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Invasive plant species threaten native biodiversity, ecosystems, agriculture, industry and human health worldwide, lending urgency to the search for predictors of plant invasiveness outside native ranges. There is much conflicting evidence about which plant characteristics best predict invasiveness. Here we use a global demographic survey for over 700 plant species to show that populations of invasive plants have better potential to recover from disturbance than non-invasives, even when measured in the native range. Invasives have high stable population growth rates in their invaded ranges, but this metric cannot be predicted based on measurements in the native ranges. Recovery from demographic disturbance is a measure of transient population amplification, linked to high levels of reproduction, and shows phylogenetic signal. Our results demonstrate that transient population dynamics and reproductive capacity can help to predict invasiveness across the plant kingdom, and should guide international policy on trade and movement of plants.

Abstract
. In this paper, a non-singular SIR model with the Mittag-Leffler law is proposed. The nonlinear Beddington-DeAngelis infection rate and Holling type II treatment rate are used. The qualitative properties of the SIR model are discussed in detail. The local and global stability of the model are analyzed. Moreover, some conditions are developed to guarantee local and global asymptotic stability. Finally, numerical simulations are provided to support the theoretical results and used to analyze the impact of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of the disease, and limitation in treatment resources on COVID-19. The graphical results show that face masks, social distancing, quarantine, lockdown, immigration, and effective treatment rates significantly reduce the infected population over time. In contrast, limitation in the availability of treatment raises the infected population.

The Bubonic Plague outbreak that wormed its way through San Francisco’s Chinatown in 1900 tells a story of prejudice guiding health policy, resulting in enormous suffering for much of its Chinese population. This article seeks to discuss the potential for hidden “prejudice” should Artificial Intelligence (AI) gain a dominant foothold in healthcare systems. Using a toy model, this piece explores potential future outcomes, should AI continue to develop without bound. Where potential dangers may lurk will be discussed, so that the full benefits AI has to offer can be reaped whilst avoiding the pitfalls. The model is produced using the computer programming language MATLAB and offers visual representations of potential outcomes. Interwoven with these potential outcomes are numerous historical models for problems caused by prejudice and recent issues in AI systems, from police prediction and facial recognition software to recruitment tools. Therefore, this research’s novel angle, of using historical precedents to model and discuss potential futures, offers a unique contribution.

Damp and high levels of relative humidity (RH), typically above 70-80%, are known to provide mould-favourable conditions. Exposure to indoor mould contamination has been associated with an increased risk of developing and/or exacerbating a range of allergic and non-allergic diseases. The VTT model is a mathematical model of indoor mould growth that was developed based on surface readings of RH and temperature on wood in a controlled laboratory chamber. The model provides a mould index based on the environmental readings. We test the generalisability of this laboratory-based model to less-controlled domestic environments across different values of model parameters. Mould indices were generated using objective measurements of RH and temperature in the air, taken from sensors in a domestic setting every 3-5 minutes over 1 year in the living room and bedroom across 219 homes. Mould indices were assessed against self-reports from occupants regarding the presence of visible mould growth and mouldy odour in the home. Logistic regression provided evidence for relationships between mould indices and occupant responses. Mould indices were most successful at predicting occupant responses when the model parameters encouraged higher vulnerability to mould growth compared with the original VTT model. A lower critical RH level, above which mould grows, a higher sensitivity, and larger increases in the mould index all consistently increased performance. Using moment-to-moment time-series data for temperature and RH, the model and its developments could help inform smart monitoring or control of RH, for example to counter risks associated with reduced ventilation in energy efficient homes.

BACKGROUND: the incidence of Acute Kidney Injury (AKI) continues to increase in the UK, with associated mortality rates remaining significant. Approximately one fifth of hospital admissions are associated with AKI and approximately a third of patients with AKI in hospital develop AKI during their time in hospital. A fifth of these cases are considered avoidable. Early risk detection remains key to decreasing AKI in hospitals, where sub-optimal care was noted for half of patients who developed AKI. METHODS: Electronic anonymised data for adults admitted into the Royal Cornwall Hospitals Trust (RCHT) between 18th March and 31st December 2015 was trimmed to that collected within the first 24 h of hospitalisation. These datasets were split according to three separate time periods: data used for training the Takagi-Sugeno Fuzzy Logic Systems (FLS) and the multivariable logistic regression (MLR) models; data used for testing; and data from a later patient spell used for validation. Three fuzzy logic models and three MLR models were developed to link characteristics of patients diagnosed with a maximum stage AKI within 7 days of admission: the first models to identify any AKI Stage (FLS I, MLR I), the second for patterns of AKI Stage 2 or 3 (FLS II, MLR II), and the third to identify AKI Stage 3 (FLS III, MLR III). Model accuracy is expressed by area under the curve (AUC). RESULTS: Accuracy for each model during internal validation was: FLS I and MLR I (AUC 0.70, 95% CI: 0.64-0.77); FLS II (AUC 0.77, 95% CI: 0.69-0.85) and MLR II (AUC 0.74, 95% CI: 0.65-0.83); FLS III and MLR III (AUC 0.95, 95% CI: 0.92-0.98). CONCLUSIONS: FLS II and FLS III (and the respective MLR models) can identify with a high level of accuracy patients at high risk of developing AKI in hospital. These two models cannot be properly assessed against prior studies as this is the first attempt at quantifying the risk of developing specific Stages of AKI for a broad cohort of both medical and surgical inpatients. FLS I and MLR I performance is comparable to other existing models.

