A numerical method for solving optimal control problems

Quantum Optimal Control Dynamics for Delay Intracellular and Multiple Chemotherapy Treatment (MCT) of Dual Delayed HIV - Pathogen Infections

“…Equations (10) and (11) indicate that with the application of the optimization methods of [16,19], we can solve the maximization problem of equation (7) by successive increase in the values of the functional Q of equation (6), based on solutions of model (2) and costate equation (11). Therefore, the boundary conditions on costate variable at the right side of the trajectory are given as: …”

“…Then, it can be said that once the trajectories model (2) passes hypersurface (13), the costate variable vector-valued function (9) satisfies the jump conditions [19] …”

“…In this considered problem, the change in functional (12), which is determined with the aid of equations (16) and (19), is obtained as follows: Of note, is the fact that for a nonlinear delay-differential equation as model (2), the method of successive increment of control measure is achieved provided the following two conditions are satisfied: i) -the trajectory iteration is superimposed by any external source, ii) -the change in the control measure values are sufficiently small. The proof is standard and could be found in [4,9].…”

“…Using the phase conjugate, we shall explore the optimization method initiated by [19] to transform the 10-Dimensional state variables to phase coordinates typical of model (2) …”

HIV vs. the Immune System: A Differential Game

“…So that a numerical resolution turns out to be crucial in order to get some enlightening results. We formulate and apply an efficient and intuitive algorithm by combining the Forward-Backward Sweep algorithm [18][19][20][21] and a 4th order Runge-Kutta routine (see e.g., [22]). We performed several simulations and we always got a unique solution to the necessary conditions.…”