On the Stability and Controllability of Degenerate Differential Systems in Hilbert Spaces

Main Article Content

Norelhouda Beghersa, Fares Yazid


We apply the famous theorem of Lyapunov for some degenerate differential systems taken the form Ax’(t) + Bx(t) = Φ(t, x(t)), where t∈R+ and A, B are linear bounded operators in Hilbert spaces, Φ is a given function. The obtained results are used to study the stabilizability and controllability of certain implicit controlled systems.

Article Details


  1. A.G. Rutkas, Spectral Methods for Studying Degenerate Differential-Operator Equations. I, J. Math. Sci. 144 (2007), 4246-4263. https://doi.org/10.1007/s10958-007-0267-2.
  2. J.L. Doletski, M.G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc. 2002.
  3. M. Benabdallah, M. Hariri, On The Stability of The Quasi-Linear Implicit Equations in Hilbert Spaces, Khayyam J. Math. 5 (2019), 105-112. https://doi.org/10.22034/kjm.2019.81222.
  4. M. Benabdallakh, A.G. Rutkas, A.A. Solov’ev, On the Stability of Degenerate Difference Systems in Banach Spaces, J. Soviet Math. 57 (1991), 3435-3439. https://doi.org/10.1007/bf01880215.
  5. A. Favini, L. Vlasenko, Degenerate Non-Stationary Differential Equations With Delay in Banach Spaces, J. Differ. Equ. 192 (2003), 93-110. https://doi.org/10.1016/s0022-0396(03)00090-1.
  6. F.R. Gantmacher, The Theory of Matrices, Vol. 1 and Vol. 2, Chelsea Publishing Company, New York, 1959.
  7. L. Baghdadi, R. Rabah, A Note on the Stabilization of Linear Systems in Hilbert Spaces, Demonstr. Math. 21 (1988), 631-641. https://doi.org/10.1515/dema-1988-0307.
  8. M. Slemrod, A Note on Complete Controllability and Stabilizability for Linear Control Systems in Hilbert Space, SIAM J. Control. 12 (1974), 500-508. https://doi.org/10.1137/0312038.
  9. W.M. Wonham, Linear Multivariable Control: A Geometric Approach, 3rd edition., Springer, New York, 1985.