Many animal taxa exhibit sex-specific variation in ecological traits, such as foraging and distribution. These differences could result in sex-specific responses to change, but such demographic effects are poorly understood. Here, we test for sex-specific differences in the demography of northern (NGP, Macronectes halli) and southern (SGP, M. giganteus) giant petrels - strongly sexually size-dimorphic birds that breed sympatrically at South Georgia, South Atlantic Ocean. Both species feed at sea or on carrion on land, but larger males (30% heavier) are more reliant on terrestrial foraging than the more pelagic females. Using multi-event mark-recapture models, we examine the impacts of long-term changes in environmental conditions and commercial fishing on annual adult survival and use two-sex matrix population models to forecast future trends. As expected, survival of male NGP was positively affected by carrion availability, but negatively affected by zonal winds. Female survival was positively affected by meridional winds and El Niño-Southern Oscillation (ENSO), and negatively affected by sea ice concentration and pelagic longline effort. Survival of SGPs did not differ between sexes; however, survival of males only was positively correlated with the Southern Annular Mode (SAM). Two-sex population projections indicate that future environmental conditions are likely to benefit giant petrels. However, any potential increase in pelagic longline fisheries could reduce female survival and population growth. Our study reveals that sex-specific ecological differences can lead to divergent responses to environmental drivers (i.e. climate and fisheries). Moreover, because such effects may not be apparent when all individuals are considered together, ignoring sex differences could underestimate the relative influence of a changing environment on demography.

We discuss the use of optimal control theory to determine the most cost-effective management strategies for insect pests. We use a stage-structured linear population projection model where the modeled control action increases the mortality in one of the stage-classes. We illustrate the method by using a published model for the root weevil Diaprepes abbreviatus, an invasive insect species having a substantial negative impact on citrus trees in regions such as Florida and California in the United States. Here control corresponds to the application of inundative biological control agents (entomopathogenic nematodes as biopesticides) which increases the mortality of the larval stage. Our approach determines levels and timing of control to minimize the economic loss caused by D. abbreviatus. We use two numerical methods to approximate the optimal control, and compare their effectiveness.

We revisit the question of when can dispersal-induced coupling between discrete sink populations cause overall population growth? Such a phenomenon is called dispersal driven growth and provides a simple explanation of how dispersal can allow populations to persist across discrete, spatially heterogeneous, environments even when individual patches are adverse or unfavourable. For two classes of mathematical models, one linear and one non-linear, we provide necessary conditions for dispersal driven growth in terms of the non-existence of a common linear Lyapunov function, which we describe. Our approach draws heavily upon the underlying positive dynamical systems structure. Our results apply to both discrete- and continuous-time models. The theory is illustrated with examples and both biological and mathematical conclusions are drawn.

Fisheries bycatch is a major threat to seabird populations, and understanding sex- and age-biases in bycatch rates is important for assessing population-level impacts. We analysed 44 studies to provide the first global assessment of seabird bycatch by sex and age, and used generalised models to investigate the effects of region and fishing method. Bycatch was highly biased by sex (65% of 123 samples) and age (92% of 114 samples), with the majority of samples skewed towards males and adults. Bycatch of adults and males was higher in subpolar regions, whereas there was a tendency for more immatures and females to be killed in subtropical waters. Fishing method influenced sex- and age-ratios only in subpolar regions. Sex- and age-biases are therefore common features of seabird bycatch in global fisheries that appear to be associated largely with differences in at-sea distributions. This unbalanced mortality influences the extent to which populations are impacted by fisheries, which is a key consideration for at-risk species. We recommend that researchers track individuals of different sex and age classes to improve knowledge of their distribution, relative overlap with vessels, and hence susceptibility to bycatch. This information should then be incorporated in ecological risk assessments of effects of fisheries on vulnerable species. Additionally, data on sex, age and provenance of bycaught birds should be collected by fisheries observers in order to identify regions and fleets where bycatch is more likely to result in population-level impacts, and to improve targeting of bycatch mitigation and monitoring of compliance.

We consider the inclusion of a static antiwindup component in a continuous-time low-gain integral controller in feedback with a multi-input multi-output stable linear system subject to an input nonlinearity (from a class of functions that includes componentwise diagonal saturation). We demonstrate that the output of the closed-loop system asymptotically tracks every constant reference vector, which is 'feasible' in a natural sense, provided that the integrator gain is sufficiently small. Robustness properties of the proposed control scheme are investigated and three examples are discussed in detail.

Using methods from classical absolute stability theory, combined with recent results on input-to-state stability (ISS) of Lur’e systems, we derive necessary and sufficient conditions for a class of Lur’e systems to have the converging-input converging-state (CICS) property. In particular, we provide sufficient conditions for CICS which are reminiscent of the complex Aizerman conjecture and the circle criterion and connections are also made with small gain ISS theorems. The penultimate section of the paper is devoted to non-negative Lur’e systems which arise naturally in, for example, ecological and biochemical applications: the main result in this context is a sufficient criterion for a so-called “quasi CICS” property for Lur’e systems which, when uncontrolled, admit two equilibria. The theory is illustrated with numerous examples.

