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This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator The first result is the tunneling estimate The main result is a stability estimate for solutions to the hypoelliptic wave equation We then prove the approximate controllability of the hypoelliptic heat equation We also explain how the analyticity assumption can be relaxed, and a boundary Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).

The quantitative understanding and precise control of complex dynamical systems can only be achieved by observing their internal states via measurement and/or estimation. In large-scale dynamical networks, it is often difficult or physically impossible to have enough sensor nodes to make the system fully observable. Even if the system is in principle observable, high dimensionality poses fundamental limits on the computational tractability and performance of a full-state observer. To overcome the curse of dimensionality, we instead require the system to be functionally observable, meaning that a targeted subset of state variables can be reconstructed from the available measurements. Here, we develop a graph-based theory of functional observability, which leads to highly scalable algorithms to 1) determine the minimal set of required sensors and 2) design the corresponding state observer of minimum order. Compared with the full-state observer, the proposed functional observer achieves the same estimation quality with substantially less sensing and fewer computational resources, making it suitable for large-scale networks. We apply the proposed methods to the detection of cyberattacks in power grids from limited phase measurement data and the inference of the prevalence rate of infection during an epidemic under limited testing conditions. The applications demonstrate that the functional observer can significantly scale up our ability to explore otherwise inaccessible dynamical processes on complex networks.

The characteristics of the goals and objectives of digitalization of pipeline systems (PS) are given, during which they acquire new properties of cyber-physical objects. A structure of tasks of general importance for PS of various types and purposes (heat, water, oil, gas supply, etc.) is proposed. The characteristic of probabilistic state models is given. On this basis, a system of indicators for analyzing the controllability, identifiability and observability of PS is proposed, and their relationship is disclosed. For the first time, analytical dependences of the parametric identifiability of PS on the composition of measurements have been obtained.

A quantitative description of a complex system is inherently limited by our ability to estimate the system's internal state from experimentally accessible outputs. Although the simultaneous measurement of all internal variables, like all metabolite concentrations in a cell, offers a complete description of a system's state, in practice experimental access is limited to only a subset of variables, or sensors. A system is called observable if we can reconstruct the system's complete internal state from its outputs. Here, we adopt a graphical approach derived from the dynamical laws that govern a system to determine the sensors that are necessary to reconstruct the full internal state of a complex system. We apply this approach to biochemical reaction systems, finding that the identified sensors are not only necessary but also sufficient for observability. The developed approach can also identify the optimal sensors for target or partial observability, helping us reconstruct selected state variables from appropriately chosen outputs, a prerequisite for optimal biomarker design. Given the fundamental role observability plays in complex systems, these results offer avenues to systematically explore the dynamics of a wide range of natural, technological and socioeconomic systems.

Distribution network state inference refers to the process of calculating the state variables of each node by using measurement data and network models in the operation of the distribution system. However, the uneven measurement layout and insufficient measurement accuracy in the distribution network have brought great challenges to the state inference of the distribution network. This paper proposes a low-observable distribution network state inference method based on a graph convolution network (GCN), which uses sparse measurement data to infer missing measurement information. Firstly, the observability of the distribution network is analyzed by the numerical probability analysis method. Secondly, the GCN is employed to extract feature information from measurement data and integrate these features. The state inference model of the distribution network based on the GCN is established. Subsequently, power flow constraints of the distribution network are incorporated into the GCN training process to enhance the precision of the generated data. Ultimately, the efficacy of the proposed method is validated using the IEEE 33-node distribution system.

Ecological systems can undergo sudden, catastrophic changes known as critical transitions. Anticipating these critical transitions remains challenging in systems with many species because the associated early warning signals can be weakly present or even absent in some species, depending on the system dynamics. Therefore, our limited knowledge of ecological dynamics may suggest that it is hard to identify those species in the system that display early warning signals. Here, we show that, in mutualistic ecological systems, it is possible to identify species that early anticipate critical transitions by knowing only the system structure—that is, the network topology of plant–animal interactions. Specifically, we leverage the mathematical theory of structural observability of dynamical systems to identify a minimum set of “sensor species,” whose measurement guarantees that we can infer changes in the abundance of all other species. Importantly, such a minimum set of sensor species can be identified by using the system structure only. We analyzed the performance of such minimum sets of sensor species for detecting early warnings using a large dataset of empirical plant–pollinator and seed-dispersal networks. We found that species that are more likely to be sensors tend to anticipate earlier critical transitions than other species. Our results underscore how knowing the structure of multispecies systems can improve our ability to anticipate critical transitions.

This paper investigates the state estimation problem for linear time-invariant systems where sensors and controllers are connected by a stationary memoryless digital communication channel. Time delay, packet dropout, and data rate limitation occur simultaneously in such a channel. We discuss observability of such systems under communication constraints. A real-time transport protocol (RTP) is used for data transmission. A quantization, coding, and control scheme on the basis of RTP is proposed in order to overcome such difficulties. Sufficient conditions for observability are provided in this case. Our result contains some existing results, and shows that there exists the inherent tradeoff between control and communication costs. Illustrative examples are given to demonstrate the effectiveness of the proposed quantization, coding, and control scheme.

The observation of behaviour is a key theoretical parameter underlying a number of models of prosociality. However, the empirical findings showing the effect of observability on prosociality are mixed. In this meta-analysis, we explore the boundary conditions that may account for this variability, by exploring key theoretical and methodological moderators of this link. We identified 117 papers yielding 134 study level effects (total n = 788 164) and found a small but statistically significant, positive association between observability and prosociality (r = 0.141, 95% confidence interval = 0.106, 0.175). Moderator analysis showed that observability produced stronger effects on prosociality: (i) in the presence of passive observers (i.e. people whose role was to only observe participants) versus perceptions of being watched, (ii) when participants’ decisions were consequential (versus non-consequential), (iii) when the studies were performed in the laboratory (as opposed to in the field/online), (iv) when the studies used repeated measures (instead of single games), and (v) when the studies involved social dilemmas (instead of bargaining games). These effects show the conditions under which observability effects on prosociality will be maximally observed. We describe the theoretical and practical significance of these results.

Observability and controllability are essential concepts to the design of predictive observer models and feedback controllers of networked systems. For example, noncontrollable mathematical models of real systems have subspaces that influence model behavior, but cannot be controlled by an input. Such subspaces can be difficult to determine in complex nonlinear networks. Since almost all of the present theory was developed for linear networks without symmetries, here we present a numerical and group representational framework, to quantify the observability and controllability of nonlinear networks with explicit symmetries that shows the connection between symmetries and nonlinear measures of observability and controllability. We numerically observe and theoretically predict that not all symmetries have the same effect on network observation and control. Our analysis shows that the presence of symmetry in a network may decrease observability and controllability, although networks containing only rotational symmetries remain controllable and observable. These results alter our view of the nature of observability and controllability in complex networks, change our understanding of structural controllability, and affect the design of mathematical models to observe and control such networks.