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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.

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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.

3

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).

4

Optimality conditions for a class of optimal boundary control problems with quasilinear elliptic equations

Casas E., Dhamo V.

Control and Cybernetics

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2011

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Vol. 40, no 2

457-490

EN

First- and second-order optimality conditions are established for the boundary optimal control of quasilinear elliptic equations with pointwise constraints on the control. The theory is developed for Neumann controls in polygonal domains of dimension two. For the derivation of second-order sufficient optimality conditions, which is the main goal of this paper, the regularity of the solutions to the state equation and its linearization is studied in detail. Moreover, a Pontryagin principle is proved. The main difficulty in the analysis of these problems is the nonmonotone character of the state equation.

5

Revisiting the analysis of optimal control problems with several state constraints

Bonnans J. F., Hermant A.

Control and Cybernetics

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2009

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Vol. 38, no 4A

1021-1052

EN

This paper improves the results of and gives shorter proofs for the analysis of state constrained optimal control problems than presented by the authors in Bonnans and Hermant (2009b), concerning second order optimality conditions and the well-posedness of the shooting algorithm. The hypothesis for the second order necessary conditions is weaker, and the main results are obtained without reduction to the normal form used in that reference, and without analysis of high order regularity results for the control. In addition, we provide some numerical illustration. The essential tool is the use of the "alternative optimality system".

6

Second order convexity and a modified objective function method in mathematical programming

Antczak T.

Control and Cybernetics

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2007

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Vol. 36, no 1

161-182

EN

An approach to nonlinear constrained mathematical programming problems which makes use of a second order derivative is presented. By using a second order modified objective function method, a modified optimization problem associated with a primal mathematical programming problem is constructed. This auxiliary optimization problem involves a second order approximation of an objective function constituting the primal mathematical programming problem. The equivalence between the original mathematical programming problem and its associated modified optimization problem is established under second order convexity assumption. Several practical O.R. applications show that our method is efficient. Further, an iterative algorithm based on this approach for solving the considered nonlinear mathematical programming problem is given for the case when the functions constituting the problem are second order convex. The convergence theorems for the presented algorithm are established.

7

Sensitivity analysis for optimal control of problems governed by semilinear parabolic equations

Bergounioux M., Merabet N.

Control and Cybernetics

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2000

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Vol. 29, no 4

861-886

EN

We investigate an optimal control problem governed by a semilinear parabolic equation with perturbed initial data. We perform some sensitivity analysis: under polyhedricity assumption and second order optimality conditions we derive second order expansion of the optimal value function.