A result is presented describing the eigenvectors of a perturbed matrix, for a class of structured perturbations. One motivation for doing so is that positive eigenvectors of nonnegative, irreducible matrices are known to induce norms — acting much like Lyapunov functions — for linear positive systems, which may help estimate or control transient dynamics. The results apply to both discrete- and continuous-time linear positive systems. The theory is illustrated with several examples.

Population managers will often have to deal with problems of meeting multiple goals, for example, keeping at specific levels both the total population and population abundances in given stage-classes of a stratified population. In control engineering, such set-point regulation problems are commonly tackled using multi-input, multi-output proportional and integral (PI) feedback controllers. Building on our recent results for population management with single goals, we develop a PI control approach in a context of multi-objective population management. We show that robust set-point regulation is achieved by using a modified PI controller with saturation and anti-windup elements, both described in the paper, and illustrate the theory with examples. Our results apply more generally to linear control systems with positive state variables, including a class of infinite-dimensional systems, and thus have broader appeal.

© 2016 SIAM. A stability/instability trichotomy for a class of nonnegative continuous-time Lur'e systems is derived. Asymptotic, exponential, and input-to-state stability concepts are considered. The presented trichotomy rests on Perron-Frobenius theory, absolute stability theory, and recent input-to-state stability results for Lur'e systems. Applications of the results derived arise in various fields, including density-dependent population dynamics, and two examples are discussed in detail.

The dynamics of structured plant populations in variable environments can be decomposed into the 'asymptotic' growth contributed by vital rates, and 'transient' growth caused by deviation from stable stage structure. We apply this framework to a large, global data base of longitudinal studies of projection matrix models for plant populations. We ask, what is the relative contribution of transient boom and bust to the dynamic trajectories of plant populations in stochastic environments? is this contribution patterned by phylogeny, growth form or the number of life stages per population and per species? We show that transients contribute nearly 50% or more to the resulting trajectories, depending on whether transient and stable contributions are partitioned according to their absolute or net contribution to population dynamics. Both transient contributions and asymptotic contributions are influenced heavily by the number of life stages modelled. We discuss whether the drivers of transients should be considered real ecological phenomena, or artefacts of study design and modelling strategy. We find no evidence for phylogenetic signal in the contribution of transients to stochastic growth, nor clear patterns related to growth form. We find a surprising tendency for plant populations to boom rather than bust in response to temporal changes in vital rates and that stochastic growth rates increase with increasing tendency to boom. Synthesis. Transient dynamics contribute significantly to stochastic population dynamics but are often overlooked in ecological and evolutionary studies that employ stochastic analyses. Better understanding of transient responses to fluctuating population structure will yield better management strategies for plant populations, and better grasp of evolutionary dynamics in the real world. Transient dynamics contribute significantly to stochastic population dynamics but are often overlooked in ecological and evolutionary studies that employ stochastic analyses. Better understanding of transient responses to fluctuating population structure will yield better management strategies for plant populations, and better grasp of evolutionary dynamics in the real world.

We present a novel management methodology for restocking a declining population. The strategy uses integral control, a concept ubiquitous in control theory which has not been applied to population dynamics. Integral control is based on dynamic feedback-using measurements of the population to inform management strategies and is robust to model uncertainty, an important consideration for ecological models. We demonstrate from first principles why such an approach to population management is suitable via theory and examples.

Deterministic dynamic models for coupled resident and invader populations are considered with the purpose of finding quantities that are effective at predicting when the invasive population will become established asymptotically. A key feature of the models considered is the stage-structure, meaning that the populations are described by vectors of discrete developmental stage- or age-classes. The vector structure permits exotic transient behaviour-phenomena not encountered in scalar models. Analysis using a linear Lyapunov function demonstrates that for the class of population models considered, a large so-called population inertia is indicative of successful invasion. Population inertia is an indicator of transient growth or decline. Furthermore, for the class of models considered, we find that the so-called invasion exponent, an existing index used in models for invasion, is not always a reliable comparative indicator of successful invasion. We highlight these findings through numerical examples and a biological interpretation of why this might be the case is discussed.

Sink populations are doomed to decline to extinction in the absence of immigration. The dynamics of sink populations are not easily modelled using the standard framework of per capita rates of immigration, because numbers of immigrants are determined by extrinsic sources (for example, source populations, or population managers). Here we appeal to a systems and control framework to place upper and lower bounds on both the transient and future dynamics of sink populations that are subject to noisy immigration. Immigration has a number of interpretations and can fit a wide variety of models found in the literature. We apply the results to case studies derived from published models for Chinook salmon (Oncorhynchus tshawytscha) and blowout penstemon (Penstemon haydenii).

Controllability of positive systems by positive inputs arises naturally in applications where both external and internal variables must remain positive for all time. In many applications, particularly in population biology, the need for positive inputs is often overly restrictive. Relaxing this requirement, the notion of positive state controllability of positive systems is introduced. A connection between positive state controllability and positive input controllability of a related system is established and used to obtain Kalman-like controllability criteria. In doing so we aim to encourage further study in this underdeveloped area. © 2013 Elsevier B.V. All rights reserved.

