2012.
[4] S.L. Campbell, Linearization of DAEs along trajectories, Z. Angew. Math. Phys. 46 (1995) 70–84.
[5] S.L. Campbell, C.D. Meyer and N.J. Rose, Application of the drazin inverse to linear systems of differential equations with singular constant coefficients, SIMA J. Appl. Math. 31 (1976) 411–425.
[6] V.F. Chistyakov, Selected chapters of the theory of algebraic–differential systems, in Russian, Nauka, Moscow, 2003.
[7] L. Dai, Singular Control Systems, Lecture Notes in Control and Information Sciences, Springer Berlin., 1989.
[8] G. Denk, R. Winkler, Modelling and simulation of transient noise in circuit simulation, Int. J. Math. Comput. Model. Dyn. Syst. 13 (2007) 383–394.
[9] E. Hairer and G. Wanner,Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems, Springer Series in Computational Mathematics, 2004.
[10] M. Hiller and K. Hirsch, Multibody system dynamics and mechatronics, Z. Angew. Math. Mech. 86 (2006) 87–109.
[11] A. Ilchmann and T. Reis, Surveys in differential-algebraic equations III, Springer-Verlag New York Inc., 2017.
[12] P. Kunkel and V. Mehrmann. Differential-Algebraic Equations. Analysis and Numerical Solution, EMS Publishing House, Z¨urich, Switzerland, 2006.
[13] P. Kunkel and V. Mehrmann, Canonical forms for linear differential-algebraic equations with variable coefficients, J. Comput. Appl. Math. 56 (1994) 225—259.
[14] R. Lamour, R. M¨arz and C. Tischendorf, Differential-algebraic equations: a projector based analysis, Springer Science & Business Media, 2013.
[15] E. L. Lewis, A Survey of Linear Singular System, Circuits Proc. Int. Symp. Singular System, Atlanta, 1987.
[16] F. Magri, Variational formulation for every linear equation, Int. J. Eng. Sci. 12 (1974) 537–549.
[17] R. Marz, Numerical methods for differential-algebraic equations, Acta Numerica 1 (1992) 141—198.
[18] R. Marz, Differential-algebraic equation from a functional-analytic viewpoint, A Survey., 2013.
[19] V.K. Mishra and N.K. Tomar, On complete and strong controllability for rectangular descriptor systems, Circuits Syst. Signal Proc. 35 (2016) 1395—1406.
[20] V.K. Mishra, N.K. Tomar and M.K. Gupta, On controllability and normalizability for linear descriptor systems, J. Control Aut. Elect. Syst. 27 (2016) 19-–28.
[21] P. Rabier and W. Rheinboldt, Techniques of Scientific Computing (part 4)- Theoretical and Numerical Analysis of Differential–Algebraic Equations, Handbook of Numerical Analysis, VIII, pp. 183–540. North Holland/Elsevier, Amsterdam, 2002.
[22] R.K. Rektorys, Variational Methods in Mathematics, Science and Engineering, Reidel Pub. Company, London,1980.
[23] G. Ren Duan, Analysis and Design of Descriptor Linear System, Springer, New York Dordrecht Heidelberg London, 2010.
[24] E. Tonti, Variational formulation of nonlinear differential equations, Bull. Acad. Roy. Belgique, 5 Serie, 137–165 (first part); 262-278 (second part), 1969.
[25] E. Zeidler, Applied Functional Analysis, Main Principles and Their Applications, Applied Mathematical Sciences 109, Springer-Verlag New York Inc., 1995.
[26] S.M. Zhuk, Closedness and normal solvability of an operator generated by a degenerate linear differential equation.
Volume 13, Issue 1
March 2022
Pages 365-378
  • Receive Date: 02 March 2021
  • Revise Date: 14 April 2021
  • Accept Date: 18 May 2021