On Structured Distance to Uncontrollability of General Linear Retarded Systems

  • Nguyen Khoa SonEmail author
  • Nguyen Thi Hong


In this paper, we study the robustness of controllability in the state space \(M_{p}=\mathbb {K}^{n}\times L_{p}([-h,0],\mathbb {K}^{n}), 1<p<\infty , \) for retarded systems described by linear functional differential equations (FDE) of the form \( \dot x(t)=A_{0}x(t) + {\int }_{-h}^{0}d[\eta (\theta )]x(t+\theta )+B_{0}u(t), x(t)\in \mathbb {K}^{n}, u(t)\in \mathbb {K}^{m}, \mathbb {K}=\mathbb {C}\), or \(\mathbb {R}\). Some formulas for estimating and computing the distance to uncontrollability of a controllable FDE system are obtained under the assumption that the system’s matrices A0, η, B0 are subjected to structured perturbations. An example is provided to illustrate the obtained results.


Retarded systems Function space controllability Multi-valued linear operators Structured perturbations Distance to matrix non-surjectivity Controllability radius 

Mathematics Subject Classification (2010)

34K06 47A06 49A55 90C31 93B05 93B35 


Funding Information

This work is supported financially by the National Foundation for Science and Technology Development, NAFOSTED, under the research project 101.01-2017.20. The final version of the paper has been completed during the research stay of the authors from April to July, 2018, at the Vietnam Institute for Advanced Study in Mathematics, VIASM.


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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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