By Andrew Clark, Basel Alomair, Linda Bushnell, Radha Poovendran, John Baillieul

ISBN-10: 3319269755

ISBN-13: 9783319269757

Foreword by way of John Baillieul

This e-book offers a framework for the keep an eye on of networked platforms using submodular optimization strategies. the focus is on picking enter nodes for the keep an eye on of networked platforms, an inherently discrete optimization challenge with purposes in energy process balance, social effect dynamics, and the keep an eye on of auto formations. the 1st a part of the e-book is dedicated to history info on submodular features, matroids, and submodular optimization, and provides algorithms for dispensed submodular optimization which are scalable to giant networked systems.

In flip, the second one half develops a unifying submodular optimization method of controlling networked structures according to a number of functionality and controllability standards. strategies are brought for choosing enter nodes to make sure delicate convergence, synchronization, and robustness to environmental and adverse noise. Submodular optimization is the 1st unifying method in the direction of making certain either functionality and controllability with provable optimality bounds in static in addition to time-varying networks. in the course of the textual content, the submodular framework is illustrated with the aid of numerical examples and application-based case experiences in organic, strength and vehicular systems.

The booklet successfully combines parts of becoming curiosity, and may be specially important for researchers up to speed conception, utilized arithmetic, networking or computer studying with adventure in submodular optimization yet who're much less acquainted with the issues and instruments to be had for networked structures (or vice versa). it is going to additionally profit graduate scholars, supplying constant terminology and notation that vastly reduces the preliminary attempt linked to starting a process examine in a brand new area.

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**Additional info for Submodularity in Dynamics and Control of Networked Systems**

**Example text**

Since X1 and Y1 are both independent in M1 , there exists y ∈ Y1 \X1 such that (X1 ∪ / X2 , then (X ∪ {y}) ∈ M1 ∪ M2 , with decomposition {y}) ∈ M1 (Fig. 8). If y ∈ X ∪ {y} = (X1 ∪ {y}) ∪ X2 . It remains to consider the case where y ∈ X2 . If this holds, then choose X = (X1 ∪ {y}) ∪ (X2 \{y}), with (X1 ∪ {y}) ∈ M1 by assumption and (X2 \{y}) ∈ M2 by (M2). Thus |(X1 ∪ {y}) ∩ Y2 | = |X1 ∩ Y2 |, since y ∈ Y1 and Y = Y1 ∪ Y2 is a disjoint partition. At the same time, |(X2 \{y}) ∩ Y1 | < |X2 ∩ Y1 |, since y ∈ X2 ∩ Y1 .

A6 }, and define A1 = {a1 , a2 }, A2 = {a1 , a4 }, A3 = {a2 , a3 }, A4 = {a1 , a6 }, and A5 = {a2 , a5 }. Let k = 2. By following the greedy algorithm, we first set S = {1}, with f (S) = 2. All remaining elements will then satisfy f (S ∪ {v}) − f (S) = 1, so let S = {1, 2} with f (S) = 3. Note that S is not the optimal solution; we have f ({2, 3}) = 4, for example. However, we have 24 2 Centralized Submodular Optimization 3 = f (S) ≥ 1 − 1 1 4= 1− e e f (S ∗ ), verifying that the (1 − 1/e) bound is satisfied.

Proof First, since V is a spanning set for any matroid, ∅ ∈ M ∗ , thus proving (M1). To show (M2), suppose that B ∈ I ∗ and A ⊆ B. Conversely, V \B ⊆ V \A, which implies that V = cl(V \B) ⊆ cl(V \A). Hence, V \A is a spanning set of M and A ∈ M ∗ . Finally, to show (M3), let A∗1 , A∗2 ∈ I ∗ , with |A∗1 | < |A∗2 |. Define A1 = V \A∗1 and A2 = V \A∗2 . Since A1 and A2 span M , there exist bases B1 ⊆ A1 and B2 ⊆ A2 . Furthermore, B2 \A∗1 can be extended to a basis B2 that is disjoint from A∗1 . First, it is shown that there exists v ∈ (A∗2 − A∗1 ) − B2 .

### Submodularity in Dynamics and Control of Networked Systems by Andrew Clark, Basel Alomair, Linda Bushnell, Radha Poovendran, John Baillieul

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