<abstract xmlns="http://www.w3.org/1999/xhtml">

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 <italic>local</italic> 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 <italic>x</italic>i varying within their intervals [0, <italic>l</italic>i], associated with the corresponding beams at network edges. The lengths <italic>l</italic>i 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.
</abstract>

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.

Network optimality conditions

Control and Cybernetics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Network optimality conditions"}

Optimality conditions for optimal control problems arising in network modeling are discussed. We confine ourselves to the steady state network models....

Network optimality conditions

Publié en ligne: 11 avr. 2024

Pages: 129 - 180

Reçu: 01 oct. 2023

Accepté: 01 déc. 2023

DOI: https://doi.org/10.2478/candc-2023-0035

Mots clés
network, optimal control problem, weak local minimum, Pontryagin’s maximum principle, critical cone, quadratic form, second order optimality conditions, Riccati equation

© 2023 Nikolai P. Osmolovskii et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Network optimality conditions

Publié en ligne: 11 avr. 2024

Pages: 129 - 180

Reçu: 01 oct. 2023

Accepté: 01 déc. 2023

DOI: https://doi.org/10.2478/candc-2023-0035

Mots clésnetwork, optimal control problem, weak local minimum, Pontryagin’s maximum principle, critical cone, quadratic form, second order optimality conditions, Riccati equation

© 2023 Nikolai P. Osmolovskii et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Mots clés
network, optimal control problem, weak local minimum, Pontryagin’s maximum principle, critical cone, quadratic form, second order optimality conditions, Riccati equation