There is a growing interest in predicting the social and ecological contexts that favor the evolution of maternal effects. Most predictions focus, however, on maternal effects that affect only a single character, whereas the evolution of maternal effects is poorly understood in the presence of suites of interacting traits. To overcome this, we simulate the evolution of multivariate maternal effects (captured by the matrix M) in a fluctuating environment. We find that the rate of environmental fluctuations has a substantial effect on the properties of M: in slowly changing environments, offspring are selected to have a multivariate phenotype roughly similar to the maternal phenotype, so that M is characterized by positive dominant eigenvalues; by contrast, rapidly changing environments favor Ms with dominant eigenvalues that are negative, as offspring favor a phenotype which substantially differs from the maternal phenotype. Moreover, when fluctuating selection on one maternal character is temporally delayed relative to selection on other traits, we find a striking pattern of cross-trait maternal effects in which maternal characters influence not only the same character in offspring, but also other offspring characters. Additionally, when selection on one character contains more stochastic noise relative to selection on other traits, large cross-trait maternal effects evolve from those maternal traits that experience the smallest amounts of noise. The presence of these cross-trait maternal effects shows that individual maternal effects cannot be studied in isolation, and that their study in a multivariate context may provide important insights about the nature of past selection. Our results call for more studies that measure multivariate maternal effects in wild populations.

The genetic variance-covariance (G) matrix describes the variances and covariances of genetic traits under strict genetic inheritance. Genetically expressed traits often influence trait expression in another via nongenetic forms of transmission and inheritance, however. The importance of non-genetic influences on phenotypic evolution is increasingly clear, but how genetic and nongenetic inheritance interact to determine the response to selection is not well understood. Here, we use the 'reachability matrix' - a key analytical tool of geometric control theory - to integrate both forms of inheritance, capturing how the consequences of generation-lagged maternal effects accumulate. Building on the classic Lande and Kirkpatrick model that showed how nongenetic (maternal) inheritance fundamentally alters the expected path of phenotypic evolution, we make novel inferences through decomposition of the reachability matrix. In particular, we quantify how nongenetic inheritance affects the distribution (orientation and shape) of ellipses of phenotypic change and how these distributions influence subsequent evolution. This interweaving of phenotypic means and variances accumulates generation by generation and is described analytically by the reachability matrix, which acts as an analogue of G when genetic and nongenetic inheritance both act.

Reactivity (a.k.a initial growth) is necessary for diffusion driven instability (Turing instability). Using a notion of common Lyapunov function we show that this necessary condition is a special case of a more powerful (i.e. tighter) necessary condition. Specifically, we show that if the linearised reaction matrix and the diffusion matrix share a common Lyapunov function, then Turing instability is not possible. The existence of common Lyapunov functions is readily checked using semi-definite programming. We apply this result to the Gierer-Meinhardt system modelling regenerative properties of Hydra, the Oregonator, to a host-parasite-hyperparasite system with diffusion and to a reaction-diffusion-chemotaxis model for a multi-species host-parasitoid community. © 2012 Elsevier Inc.

We use feedback control methods to prove a trichotomy of stability for nonlinear (density dependent) discrete-time population dynamics defined on a natural state space of non-negative vectors. Specifically, using comparison results and small gain techniques we obtain a computable formula for parameter ranges when one of the following must hold: there is a positive, globally asymptotically stable equilibrium; zero is globally asymptotically stable or all solutions with non-zero initial conditions diverge. We apply our results to a model for Chinook Salmon. © 2011 Elsevier B.V. All rights reserved.

Many stage-structured density dependent populations with a continuum of stages can be naturally modeled using nonlinear integral projection models. In this paper, we study a trichotomy of global stability result for a class of density dependent systems which include a Platte thistle model. Specifically, we identify those systems parameters for which zero is globally asymptotically stable, parameters for which there is a positive asymptotically stable equilibrium, and parameters for which there is no asymptotically stable equilibrium.

We demonstrate that deterministic sea wave prediction (DSWP) combined with constrained optimal control can dramatically improve the efficiency of sea wave energy converters (WECs), while maintaining their safe operation. We focus on a point absorber WEC employing a hydraulic/electric power take-off system. Maximizing energy take-off while minimizing the risk of damage is formulated as an optimal control problem with a disturbance input (the sea elevation) and with both state and input constraints. This optimal control problem is non-convex, which prevents us from using quadratic programming algorithms for the optimal solution. We demonstrate that the optimum can be achieved by bang-bang control. This paves the way to adopt a dynamic programming (DP) algorithm to resolve the on-line optimization problem efficiently. Simulation results show that this approach is very effective, yielding at least a two-fold increase in energy output as compared with control schemes which do not exploit DSWP. This level of improvement is possible even using relatively low precision DSWP over short time horizons. A key finding is that only about 1 second of prediction horizon is required, however, the technical difficulties involved in obtaining good estimates necessitate a DSWP system capable of prediction over tens of seconds. © 2012 Elsevier Ltd.

Empirical models are central to effective conservation and population management, and should be predictive of real-world dynamics. Available modelling methods are diverse, but analysis usually focuses on long-term dynamics that are unable to describe the complicated short-term time series that can arise even from simple models following ecological disturbances or perturbations. Recent interest in such transient dynamics has led to diverse methodologies for their quantification in density-independent, time-invariant population projection matrix (PPM) models, but the fragmented nature of this literature has stifled the widespread analysis of transients. We review the literature on transient analyses of linear PPM models and synthesise a coherent framework. We promote the use of standardised indices, and categorise indices according to their focus on either convergence times or transient population density, and on either transient bounds or case-specific transient dynamics. We use a large database of empirical PPM models to explore relationships between indices of transient dynamics. This analysis promotes the use of population inertia as a simple, versatile and informative predictor of transient population density, but criticises the utility of established indices of convergence times. Our findings should guide further development of analyses of transient population dynamics using PPMs or other empirical modelling techniques.

