Executive Summary : | Optimal control problems subject to monodomain equations have important practical applications in cardiac medicine. A cardiovascular arrhythmia is an irregular heartbeat that occurs when the electrical signals that coordinate the heart rhythm stop or behave abruptly. Applying a strong electric shock is the only way to return the heart rhythm to normal. This project aims in finding the optimal location of control devices of a defibrillator on the heart. This can be done by carefully selecting the cost function of the optimal control problem. This proposal is intended to study the sparse optimal control of monodomain equations in cardiac electrophysiology by including a nonsmooth term to the cost function in addition to the Tikhonov term. This study is not found in the literature. The objective function is non-smooth due to the presence of L1 norm of control. Minimizing a non-smooth cost function leads to an optimal control that vanishes on major parts of the control domain (sparsity). This is useful in placing the control in optimal manner over the control domain. The existence of optimal control can be guaranteed due to the existence of a weak solution of the monodomain model and weakly lower semicontinuity of the nonsmooth term in the cost function. The differentiability of the control-to-state operator shall be proved. Since the cost remains not differentiable, the concept of subdifferential has to be used while deriving the first-order necessary optimality condition for the local solution of the problem. To solve the discretized optimization problem, we propose to use the nonlinear projected gradient method with Hanger-Zhang update with and without active sets. As the available capacities of control devices are usually restricted, it is required to suppose a bound for the control variable. Thus, the problem is to minimize the cost over a convex set. Here we use the Karush-Kuhn-Tucker theory (KKT) in function spaces to prove the existence of multipliers. Due to the two-sided pointwise box constraints for the control, there exist two multipliers that satisfy complementarity slackness conditions. Active sets can be derived from the optimality system of the problem which plays an important role in developing numerical methods for finding the optimal control. Similar bounds can be assumed for the state. This makes the optimization problem more difficult. The Lagrange multipliers associated with the state constraints are regular Borel measures, which are difficult to analyze. One way to deal with such a state-constrained problem is by regularizing the cost function. We will be using Moreau-Yosida regularization to treat the state-constrained problem. The regularizing parameters in the cost play a major role in sparsity. We are also interested in the analysis and computation of problems with only nonsmooth term in the cost function without Tikhonov term. |