scholarly journals SOLUTION OF THE BOUNDARY-VALUE PROBLEM OF OPTIMAL CONTROL IN THE CRITICAL CASE

2013 ◽  
Vol 0 (2) ◽  
Author(s):  
Y. V. Panasenko
Keyword(s):  
Optimal Control ◽  
Boundary Value Problem ◽  
Critical Case ◽  
Boundary Value

scholarly journals NONSINGULAR INTEGRO-DIFFERENTIAL BOUNDARY VALUE PROBLEM NOT SOLVED WITH RESPECT TO THE DERIVATIVE

2020 ◽  
Vol 8 (2) ◽  
pp. 127-138
Author(s):  
S. Chuiko ◽  
O. Chuiko ◽  
V. Kuzmina
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Systematic Study ◽  
Critical Case ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Linear Boundary ◽  
Actual Problem ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boi- chuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of linear boundary-value problems for ordinary di- fferential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the linear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear osci- llations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the linear boundary value problems for the integro-differential boundary value problem not solved with respect to the derivative, in parti- cular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear integro-differential boundary value problem not solved with respect to the derivative. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian integro-differential boundary value problem not solved with respect to the derivative. The proposed scheme of the research of the nonlinear Noetherian integro-differential boundary value problem not solved with respect to the derivative in the critical case in this article can be transferred to the seminonlinear integro-differential boundary value problem not solved with respect to the derivative.

scholarly journals The Periodic Boundary Value Problem for the Weakly Dissipativeμ-Hunter-Saxton Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Zhengyong Ouyang ◽  
Xiangdong Wang ◽  
Haiwu Rong
Keyword(s):  
Boundary Value Problem ◽  
Besov Space ◽  
Periodic Boundary ◽  
Critical Case ◽  
Boundary Value ◽  
Periodic Boundary Value Problem ◽  
Well Posedness

We study the periodic boundary value problem for the weakly dissipativeμ-Hunter-Saxton equation. We establish the local well-posedness in Besov spaceB2,13/2, which extends the previous regularity range to the critical case.

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.

A new approach for the optimal control of time-varying delay systems with external persistent matched disturbances

2017 ◽  
Vol 24 (19) ◽  
pp. 4505-4512 ◽  
Author(s):  
Amin Jajarmi ◽  
Mojtaba Hajipour ◽  
Dumitru Baleanu
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Necessary Optimality Conditions ◽  
Delay Systems ◽  
Time Varying ◽  
Time Varying Delay ◽  
Augmented System ◽  
Varying Delay

The aim of this study is to develop an efficient iterative approach for solving a class of time-delay optimal control problems with time-varying delay and external persistent disturbances. By using the internal model principle, the original time-delay model with disturbance is first converted into an augmented system without any disturbance. Then, we select a quadratic performance index for the augmented system to form an undisturbed time-delay optimal control problem. The necessary optimality conditions are then derived in terms of a two-point boundary value problem involving advance and delay arguments. Finally, a fast iterative algorithm is designed for the latter advance-delay boundary value problem. The convergence of the new iterative technique is also investigated. Numerical simulations verify that the proposed approach is efficient and provides satisfactory results.

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