The cost of approximate controllability of heat equation with general dynamical boundary conditions

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

“…Similar results for semilinear problems. In this section, following [6,9], and using the results obtained in the linear case with a fixed point argument, we deduce similar results for the following semilinear problem Since the method is standard, we only give the main ideas. In the semilinear framework, the corresponding functionals J 1 and J 2 are not convex in general.…”

“…We have also (L 2 , H 2 ) 1 2 ,2 = H 1 . The following existence and uniqueness results hold, see [9] for the proof.…”

“…We have the following Carleman estimates, see [1] and [9] for the proof. (ii) If the functions F and F Γ are not necessarily zero, then there exist λ 2 ≥ 1, s 2 ≥ 1 and C 2 = C 2 (ω, Ω) > 0 such that the solution Q = (q, q Γ ) to (27) satisfies…”

“…This type of boundary conditions has been considered in the context of null controllability, when only one control is acting on the system, in [31,37]. The cost of approximate controllability and an inverse source problem of such equations were also studied in [9] and [1], respectively. For some controllability results of hyperbolic equation with dynamic boundary condition, the reader can see [22].…”

Impulse null approximate controllability for heat equation with dynamic boundary conditions

“…We mention the papers [15,32], which physically derive the dynamic boundary conditions. Furthermore, the controllability and inverse problems of heat equation with dynamic boundary conditions have recently been investigated in [2,3,7,19,25], where the authors have proven controllability and stability results by proving new Carleman estimates.…”