Population dynamics often defy predictions based on empirical models, and explanations for noisy dynamics have ranged from deterministic chaos to environmental stochasticity. Transient (short-term) dynamics following disturbance or perturbation have recently gained empirical attention from researchers as further possible effectors of complicated dynamics. Previously published methods of transient analysis have tended to require knowledge of initial population structure. However, this has been overcome by the recent development of the parametric Kreiss bound (which describes how large a population must become before reaching its maximum possible transient amplification following a disturbance) and the extension of this and other transient indices to simultaneously describe both amplified and attenuated transient dynamics. We apply the Kreiss bound and other transient indices to a data base of matrix models from 108 plant species, in an attempt to detect ecological and mathematical patterns in the transient dynamical properties of plant populations. We describe how life history influences the transient dynamics of plant populations: species at opposite ends of the scale of ecological succession have the highest potential for transient amplification and attenuation, whereas species with intermediate life history complexity have the lowest potential. We find ecological relationships between transients and asymptotic dynamics: faster-growing populations tend to have greater potential magnitudes of transient amplification and attenuation, which could suggest that short- and long-term dynamics are similarly influenced by demographic parameters or vital rates. We describe a strong dependence of transient amplification and attenuation on matrix dimension: perhaps signifying a potentially worrying artefact of basic model parameterization. Synthesis. Transient indices describe how big or how small plant populations can get, en route to long-term stable rates of increase or decline. The patterns we found in the potential for transient dynamics, across many species of plants, suggest a combination of ecological and modelling strategy influences. This better understanding of transients should guide the formulation of management and conservation strategies for all plant populations that suffer disturbances away from stable equilibria. © 2010 the Authors. Journal compilation © 2010 British Ecological Society.

Sensitivity and elasticity analysis of population projection matrices (PPMs) are established tools in the analysis of structured populations, allowing comparison of the contributions made by different demographic rates to population growth. In some commonly used structures of PPM, however, there are mathematically inevitable patterns in the relative sensitivity and elasticity of certain demographic rates. We take a simulation approach to investigate these mathematical constraints for a range of PPM models. Our results challenge some previously proposed constraints on sensitivity and elasticity. We also identify constraints beyond those that have already been proven mathematically and promote them as candidates for future mathematical proof. A general theme among these rules is that changes to the demographic rates of older or larger individuals have less impact on population growth than do equivalent changes among younger or smaller individuals. However, the validity of these rules in each case depends on the choice between sensitivity and elasticity, the growth rate of the population, and the PPM structure used. If the structured population conforms perfectly to the assumptions of the PPM used to model it, the rules we describe represent biological reality, allowing us to prioritize management strategies in the absence of detailed demographic data. Conversely, if the model is a poor fit to the population (specifically, if demographic rates within stages are heterogeneous), such analyses could lead to inappropriate management prescriptions. Our results emphasize the importance of choosing a structured population model that fits the demographics of the population.

1. Perturbation analyses of population projection matrices predict the response of a population's growth rate to changes in lifestage-specific vital rates. Such predictions have been widely used in population management but their reliability remains hotly debated. 2. We grew replicate populations of the water flea Daphnia magna in controlled, density-independent conditions and subjected treatment populations to harvesting of the largest lifestage. We predicted the growth rate of treatment populations using sensitivity analysis (a linear approximation), and transfer function analysis (TFA; which captures nonlinear responses) applied to projection matrix models parameterized from the control populations. 3. When perturbation analyses considered only the direct effect of harvesting on adult survival, the growth rate of harvested populations (averaging 0.051) was significantly overestimated (average of 0.112) by TFA and non-significantly underestimated (average of 0.012) by sensitivity. 4. When the indirect effects of harvesting on other vital rates were accounted for in a structured perturbation, TFA gave accurate predictions (average growth rate of 0.068), while sensitivity gave significant underestimates (average of -0.043). 5. Our results demonstrate two crucial sources of error that may influence predictions of the impacts of demographic perturbations on population dynamics. First, impacts of stage-specific harvesting are inherently nonlinear, hence predictions based on sensitivity must be treated with caution. Second, stage-specific perturbations can change non-target demographic rates, even in the absence of adaptation. 6. Population managers should consider both nonlinear and indirect effects of perturbations when designing management interventions. We encourage the development of methods to assess the robustness of predictions to unforeseen perturbation structures and indirect harvesting impacts.

1. Not all members of natural populations contribute equally to population growth or decline. Populations that are disturbed away from stable stage structure will amplify (i.e. get bigger than expected) and/or attenuate (i.e. get smaller than expected) in the short term. 2. We provide mathematical bounds for the magnitude of this amplification and attenuation, both in terms of absolute population change and population change relative to the long-term rate of population increase. 3. Our results correct an important error in an earlier analysis of transient population amplification, and provide new transient bounds for the analysis of population attenuation. 4. Synthesis and applications. Bounds on transient amplification and attenuation help population managers to gauge 'worst case' and 'best case' scenarios for the response of stage-structured populations to disturbance and management strategies. Such bounds help to create an envelope of possible future population scenarios around the mean, long-term predictions made by eigenvalues and eigenvectors of projection matrix models. Transient amplification, caused by stage structures biased towards reactive life stages, may be exploited by conservation managers wishing to boost population densities in the short term and may be avoided in pest species by stage-specific control strategies. Similarly, transient attenuation should be avoided by conservation managers and exploited by pest managers. © 2008 the Authors.

