Optimal control of the bidomain system (II): uniqueness and regularity theorems for weak solutions

Abstract

Φ tr (x, 0) = Φ 0 (x) and W (x, 0) = W 0 (x) for a. a. x ∈ Ω (1.10)arising as a special case of (1.1) − (1.6) if the conductivity tensors satisfy M e = λ M i with a constant parameter λ > 0, thus allowing to eliminate Φ e as an independent variable. In a series of papers, 02) the authors investigated optimal control problems related to the dynamics (1.1) − (1.6) and (1.7) − (1.10) together with standard two-variable ionic models, namely the Rogers-McCulloch, FitzHugh-Nagumo and the linearized Aliev-Panfilov model (see Subsection 2.a) below). Using I e as control variable while I i = o, 03) and relying on a weak solution concept for the monodomain as well as for the bidomain system, the authors studied existence of minimizers and derived first-order necessary optimality conditions. 01) The bidomain model has been considered first in [ Tung 78 ] . A detailed introduction may be found e. g. in [ Sundnes/ Lines/Cai/Nielsen/Mardal/Tveito 06 ] , pp. 21 − 56. 02) [ Kunisch/Wagner 12 ] , [ Kunisch/Wagner 13a ] and [ Kunisch/Wagner 13b ] . 03) This setting is due to physiological reasons. 11) [ Simon 87 ] , p. 90, Corollary 8. 12) [ FitzHugh 61 ] , together with [ Nagumo/Arimoto/Yoshizawa 62 ] . 13) [ Rogers/McCulloch 94 ] . 14) This model is taken from [ Bourgault/Coudière/Pierre 09 ] , p. 480. Instead, the original model from [ Aliev/ Panfilov 96 ] contains a Riccati equation for the gating variable. * , (2.62)

Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions

“…(together with p. 152), Theorem IV.4.11. [12], page 7, Theorem 2.6, we obtain uniform bounds with respect to N for the norms of Φ tr N , Φ e N , W N , ∂Φ tr N /∂t and ∂W N /∂t, implying weak convergence of ∂Φ tr N /∂t, ∇Φ tr N and ∂W N /∂t as well as a.e. pointwise convergence of Φ tr N on Ω T along a suitable subsequence.…”

“…The first one is a stability estimate for the primal equations (Thm. 2.4), whose proof has been already provided in the previous publication [12]. Secondly, we need an existence proof for weak solutions of the adjoint system, which is contained in the present paper (Thm.…”

“…Due to the uniform boundedness of I e N X4 with respect to N , the norms Φ tr N X1 , Φ e N X2 and W N X3 are uniformly bounded as well (cf. again [12], p. 7, Thm. 2.6), and we may pass to a subsequence { (Φ tr N , Φ e N , W N , I e N ) } , which converges to a feasible quadruple (Φ tr ,Φ e ,Ŵ ,Î e ) in the sense of Proposition 3.3.…”

“…Parabolic-elliptic form of the bidomain system; strong and weak solutions It is well-known that the bidomain system (1.1)-(1.6) can be equivalently stated in parabolic-elliptic form, cf. [4], page 459, and [12], page 4, (2.1)-(2.9). In its weak formulation, the system reads as follows:…”

“…After the study of the monodomain approximation of the equations and a thorough stability and regularity analysis of weak solutions for the full bidomain equations, as contained in the previous papers [11,12], we are now in position to analyze the related control problems with respect to the existence of minimizers as well as to provide a rigorous proof of the first-order necessary optimality conditions.…”