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 (cid:26) R n , the initial state x 0 2 L 2 ( Ω ) is supposed known on one part of the domain Ω and uncertain on the rest. That means Ω = ! 1 [ ! 2 [ : : : [ ! t with ! i \ ! j = ∅ , 8 i ; j 2 f 1 ; : : :; t g , i , j where ! i , ∅ and x 0 ( (cid:18) ) = (cid:11) i for (cid:18) 2 ! i , 8 i , i.e., x 0 ( (cid:18) ) = t ∑ i = 1 (cid:11) i 1 ! i ( (cid:18) ) where the values (cid:11) 1 ; : : :; (cid:11) r are supposed known and (cid:11) r + 1 ; : : :; (cid:11) t unknown and 1 ! i is the indicator function. The uncertain part ( (cid:11) 1 ; : : :; (cid:11) r ) of the initial state x 0 is said to be ( " 1 ; : : :; " r ) -admissible if the sensitivity of corresponding output signal ( y i ) i 0 relatively to uncertainties ( (cid:11) k ) 1 ¬ k ¬ r is less to the treshold " k , i.e., (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) @ y i @(cid:11) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ¬ " k , 8 i 0, 8 k 2 f 1 ; : : :; r g . 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.