Sensitivity and elasticity analyses of population projection matrices (PPMs) are established tools in the analysis of structured populations, but they make a linear approximation of the usually nonlinear relationship between population growth and matrix elements. The evaluation of alternative population management interventions may be particularly vulnerable to error if the degree of nonlinearity depends on the element of the PPM that is targeted. The second self-derivative and the self-sensitivity of elasticity can be used to estimate the degree and sign of nonlinearity in sensitivity and elasticity analyses, respectively. Using simulated matrices, we demonstrate that the magnitude and sign of the second self-derivative and the self-sensitivity of elasticity vary systematically, according to the element of the PPM subject to perturbation. Population management prescriptions based on sensitivity and elasticity analysis should always be backed up by nonlinear analyses.

In this paper we present a simple method for identifying life-history perturbations in population projection matrices that yield an accelerating population growth rate. Accelerating growth means that the dependence of the growth rate on the perturbation is convex. Convexity, when the second sensitivity of the growth rate is positive, is calculated using a new formula derived from the transfer function of the perturbed system. This formula is used to explore the relationship between stasis and growth probabilities from stage-structured population projection matrices.

Structured population models are increasingly used in decision making, but typically have many entries that are unknown or highly uncertain. We present an approach for the systematic analysis of the effect of uncertainties on long-term population growth or decay. Many decisions for threatened and endangered species are made with poor or no information. We can still make decisions under these circumstances in a manner that is highly defensible, even without making assumptions about the distribution of uncertainty, or limiting ourselves to discussions of single, infinitesimally small changes in the parameters. Suppose that the model (determined by the data) for the population in question predicts long-term growth. Our goal is to determine how uncertain the data can be before the model loses this property. Some uncertainties will maintain long-term growth, and some will lead to long-term decay. The uncertainties are typically structured, and can be described by several parameters. We show how to determine which parameters maintain long-term growth. We illustrate the advantages of the method by applying it to a Peregrine Falcon population. The U.S. Fish and Wildlife Service recently decided to allow minimal harvesting of Peregrine Falcons after their recent removal from the Endangered Species List. Based on published demographic rates, we find that an asymptotic growth rate lambda > 1 is guaranteed with 5% harvest rate up to 3% error in adult survival if no two-year-olds breed, and up to 11% error if all two-year-olds breed. If a population growth rate of 3% or greater is desired, the acceptable error in adult survival decreases to between 1% and 6% depending of the proportion of two-year-olds that breed. These results clearly show the interactions between uncertainties in different parameters, and suggest that a harvest decision at this stage may be premature without solid data on adult survival and the frequency of breeding by young adults.

Matrix-based models lie at the core of many applications across the physical, engineering and life sciences. In ecology, matrix models arise naturally via population projection matrices (PPM). The eigendata of PPMs provide detailed quantitative and qualitative information on the dynamic behaviour of model populations, especially their asymptotic rates of growth or decline. A fundamental task in modern ecology is to assess the effect that perturbations to life-cycle transition rates of individuals have on such eigendata. The prevailing assessment tools in ecological applications of PPMs are direct matrix simulations of eigendata and linearised extrapolations to the typically non-linear relationship between perturbation magnitude and the resulting matrix eigenvalues. In recent years, mathematical systems theory has developed an analytical framework, called 'Robustness Analysis and Robust Control', encompassing also algorithms and numerical tools. This framework provides a systematic and precise approach to studying perturbations and uncertainty in systems represented by matrices. Here we lay down the foundations and concepts for a 'robustness' inspired approach to predictive analyses in population ecology. We treat a number of application-specific perturbation problems and show how they can be formulated and analysed using these robustness methodologies.

We consider digital input-constrained adaptive and non-adaptive output feedback control for a class of nonlinear systems which arise as models for controlled exothermic chemical reactors. Our objective is set-point control of the temperature of the reaction, with pre-specified asymptotic tracking accuracy set by the designer. Our approach is based on λ-tracking controllers, but in a context of piecewise constant sampled-data output feedbacks and possibly adapted sampling periods. The approach does not require any knowledge of the systems parameters, does not invoke an internal model, is simple in its design, copes with noise corrupted output measurements, and requires only a feasibility assumption in terms of the reference temperature and the input constraints. © 2005 Elsevier B.V. All rights reserved.

In this paper we show that the concept of an implemented semigroup provides a natural mathematical framework for analysis of the infinite-dimensional differential Lyapunov equation. Lyapunov equations of this form arise in various system-theoretic and control problems with a finite time horizon, infinite-dimensional state space and unbounded operators in the mathematical model of the system. The implemented semigroup approach allows us to derive a necessary and sufficient condition for the differential Lyapunov equation with an unbounded forcing term to admit a bounded solution in a suitable space. Whilst our focus is on the differential Lyapunov equation, we show that the same framework is also appropriate for the algebraic version of this equation. As an application we show that the approach can be used to solve a simple decoupling problem arising in optimal control. The problem of infinite time admissibility of the control operator and an infinite-dimensional version of the Lyapunov theorem serve as additional illustrations. © 2005 Taylor & Francis Group Ltd.

