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

Abstract

We consider the heat equation with dynamic boundary conditions involving gradient terms in a bounded domain. In this paper we study the cost of approximate controllability for this equation. Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.

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.…”