Some tools for the direct solution of optimal control problems

Abstract

In the work the general methodology of control is used for obtaining the solution of the problem of optimal annealing stage in a polymerase chain reaction in order to effectively conduct the study and the possibility of providing a multi-stage cyclic regime of temperature change. The annealing stage should occur at certain temperatures and over time, because otherwise the necessary transformations of DNA molecules may not occur. The developed model of annealing stage of the polymerase chain reaction, which takes into account the ratio of the number of single-stranded DNA, primer, single-stranded DNA bound to the primer, direct and reverse reaction rate for annealing, was used. In the model under study, the Arrhenius equation is used, which takes into account the dependence of the reaction rate on absolute temperature. The principle of Pontryagin's maximum is applied to the problem of optimal control of annealing stage and the necessary optimality condition is formulated. The direct method of numerical solving of the problem of optimal annealing control, which is implemented in the package of Java classes, is developed. In the form of graphs are presented the results of numerical simulation of the problem of optimal control of the annealing stage polymerase chain reaction. The results of numerical modeling of the optimal control of the annealing stage of polymerase chain reaction for changing the number of single-stranded DNA, the number of primers, changes in the number of single-stranded DNA that are connected to the primer and the optimal temperature value of the investigated stage are constructed. The obtained results of numerical simulation of the problem of optimal control of the annealing stage of polymerase chain reaction will help to minimize the necessary time for the implementation of this stage. The scheme of temperature setting thus constructed minimizes the required time of implementation of the annealing stage, which in the general case will ensure the achievement of the minimum time for polymerase chain reaction.

Integrating PSO with modified hybrid GA for solving nonlinear optimal control problems

“…Firstly the control variables are parameterized. Next, NOCP is changed into a finite dimensional NLP (See [11]). Now, we can imply an optimization algorithm to find the global solution of the corresponding NLP.…”

“…Let iter = 1. while stopping conditions are not satisfied do {Evaluation} Evaluate the fitness value of each particle by (9). {Update} Update the inertia weight and acceleration coefficients, w and c k , k = 1, 2, by (12) and (13), new Gbest iter and new position of particles by (11).…”

“…the proposed algorithm is compared with some recently proposed algorithms. These methods consist of SUMT and SQP, proposed in [11], continuous genetic algorithm, CGA [1], (better than SUMT), IPSO-SQP [21], (more accurate than some heuristic algorithms such as GA [22], DE [8], and PSO [15]) and some numerical methods [12,14,29]. Because of the stochastic nature of the proposed algorithm, 10 different runs were made for each result.…”

Autonomous Helicopter Formation Using Model Predictive Control

“…Direct methods, on the other hand, discretize continuous dynamics and cost functions using a high-order Runge-Kutta method 28,29 or direct collocation, 30 convert it into a finite-dimensional nonlinear optimization problem, and obtain an optimal solution through nonlinear programming techniques. These provide approximate solutions, but they are robust against arbitrary initial conditions, and optimal solutions with reasonable accuracy can be achieved using less intensive numerical procedures than indirect methods.…”

“…29 This package uses the 4th order RungeKutta method for discretization of the continuous-time dynamics and cost functions, and achieves an optimal solution using the sequential unconstrained minimization technique (SUMT). Since the package contains the SUMT algorithm and it is tightly integrated with discretization procedures, DynOpt allows for compact and versatile implementation.…”