On Structured Distance to Uncontrollability of General Linear Retarded Systems

Abstract

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.

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Funding

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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Correspondence to Nguyen Khoa Son.

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Son, N.K., Hong, N.T. On Structured Distance to Uncontrollability of General Linear Retarded Systems. Acta Math Vietnam 45, 411–433 (2020). https://doi.org/10.1007/s40306-019-00337-2

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Keywords

  • 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