Executive Summary : | The study aims to develop mathematical analysis and efficient numerical techniques for optimal boundary control of bidomain-bath models with mixed state and control constraints in cardiac defibrillation. Cardiac fibrillation is a breakdown of organized electrical activity in the heart into disorganized self-sustained electrical activation patterns, affecting the main pumping chambers and leading to loss of cardiac output and death. The dynamics of reaction-diffusion systems model complex electrical activity, known as bidomain-bath equations. The study investigates the well-posedness of the complete bidomain-bath model using the Faedo-Galerkin approach followed by compactness, which helps in proving the existence and uniqueness of the adjoint system. The natural optimal control approach to cardiac defibrillation is to determine the control variable to minimize undesired transmembrane voltage values or bring the arrhythmia state to a desired state. The study investigates the feasibility study of $L^1$ type cost functional for the bidomain-bath model and derives weak and strong solution concepts for the adjoint system. The computational complexity of solving bidomain equations and the state solution is highlighted, highlighting the need for a novel numerical approach and efficient optimization algorithms to solve PDE constrained optimal control problems. The study also investigates the semismooth Newton method and sequential quadratic programming methods, including mixed $L^2$ and $L^1$ norm of the control and mixed state and control constraints for optimization. |