Local exact boundary controllability of the Navier-Stokes system

Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

“…We shall refer to this type of estimate as to a local Carleman estimate. Here, following Fursikov-Imanuvilov [FI96], the weight function we consider is chosen singular at time t = 0 and t = T , and of the form η(t)ϕ(x)/h, where ϕ(x) is negative and satisfies a sub-ellipticity condition, and η(t) = T 2 t(T −t) . At times t → 0 + and t → T − , the exponential of the weight function, e η(t)ϕ(x)/h , thus vanishes at all orders.…”

“…Carleman estimates for parabolic operators with smooth coefficients were proven in [FI96]. The proof is based on the construction of a suitable smooth weight function ϕ.…”

Remarks on the Controllability of Some Parabolic Equations and Systems

“…Some controllability properties of (3.9) have been studied in [14] (see Chapter 1, Theorems 6.3 and 6.4). There, it is shown that one cannot reach (not even approximately) stationary solutions of (3.9) with large L 2 (0, 1)-norm at any time T .…”

“…These have been introduced in the context of the controllability of PDEs by Fursikov and Imanuvilov, see [18,14]. When they are applied to the solutions of the adjoint systems (3.4), they take the form…”

“…In particular, the null controllability of linear PDEs was first analyzed in [25,26,22,23,18,21]. For semilinear systems, the first contributions have been given in [27,6,14].…”