## Abstract

In this paper,we consider an infinite dimensional linear systems. It is assumed that the initial state of system is not known throughout all the domain Ω ⊂ ℝ^{n}, the initial state x_{0} ϵ L^{2}(Ω) is supposed known on one part of the domain Ω and uncertain on the rest. That means Ω = ω_{1} [ ω_{2} [ ⋯ [ ω_{t} with ω_{i} \ ω_{j} = Ø, ∀i, j ϵ {1, ⋯ ,t}, i ≠ j where ωi ≠ Ø and x_{0}(θ) = αi for θ 2 ωi, ∀i, i.e., x_{0}(θ) = Σ^{t} _{i=1} αi1ωi (θ) where the values α1, ⋯, αr are sup- posed known and αr+1; ⋯ ; αt unknown and 1ωi is the indicator function. The uncertain part (α1; ⋯ ; αr ) of the initial state x_{0} is said to be (ϵ_{1}, ⋯ , ϵ_{r} )-admissible if the sensitivity of corresponding output signal (yi )_{i≥0} relatively to uncertainties (α_{k} )1≤k≤r is less to the treshold (equation presented) The main goal of this paper is to determine the set of all possible gain operators that makes the system insensitive to all uncertainties. The char- acterization of this set is investigated and an algorithmic determination of each gain operators is presented. Some examples are given.

Original language | English |
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Pages (from-to) | 139-155 |

Number of pages | 17 |

Journal | Archives of Control Sciences |

Volume | 30 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

- Distributed system
- Gain operators
- Linear programming
- Linear system
- Ob-servability
- Stability
- Uncertain initial state

## ASJC Scopus subject areas

- Control and Systems Engineering
- Modelling and Simulation
- Control and Optimization