Insensitizing controls for a phase field system

An insensitizing control problem for a linear Stabilized Kuramoto-Sivashinsky system

“…In our insensitizing problem we have used two controls, one for each component, but determining if we can reduce its number it is not so clear. As far as we know, there are only a few papers devoted to the insensitizing control problems for coupled systems, see [7,9,12]. Similar to our case, in those works, the original problem is transformed into a control problem for an extended system of four equations (two forward and two backward in time).…”

“…This is comparable to choosing α = 0 in our case (1.2). At a first glance, it seems that we can follow the ideas similar to [7] to eliminate the extra control, but in our case, the first order couplings make things difficult. Indeed, recalling our adjoint system (2.32) (i.e.…”

“…Similar to our case, in those works, the original problem is transformed into a control problem for an extended system of four equations (two forward and two backward in time). In particular, in [7], the authors have used only one control to prove their result by using a sentinel depending (explicitly) only on one of the components of the system. This is comparable to choosing α = 0 in our case (1.2).…”

“…Under the same assumptions of Theorem 1.2, it is possible to use similar strategies as in Sections 2 and 3 for studying the observability of the corresponding linearized (extended) adjoint system (cf. (1.8)) and then use a local inversion theorem as in [7,8] to conclude that if (ξ 1 , ξ 2 ) verify (4.3)…”

“…After that, many works have been devoted to study the insensitizing problem from different perspectives: in [3,4,5], the authors study such problem for linear and semilinear heat equations with different types of nonlinearities and/or boundary conditions, while in [21] the problem of insensitizing a sentinel depending on the gradient of the solution of a linear parabolic equation is addressed. For insensitizing problems of equations in fluid mechanics we refer to the works [9,12,13,22] and for a phase field system to [7]. Most recently, the insensitizing control problem has been addressed from a numerical point of view in [6], for fourth-order parabolic equations in [25] and with respect to shape variations in [18,28].…”