GENERALIZED SENTINELS DEFINED VIA LEAST-SQUARES

We address the problem of monitoring a linear functional (c,x)E of an unknown vector x of a Hilbert space E, the available data being the ob servation z, in a Hilbert space F, of a vector Ax depending linearly o n x through some known operator A is-an-element-of L(E; F). When E = E 1 x E2, c = (c1, 0), and A is injective and defined through the soluti on of a partial differential equation, Lions ([6]-[8]) introduced sent inels s is-an-element-of F such that (s, Ax)F is sensitive to x1 is-an -element-of E1 but insensitive to x2 is-an-element-of E2. In this pape r we prove the existence, in the general case, of (i) a generalized se ntinel (s, sigma) is-an-element-of F x E, where F superset-of F with F dense in F, such that for any a priori guess x0 of x, we have <s, Ax> FF' + (sigma, x0)E = (c, x)E, where x is the least-square estimate of x closest to x0, and (ii) a family of regularized sentinels (s(n), sig ma(n)) is-an-element-of F x E which converge to (s, sigma). Generalize d sentinels unify the least-squares approach (by construction!) and th e sentinel approach (when A is injective), and provide a general frame work for the construction of ''sentinels with special sensitivity'' in the sense of Lions [8]).