Abstract
Motivated by topics and issues critical to human health, the problem studied in this work derives from the modeling and stabilizing control of electrical cardiac activity in order to maximize the efficiency and safety of treatment for cardiac disease. In this paper we consider nonlinear minimax control problems constrained by an uncertain modified bidomain model of cardiac tissue electrophysiology system, in order to take into account the influence of noises in data and time-delays in signal transmission. The state system is a degenerate nonlinear coupled system of reaction-diffusion equations in the shape of a set of delay differential equations coupled with a set of delay partial differential equations with multiple time-varying delays. The concept of our minimax control approach consists in setting the problem in the worst-case disturbances which leads to the game theory in which the controls and disturbances play antagonistic roles. The proposed strategy consists in controlling these instabilities by acting on certain data to maintain the system in a desired state. First, the mathematical model is introduced and its well-posedness is studied. Second, the minimax control problem is formulated. Afterwards the Frechet differentiability of nonlinear solution map from the couple control-disturbance input to the solution of state system is assessed as well as stability of the derived sensitive system. The existence of an optimal solution is proved and first-order necessary optimality conditions are established by using sensitivity and adjoint calculus.
Highlights
In this paper we consider nonlinear minimax control problems constrained by an uncertain modified bidomain model of cardiac tissue electrophysiology system, in order to take into account the influence of noises in data and time-delays in signal transmission
The state system is a degenerate nonlinear coupled system of reaction-diffusion equations in the shape of a set of delay differential equations coupled with a set of delay partial differential equations with multiple time-varying delays
The new feature introduced in this work concerns the study of nonlinear minimax control problem for a bidomain model with time-delays of cardiac tissue electrophysiology system, in order to take into account the influence of noises in data
Summary
The heart is an electrically controlled mechanical pump which drives blood flow through the circulatory system vessels (through deformation of its walls), where electrical impulses trigger mechanical contraction (of various chambers of heart) and whose dysfunction is incompatible with life. Fibrillation is one type of arrhythmia and is considered the most serious cardiac rhythm disturbance It occurs when the heart beats with rapid, erratic electrical impulses (highly disorganized almost chaotic activation). This causes the heart’s chambers to quiver (or fibrillate) uselessly instead of contracting normally. The evaluation of the bioelectrical activity in the heart is a very complex process which uses different phenomenological mechanism and subject to various perturbations, and physiological and pathophysiological variations This has greatly emphasized the need for methodologies capable of predicting, understanding and optimizing different complex phenomena occurring in these fields, despite different sources of uncertainty like the absence of complete or reliable data (e.g., stimulus currents, measurement data), neglected dynamics, or intrinsic physical variability. The goal of the present paper is to investigate minimax control problems for a bidomain type system, commonly used for modeling the propagation of electrophysiological waves in the myocardium, with
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