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We present a direct numerical method for the solution of an optimal controlproblem controlling the growth of LDL, HDL and plaque. The optimal controlproblem is constrained with a system of coupled nonlinear free and mixedboundary partial differential equations consisting of three parabolics oneelliptic and one ordinary differential equations. In the first step, theoriginal problem is transformed from a free boundary problem into a fixed oneand from the mixed boundary condition to a Neumann one. Then, employing a fixedpoint-collocation method, we solve the optimal control problem. In each step ofthe fixed point iteration, the problem is changed to a linear one and then, theequations are solved using the collocation method bringing about an NLP whichis solved using sequential quadratic programming. Then, the obtained solutionis verified using indirect methods originating from the first-order optimalityconditions. Numerical results are considered to illustrate the efficiency ofmethods.

Numerical solution of optimal control of atherosclerosis using direct and indirect methods with shooting/collocation approach