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1

Second-order maximum principle controlled weakly singular Volterra integral equations

Gasimov Jasarat J., Mahmudov Nazim I.

Archives of Control Sciences

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2024

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Vol. 34, No. 4

863--880

EN

This work studies a class of singular Volterra integral equations that are (controlled) and can be applied to memory-related problems. For optimum controls, we prove a second-order Pontryagin type maximal principle.

2

Pontryagin’s maximum principle for the Roesser model with a fractional Caputo derivative

Yusubov Shakir Sh., Mahmudov Elimhan N.

Archives of Control Sciences

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2024

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Vol. 34, No. 2

271--300

EN

In this paper, we study the modern mathematical theory of the optimal control problem associated with the fractional Roesser model and described by Caputo partial derivatives, where the functional is given by the Riemann-Liouville fractional integral. In the formulated problem, a new version of the increment method is applied, which uses the concept of an adjoint integral equation. Using the Banach fixed point principle, we prove the existence and uniqueness of a solution to the adjoint problem. Then the necessary and sufficient optimality condition is derived in the form of the Pontryagin’s maximum principle. Finally, the result obtained is illustrated by a concrete example.

3

Network optimality conditions

Osmolovskii Nikolai P., Qian Meizhi, Sokołowski Jan

Control and Cybernetics

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2023

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Vol. 52, No. 2

129--180

EN

Optimality conditions for optimal control problems arising in network modeling are discussed. We confine ourselves to the steady state network models. Therefore, we consider only control systems described by ordinary differential equations. First, we derive optimality conditions for the nonlinear problem for a single beam. These conditions are formulated in terms of the local Pontryagin maximum principle and the matrix Riccati equation. Then, the optimality conditions for the control problem for networks posed on an arbitrary planar graph are discussed. This problem has a set of independent variables xi varying within their intervals [0, li], associated with the corresponding beams at network edges. The lengths li of intervals are not specified and must be determined. So, the optimization problem is non-standard, it is a combination of control and design of networks. However, using a linear change of the independent variables, it can be reduced to a standard one, and we show this. Two simple numerical examples for the single-beam problem are considered.

4

Preliminary optimization technique in the design of steel girders according to Eurocode 3

Szeptyński Paweł, Mikulski Leszek

Archives of Civil Engineering

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2023

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Vol. 69, nr 1

71--89

EN

The problem of optimal design of a steel plated girder according to the Eurocode 3 is considered. Code regulations admit the Finite Element Analysis (FEA) in designing plated structures with variable cross-sections. A technique of determining an approximate solution to the optimization problem is presented. It is determined a solution of a control theory optimization task, in which Eurocode requirements regarding the Ultimate Limit State (bearing capacity, local and global stability) as well as Serviceability Limit State (flexural rigidity) are used as appropriate inequality constraints. Static analysis is performed within the framework of linear elasticity and Bernoulli-Euler beam theory making an account for second-order effects due to prescribed imperfections. Obtained solutions, after regularization, may be used for direct verification with the use of FEA or as the first guess for iterative topology optimization algorithms. Code requirements governing the determination of optimal shape are visualized in the constraint activity diagram, which is a proposed tool for analysis of optimization process.

PL

Rozważany jest problem optymalnego projektowania blachownicy stalowej zgodnie z Eurokodem 3. Zapisy normowe dopuszczają stosowanie Metody Elementów Skończonych (MES) w projektowaniu blachownic o zmiennym przekroju poprzecznym. Przedstawiono metodę wyznaczania przybliżonego rozwiązania zagadnienia optymalizacji. Jest ono wyznaczane jako rozwiązanie problemu optymalizacyjnego teorii sterowania, w którym wymagania Eurokodu dotyczące Stanu Granicznego Nośności (nośność, lokalna i globalna stateczność) i Stanu Granicznego Użytkowalności (sztywność giętna) wykorzystane są jako ograniczenia nierównościowe. Analiza statyczna przeprowadzona jest w ramach liniowej teorii sprężystości dla modelu belki Bernoulliego - Eulera z uwzględnieniem efektów drugiego rzędu z uwagi na zadane imperfekcje. Uzyskane rozwiązania, po stosownych modyfikacjach, mogą podlegać weryfikacji z wykorzystaniem MES lub mogą zostać wykorzystane jako pierwsze przybliżenie w iteracyjnych algorytmach optymalizacji topologicznej. Wymagania normowe rządzące wyznaczaniem optymalnego kształtu zostały zwizualizowane na schemacie aktywności ograniczeń, który proponowany jest jako narzędzie analizy procesu optymalizacji.

5

A second-order sufficient condition for a weak local minimum in an optimal control problem with an inequality control constraint

Osmolovskii Nikolai P.

Control and Cybernetics

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2022

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Vol. 51, No. 2

151--169

EN

This paper is devoted to a sufficient second-order condition for a weak local minimum in a simple optimal control problem with one control constraint G(u) ≤ 0, given by a C2-function. A similar second-order condition was obtained earlier by the author for a strong minimum in a much more general problem. In the present paper, we would like to take a narrower perspective than before and thus provide shorter and simpler proofs. In addition, the paper uses the first and second order tangents to the set U, defined by the inequality G(u) ≤ 0. The main difficulty of the proof, clearly shown in the paper, refers to the set, where the gradient Hu of the Hamiltonian is small, but the condition of quadratic growth of the Hamiltonian is satisfied. The paper can be valuable for self-explanation and provides a basis for extensions.

6

Extremals of the time optimal control problem for a material point moving along a straight line in the presence of friction and limitation on the velocity

Osmolovskii N. P., Figura A., Kośka M., Wójtowicz M.

Control and Cybernetics

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2017

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Vol. 46, no. 4

305--324

EN

This paper provides an analysis of the time optimal control problem for a material point moving along a straight line in the presence of strength of resistance to movement (friction) and subject to constraint on the velocity. The point is controlled by a limited traction or braking force. The analysis of the problem is based on the maximum principle for state constraints in the Dubovitskii-Milyutin form, see Dubovitskii and Milyutin (1965), and the necessary second-order optimality condition for bang-bang controls, see Milyutin and Osmolovskii (1998).

7

On the optimal control problem for two regions’ macroeconomic model

Surkov P. G.

Archives of Control Sciences

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2015

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Vol. 25, no. 4

417--427

EN

In this paper we consider a model of joint economic growth of two regions. This model bases on the classical Kobb-Douglas function and is described by a nonlinear system of differential equations. The interaction between regions is carried out by changing the balance of trade. The optimal control problem for this system is posed and the Pontryagin maximum principle is used for analysis the problem. The maximized functional represents the global welfare of regions. The numeric solution of the optimal control problem for particular regions is found. The used parameters was obtained from the basic scenario of the MERGE