<p>The problem of local control of flows in resource networks consists of finding a collection of capacities for arcs outgoing from selected controlled vertices, such that a given state <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Q'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Q</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> is a limit state for any initial state <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Q_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. The paper proposes an approach to solving this problem for the special case of cyclic networks with low resource. Necessary conditions for the stability of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Q'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Q</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> are derived. It is shown that if these stability conditions are satisfied, there exists a collection of capacities for which the given state <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Q'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Q</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> is stable. An algorithm is proposed that dynamically redistributes resource among the cyclic classes of the network in such a way that any initial state <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Q_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> converges to the stable state <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Q'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Q</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>.</p>

LOCAL CONTROL OF FLOWS IN CYCLIC RESOURCE NETWORKS WITH LOW RESOURCE

期刊论文
发表日期：

2026年4月14日

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Konstantin V. Kucherov,
Vladimir A. Skorokhodov

摘要

The problem of local control of flows in resource networks consists of finding a collection of capacities for arcs outgoing from selected controlled vertices, such that a given state $Q'$ $Q^{'}$ is a limit state for any initial state $Q_0$ $Q_{0}$ . The paper proposes an approach to solving this problem for the special case of cyclic networks with low resource. Necessary conditions for the stability of $Q'$ $Q^{'}$ are derived. It is shown that if these stability conditions are satisfied, there exists a collection of capacities for which the given state $Q'$ $Q^{'}$ is stable. An algorithm is proposed that dynamically redistributes resource among the cyclic classes of the network in such a way that any initial state $Q_0$ $Q_{0}$ converges to the stable state $Q'$ $Q^{'}$ .