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1

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.

2

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

Extension of the one-dimensional Stoney algorithm to a two-dimensional case

Gniazdowski Zenon

Zeszyty Naukowe Warszawskiej Wyższej Szkoły Informatyki

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2019

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nr 20

41--66

EN

This article presents the extension of the one-dimensional Stoney algorithm to a two-dimensional case. The proposed extension consists in modifying the method of curvature estimation. The surface proﬁle of the wafer before deposition of the thin ﬁlm and after its deposition was locally approximated by the quadric. From this quadric, a quadratic form and the ﬁrst degree surface were separated. An eigenproblem was solved for the matrix of this quadratic form. From eigenvectors a new coordinate system was created in which a new formula of the quadric was found. In this new coordinate system, the two-dimensional problem of estimating thecurvaturetensorhasbeensolvedbysolvingtwoindependentone-dimensional problems of curvature estimation. Returning to the primary coordinate system, in this primary system, a solution to the two-dimensional problem was obtained. The article proposes ﬁve versions of the two-dimensional Stoney algorithm, with diverse complexity and accuracy. The recommendation for the version of the algorithm that could be practically used was also presented.

4

Hyers-Ulam stability of quadratic forms in 2-normed spaces

Park Won-Gil, Bae Jae-Hyeong

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

496--502

EN

In this paper, we obtain Hyers-Ulam stability of the functional equations f(x+y, z+w) + f(x-y, z-w) = 2f(x, z) + 2f(y, w), f(x+y, z-w) + f(x-y, z+w) = 2f(x, z) + 2f(y, w) and f(x+y, z-w) + f(x-y, z+w) = 2f(x, z) - 2f(y, w) in 2-Banach spaces. The quadratic forms ax2+bxy+cy2, ax2+by2 and axy are solutions of the above functional equations, respectively.

5

A Horizontal Mesh Algorithm for a Class of Edge-bipartite Graphs and their Matrix Morsifications

Kaniecki M., Kosakowska J., Malicki P., Marczak G.

Fundamenta Informaticae

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2015

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Vol. 136, nr 4

345--379

EN

We construct a horizontal mesh algorithm for a study of a special type of mesh root systems of connected positive loop-free edge-bipartite graphs Δ, with n ≥ 2 vertices, in the sense of [SIAM J. Discrete Math. 27 (2013), 827–854] and [Fund. Inform. 124 (2013), 309-338]. Given such a loop-free edge-bipartite graph Δ, with the non-symmetric Gram matrix ˇGΔ ∈ Mn(Z) and the Coxeter transformation ΦA : Zn → Zn defined by a quasi-triangular matrix morsification A ∈ Mn(Z) of Δ satisfying a non-cycle condition, our combinatorial algorithm constructs a ΦA-mesh root system structure Γ(RΔ,ΦA) on the finite set of all ΦA-orbits of the irreducible root system RΔ := {v ∈ Zn; v · ˇGΔ · vtr = 1}. We apply the algorithm to a graphical construction of a ΦI - mesh root system structure Γ(RI ,ΦI ) on the finite set of ΦI -orbits of roots of any poset I with positive definite Tits quadratic form bqI : ZI → Z.

6

A mesh algorithm for principal quadratic forms

Polak A.

Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

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2011

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Vol. 11, no. 1

23--31

EN

In 1970 a negative solution to the tenth Hilbert problem, concerning the determination of integral solutions of diophantine equations, was published by Y. W. Matiyasevich. Despite this result, we can present algorithms to compute integral solutions (roots) to a wide class of quadratic diophantine equations of the form q(x) = d, where q : Z is a homogeneous quadratic form. We will focus on the roots of one (i.e., d = 1) of quadratic unit forms (q11 = ... = qnn = 1). In particular, we will describe the set of roots Rq of positive definite quadratic forms and the set of roots of quadratic forms that are principal. The algorithms and results presented here are successfully used in the representation theory of finite groups and algebras. If q is principal (q is positive semi-definite and Ker q={v ∈ Zn; q(v) = 0}= Z · h) then |Rq| = ∞. For a given unit quadratic form q (or its bigraph), which is positive semi-definite or is principal, we present an algorithm which aligns roots Rq in a Φ-mesh. If q is principal (|Rq| is less than ∞), then our algorithm produces consecutive roots in Rq from finite subset of Rq, determined in an initial step of the algorithm.

7

Second order conditions in optimal control problems with mixed equality-type constraints on a variable time interval

Osmolovskii N. P.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1535-1556

EN

This paper provides an analysis of second-order necessary or sufficient optimality conditions of Pontryagin or bounded strong minima, for optimal control problems of ordinary differential equations, considered on a nonfixed time interval, with constraints on initial-final time-state as well as mixed state-control constraints of equality type satisfying condition of linear independence of gradients w.r.t. control.

8

Sylvester inertia law in commutative Leibniz algebras with logarithms

Przeworska-Rolewicz D.

Demonstratio Mathematica

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2007

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Vol. 40, nr 3

659-669

EN

In algebras with logarithms induced by a given right invertible operator D one can define quadratic forms by means of power mappings induced by logarithmic mapping. Main results of this paper will be concerned with the case when an algebra X under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy+yDx for x, y is an element of dom D. If X is an locally m-convex algebra then these forms have the similar properties as quadratic forms in the Euclidian spaces En, including the Sylvester inertia law.

9

Second-order sufficient conditions for an extremum in optimal control

Osmolovskij N.

Control and Cybernetics

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2002

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Vol. 31, no 3

803-831

EN

Suffcient quadratic optimality conditions for a weak and a strong minimum are stated in an optimal control problem on a fixed time interval with mixed state-control constraints, under the assumption that the gradients of all active mixed constraints with respect to control are linearly independent. The conditions are stated for the cases of both continuous and discontinuous controls and guarantee in each case a lower bound of the cost function increase at t1e reference point. They are formulated in terms of an accessory problem with quadratic form, which must be positive-definite on the so-called critical cone. In the case of discontinuous control the quadratic form has some new terms related to the control discontinuity.