It is a well-known principle, for finite-dimensional systems, that applying sampled-and-hold in the feedback loop around a stabilizing state feedback (or dynamic) controller results in a stable sampled-data feedback control system if the sampling period is small enough. The principle extends to infinite-dimensional systems with compact state feedback if either the input operator is bounded or the given state-space system is analytic. In this note, we give an example for which this principle fails but which nevertheless satisfies certain necessary conditions arising in sampled-data control of infinite-dimensional systems.

In this paper we study a particular class of -node recurrent neural networks (RNN's). In the 3-node case we use monotone dynamical systems theory to show, for a well-defined set of parameters, that, generically, every orbit of the RNN is asymptotic to a periodic orbit. Then we investigate whether RNN's of this class can adapt their internal parameters so as to "learn" and then replicate autonomously (in feedback) certain external periodic signals. Our learning algorithm is similar to identification algorithms in adaptive control theory. The main feature of the algorithm is that global exponential convergence of parameters is guaranteed. We also obtain partial convergence results in the -node case.

This tutorial paper explains in detail how mathematical models of dynamical systems described by linear partial differential equations (PDEs) with controls in the boundary conditions can be generalized as abstract boundary control systems and then how the latter lead to abstract linear state equations with an unbounded input operator. We start with two simple examples: the one-dimensional heat equation with control in the Neumann boundary condition and the one-dimensional wave equation with control in the Dirichlet-Neumann boundary condition. These two examples will serve as a motivation for a more general setup for a boundary control system from which we will eventually derive the abstract state equation with an unbounded input operator. We intend to give a comprehensive introduction for control theorists who are not experts in infinite-dimensional systems theory but who want to read and understand papers in this field. © 2000 EUCA.

It is well known that proportional output feedback control can stabilize any relative-degree one, minimum-phase system if the sign of the feedback is correct and the proportional gain is high enough. Moreover, there exists simple adaptation laws for tuning the proportional gain (the so-called high-gain adaptive controllers) which are not based on system identification or plant parameter estimation algorithms or injection of probing signals. If tracking of signals is desired, then these simple controllers are also applicable without invoking an internal model if the tracking error is not necessarily supposed to converge to zero but towards a ball around zero of arbitrarily small but prespecified radius λ > 0. In this note we consider a sampled version of the high-gain adaptive λ-tracking controller. The motivation for sampling arises from the possibility that the output of a system may not be available continuously, but only at discrete time instants. The problem is that the stiffness of the system increases as the proportional gain is increased. Our result shows that adaptive sampling tracking is possible if the product hk of the decreasing sampling rate h and the increasing proportional gain k decreases at a rate proportional to 1/log k. © 1999 Elsevier Science B.V. All rights reserved.

It is well known that proportional output feedback control can stabilize any relative-degree one, minimum-phase system if the sign of the feedback is correct and the proportional gain is high enough. Moreover, there exist simple adaptation laws for tuning the proportional gain (so-called high-gain adaptive controllers) which do not need to know the system and do not attempt to identify system parameters. In this paper the authors consider sampled versions of the high-gain adaptive controller. The motivation for sampling arises from the possibility that the output of a system may not be available continuously, but only at sampled times. The main point of interest is the need to develop techniques for adapting the sampling rate, since the stiffness of the system increases as the proportional gain is increased. Our main result shows that adaptive sampling stabilization is possible if the product hk of the decreasing sampling interval h and the increasing proportional gain k decreases at a rate proportional to 1/log k.

In this note, we consider the question of whether an adaptive controller can converge to a nonadaptive stabilizing controller. Specifically, we show, for a class of back-stepping controllers with adaptive tuning functions, that the set of initial conditions in a state and estimation parameter for which the estimation parameter converges to a parameter which produces a destabilizing controller can have a nonempty interior and, consequently, a nonzero Lebesgue measure. This surprising result is proved by way of a simple example with a quadratic nonlinearity.

Closing the loop around an exponentially stable single-input/single-output regular linear system, subject to a globally Lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller, is known to ensure asymptotic tracking of constant reference signals, provided that: 1) the steady-state of the linear part of the plant is positive; 2) the positive integrator gain is sufficiently small; and 3) the reference value is feasible in a very natural sense. Here lower bounds are derived for the maximal regulating gain for various special cases including systems with nonovershooting step-response and second-order systems with a time-delay in the input or output. The lower bounds are given in terms of open-loop frequency/step response data and the Lipschitz constant of the nonlinearity, and are hence readily obtainable.

In this paper we address the question of whether the open-loop exponential growth rate of a linear system can be improved by a feedback in such a way that this improvement is robust with respect to small delays in the feedback loop. When the input operator is admissible, and the class of possible feedbacks consists of compact operators, we find that if a feedback can improve the exponential growth rate, then it can do so robustly. Furthermore, we find that if the control space is finite dimensional and a bounded feedback cannot be found to improve exponential stability, then a large class of unbounded feedbacks cannot improve the exponential growth rate robustly, even if such feedbacks can improve the exponential growth rate in the absence of delays.

A class of recurrent neural networks is shown to possess a stable limit cycle. A gradient type algorithm is used to modify the parameters of the network so that it learns and replicates autonomously a time varying periodic signal. The results are applied to controlling the repetitive motion of a two-link robot manipulator.

