Splitting of a two-point boundary-value problem that arises in the theory of optimal control

1992 ◽  
Vol 44 (5) ◽  
pp. 637-642 ◽  
Author(s):  
G. A. Kurina
Keyword(s):  
Optimal Control ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Point Boundary ◽  
Theory Of Optimal Control

Comparison of the Direct Method and the Maximum Principle in the Problem of the Aircraft Program Control Design

2019 ◽  
Vol 20 (6) ◽  
pp. 367-375
Author(s):  
O. N. Korsun ◽  
A. V. Stulovskii
Keyword(s):  
Optimal Control ◽  
Boundary Value Problem ◽  
Direct Method ◽  
Boundary Value ◽  
Control Design ◽  
Population Based ◽  
Program Control ◽  
Point Boundary ◽  
Hamilton Function

The article deals with the problem of program control design for a dynamic object defined by a nonlinear system of differential equations. Known methods of optimal control require the two-point boundary value problem solution, which in general is coupled with fundamental difficulties. Therefore, this paper proposes a technique that uses the direct method, in which the functional is minimized directly using a population-based algorithm. The use of direct methods is based on the assumption that control signals may be defined by a finite set of parameters. Then a scalar functional is formed, the numerical value of which measures the quality of the obtained solutions. In this case, the search for optimal control is reduced to the problem of single-criterion multi-parameter optimization. The practical importance of this approach is that it eliminates the need to solve a two-point boundary value problem. However, this results in another difficulty, since the approximation of control, in general, requires a large number of parameters. It is known that in this case, the effectiveness of conventional gradient numerical optimization methods decreases markedly. Therefore, it is proposed to take the next step and apply genetic or population-based optimization algorithms that have confirmed their performance in solving this class of problems. For this purpose the paper uses one of the modifications of the particle swarm algorithm. The technique is applied to a test problem describing the spatial movement of a maneuverable aircraft. The direct method is compared with two classical solutions based on the condition that the partial control derivatives of the Hamilton function are equal to zero and with the condition of Hamilton function maximum over controls (Pontryagin’s maximum principle). The presented results show the high degree of similarity between obtained controls for all considered methods of selecting the target functional. At the same time, the accuracy of classical algorithms turns out to be slightly worse, and they show a higher sensitivity to the quality of the initial approximation. Thus, the obtained results confirm the approximate equivalence of the direct method and the classical methods of program control design, at least for the class of problems under consideration. The practical significance of this research is that the use of the direct method is much simpler than solving a two-point boundary value problem necessary for classical algorithms.

Optimal Control of Linear Distributed Parameter Systems by Shifted Legendre Polynomial Functions

1983 ◽  
Vol 105 (4) ◽  
pp. 222-226 ◽  
Author(s):  
Maw-Ling Wang ◽  
Rong-Yeu Chang
Keyword(s):  
Optimal Control ◽  
Boundary Value Problem ◽  
Legendre Polynomial ◽  
Boundary Value ◽  
Distributed Parameter ◽  
Polynomial Functions ◽  
Distributed Parameter System ◽  
Point Boundary ◽  
Parameter System ◽  
The Maximum Principle

The optimal control problem of a linear distributed parameter system is studied by employing the technique of shifted Legendre polynomial functions. A partial differential equation, which represents the linear distributed parameter system, is expanded into a set of ordinary differential equations for coefficients in the shifted Legendre polynomial expansion of the input and output signals. Expressing the performance index in terms of the expansion coefficients, we transformed an optimal control gain problem into a two point boundary value problem by applying the maximum principle. The two-point boundary value problem is reduced into an initial value problem, the solution of which can be easily obtained by the proposed computational algorithm. An illustrative example will be used to prove this point.

Aviation radio control systems as complex technical systems. Part 2. Multicriteria optimization based on a local modification of the statistical theory of optimal control

2021 ◽  
Author(s):  
V.S. Verba ◽  
V.I. Merkulov
Keyword(s):  
Optimal Control ◽  
Statistical Theory ◽  
Boundary Value Problem ◽  
Multicriteria Optimization ◽  
Quadratic Forms ◽  
Boundary Value ◽  
State Model ◽  
Main Task ◽  
Theory Of Optimal Control ◽  
Time Of Operation

The analysis of the known modifications of the statistical theory of optimal control (STOC) in the application to the problem of multicriteria optimization of radio control (RC) systems showed that for its solution, a variant that takes into account the measured perturbations in the state model can be used as a basis. The physical meaning of this approach is that of all the alternatives, one is chosen, conditionally the main one, for which a state model is formed. The remaining participants in the general synthesis procedure are considered as a set of perturbations acting on this model. Accordingly, in the quality functional used, the terms that ensure the minimization of these perturbations should be taken into account in the form of an additional linear combination of quadratic forms. Management that minimizes such functionality, to one degree or another, will be jointly the best for the entire set of requirements. The choice of the local optimization option is based on the desire to avoid the need to solve a very computationally expensive twopoint boundary value problem, which is characteristic of the optimization procedures of systems for the entire time of operation, and to ensure invariance to the time of operation. The analysis of the obtained control law allows us to formulate the following conclusions. The control contains two terms, one of which implements the main purpose of the system, providing an approximation of the real state to the required one, and the second takes into account the influence of alternative requirements. The optimized system should include optimal filters that form estimates of the necessary state coordinates, and a controller that forms a control signal. The obtained result indicates that it is possible to optimize management within the framework of solving a multicriteria task (MCT) without solving a complex two-point boundary value problem. If necessary, you can choose another version of the main model with a different set of auxiliary tasks and the corresponding functionality as the main task, with a different control option. In the process of comparing the obtained control options, you can choose the best solution to the general optimization problem. Considered as an example, the law of controlling a group of three initially geographically separated and moving in different directions unmanned aerial vehicles used for radar monitoring of large areas of the earth (water) surface, which at the same time should provide the collection of the group, the required trajectory of its flight and the required topology of participants implementing a given control band, confirmed the possibility of solving the MCT based on STOC.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

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