An output sensitivity problem for a class of linear distributed systems with uncertain initial state

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 C Rⁿ, the initial state x₀ ϵ L²(Ω) is supposed known on one part of the domain Ω and uncertain on the rest. That means Ω = ω₁ U ω₂ U... U ω_t with ω_i ∩ ω_j = ∅, ∀_i ≠ j ϵ {1,...,t}, i ≠ j where ω_i ≠ ∅ and x₀(θ) = α_i for θ ϵ ω_i, ∀_i, i.e., x₀(θ) = [wzór] (θ) where the values α₁,...,α_r are supposed known and α_r+1,...,α_t unknown and 1ω_i is the indicator function. The uncertain part (α₁,...,(α)_rof the initial state x₀ is said to be (ɛ₁,...,ɛ_r )-admissible if the sensitivity of corresponding output signal (y_i)_i≥0 relatively to uncertainties (α_k)_1≤k≤r is less to the treshold ɛ_k, i.e., ∥∂y_i)/(∂α_k∥ ≤ ɛ_k, ∀_i≥ 0, ∀_k ϵ {1,...,r]. 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 characterization of this set is investigated and an algorithmic determination of each gain operators is presented. Some examples are given.