Bol. Soc. Paran. Mat. (3s.) v. 2024 (42) : 1–5. ©SPM –ISSN-2175-1188 on line ISSN-0037-8712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.64709 Sentinels of nth-Order Insensitivity for Identification Problems With High-Order Incomplete Data Abdelhak Hafdallah abstract: This study generalizes the definition of the sentinel function, which was introduced by J.L. Lions to study identification problems, to a more insensitive kind of sentinel, which is applied to identify pollution terms of the polynomial form w. r. t. the real λ. The main idea is to reconstruct the sentinel function to be nth-order independent of the incomplete data. Contrary to the original definition, information about the pollution term with an error of order n + 1 is given by the sentinel of nth-order insensitivity. Key Words:Sentinels, identification problem, incomplete data, null controllability. Contents 1 Introduction 1 2 Preliminaries 2 3 Existence of nth-order insensitive sentinel is equivalent to a null controllability prob- lem 3 4 Solving the controllability problem (6) − (7) 3 5 Information given by nth-order insensitive sentinel (pollution term detection) 4 1. Introduction Identifying missing terms of PDEs from given observations on the state was a very attractive part of the studies on inverse problems. In 1988, J. L. Lions have introduced a function called "sentinel" to treat some kind of identification problems with incomplete data, where he distinguishes between important one (pollution term) and the unimportant terms. The sentinel provides information about the pollution terms via a given adjoint state [6] and [7]. The study of several identification problems by the use of sentinel method has received much attention in the last two decades (see for instance [1], [2], [4,5,6], and [11]). Many theoretical and numerical results exist as well as applications of real physical problems motivated by researchers and industrialists, we can cite as an example the work [1,3,4] and [5]. Moreover, the method has applied to study several problems in ecology and meteorology. For example, we refer to [9] and [10] for the of problems of identification of pollution terms in distributed systems, the detection of pollution in an aquifer, the determination of missing parameters in a lake and the search for pollution in a river, also, sentinels are adapted the determination of pollution in environment. All those previous works have only focused on the missing terms of the form f 0 (x)+ λf (x) unlike the present paper is devoted to defining a more insensitive sentinel functional, this function is expected to provide information about the pollution term of the higher order form ∑ n k=1 λ k f k in contrast to the classic definition of sentinel [6], where only pollution terms of the following form λf have been considered. Technically, we prove that the construction of an nth-order insensitive sentinel, which is linked to a control function with a specific property, is equivalent to a null controllability problem for the adjoint state. 2010 Mathematics Subject Classification: 35R30, 49N30. Submitted November 03, 2022. Published March 06, 2023 1 Typeset by B S P M style. © Soc. Paran. de Mat.