We present a direct numerical method for the solution of an optimal control problem controlling the growth of LDL (Low-density Lipoprotein), HDL (High-density Lipoprotein) and plaque. The optimal control problem is constrained with a system of coupled nonlinear free and mixed boundary partial differential equations consisting of three parabolics one elliptic and one ordinary differential equations. In the first step, the original problem is transformed from a free boundary problem into a fixed one and from the mixed boundary condition to a Neumann one. Then, employing a fixed point-collocation method, we solve the optimal control problem. In each step of the fixed point iteration, the problem is changed to a linear one and then, the equations are solved using the collocation method bringing about an NLP which is solved using sequential quadratic programming. Then, the obtained solution is verified using indirect methods originating from the first-order optimality conditions. Numerical results are considered to illustrate the efficiency of methods.

我们提出了一种直接数值方法，用于解决控制 LDL（低密度脂蛋白）、HDL（高密度脂蛋白）和斑块生长的最佳控制问题。最优控制问题受到耦合非线性自由和混合边界偏微分方程组的约束，该方程组由三个抛物线一个椭圆和一个常微分方程组成。第一步，将原始问题从自由边界问题转化为固定问题，从混合边界条件转化为诺依曼问题。然后，采用固定点配置方法，我们解决了最优控制问题。在定点迭代的每一步中，问题都变为线性问题，然后，使用搭配方法求解方程，从而产生使用顺序二次规划求解的 NLP。然后，使用源自一阶最优条件的间接方法验证获得的解决方案。数值结果被认为是说明方法的效率。