In the sampled-data control literature there are necessary conditions and sufficient conditions for stabilizability of distributed parameter systems by generalized sampled-data control. For finite-dimensional systems the necessary conditions are also known to be sufficient. We show that this equivalence extends to the infinite-dimensional case if the underlying semigroup is analytic. However, for general systems, the necessary conditions are not sufficient, nor are the sufficient conditions necessary. We prove this by a single example with a free parameter - one choice of parameter shows that the necessary conditions are too weak, and another choice shows that the sufficient conditions are too strong. © 1998 Elsevier Science B.V. All rights reserved.

Closing the loop around an exponentially stable single-input single-output regular linear system, subject to a globally Lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller, is shown to ensure asymptotic tracking of constant reference signals, provided that (a) the steady-state gain of the linear part of the plant is positive, (b) the positive integrator gain is sufficiently small, and (c) the reference value is feasible in a very natural sense. The class of actuator nonlinearities under consideration contains standard nonlinearities important in control engineering such as saturation and deadzone.

This paper contains an overview of adaptive control of infinite-dimensional systems without parameter estimation or identification. We describe the main problems and present the most comprehensive results. High-gain, low-gain and switching controllers are considered. The literature is discussed and a number of open problems are posed.

It is well known that if the steady-state gain G(0) of a stable lumped system, with transfer function G(s), is positive, then compensating the system by an integral controller k/s, where k is a gain parameter, leads to a stable closed-loop system which achieves tracking of arbitrary constant reference signals, provided that the gain parameter k is positive and sufficiently small. It is also well known that this result extends to certain classes of differential-delay and distributed parameter systems. The authors derive an adaptive version of the above result for the class of stable lumped systems with output delay, i.e. they show that the gain parameter k can be tuned adaptively, so that tracking is achieved for any system of this class. The resulting adaptive tracking controller is not based on system identification or parameter estimation algorithms, nor is the injection of probing signals required. © IEE, 1997.

Using a frequency-domain analysis, it is shown that the application of a feedback controller of the form k/(z - 1) or kz/(z - 1), where k ∈ IR, to a power-stable infinite-dimensional discrete-time system with square transfer-function matrix G(z) will result in a power-stable closed-loop system which achieves asymptotic tracking of arbitrary constant reference signals, provided that i) all the eigenvalues of G(1) have positive real parts, and ii) the gain parameter k is positive and sufficiently small. Moreover, if G(1) is positive definite, we show how the gain parameter gain k can be tuned adaptively. The resulting adaptive tracking controllers are universal in the sense that they apply to any power-stable system with G(1) > 0; in particular, they are not based on system identification or plant parameter estimation algorithms, nor is the injection of probing signals required. Finally, we apply these discrete-time results to obtain adaptive sampled-data low-gain controllers for the class of regular systems, a rather general class of infinite-dimensional continuous-time systems for which convenient representations are known to exist, both in state space and in frequency domain. We emphasize that our results guarantee not only asymptotic tracking at the sampling instants but also in the sampling interval.

It is well known that closing the loop around an exponentially stable, finite-dimensional, linear, time-invariant plant with square transfer-function matrix G(s) compensated by a controller of the form (k/s)Γ0, where k ∈ ℝ and Γ0 ∈ ℝmxm, will result in an exponentially stable closed-loop system which achieves tracking of arbitrary constant reference signals, provided that (i) all the eigenvalues of G(0)Γ0 have positive real parts and (ii) the gain parameter k is positive and sufficiently small. In this paper we consider a rather general class of infinite-dimensional linear systems, called regular systems, for which convenient representations are known to exist, both in time and in frequency domain. The purpose of the paper is twofold: (i) we extend the above result to the class of exponentially stable regular systems and (ii) we show how the parameters k and Γ0 can be tuned adaptively. The resulting adaptive tracking controllers are not based on system identification or parameter estimation algorithms, nor is the injection of probing signals required.

In this paper we consider the stabilization of continuous-time distributed parameter systems by piecewise polynomial controls. We obtain necessary and sufficient conditions for stabilization by piecewise polynomial controls.

In this paper we consider the problem of integral low-gain control of stable finite-dimensional, single-input single-output systems with positive steady-state gain. The linear plant is subject to a sector-bounded actuator nonlinearity such as saturation or dead-zone. We show that provided the integral gain is small enough then the closed-loop system is globally stable, and arbitrary stepped reference signals are asymptotically tracked by the system output. The value of the gain needed is smaller than that required for closed loop stability without the actuator nonlinearity and is computed graphically from a Nyquist contour plot of the series connection of the system and an integrator.

A class of recurrent neural network configurations that are related to control systems have been introduced. The main result states that this class of recurrent neural networks generates a limit cycle for a broad range of parameter values. When the parameters of the network cross a specified area in the parameter space, the origin becomes an asymptotic equilibrium point. The fact that the recurrent network possesses a stable limit cycle enhances the robustness properties of the network. This stable limit cycle can be achieved by a suitable choice of the parameters of the network and is independent of the initial condition of the network. In this paper, a recurrent neural network is made to learn and replicate a desired periodic signal within a certain class. This learning is achieved by using a gradient descent algorithm to adjust the internal parameters of the recurrent network.

A low gain tracking problem for a general class of open loop stable systems or plants is considered. As a first step, the low-gain tracking problem is outlined, and some preliminaries on abstract linear systems are reviewed. Further, it is shown that for this class of plants, as in the finite dimensional case, integral control, with small enough integral gain will result in an exponentially stable feedback system achieving asymptotic tracking of arbitrary constant reference signals, provided that the initial state is in the domain of the generator of the open-loop system.

This paper considers two classes of infinite-dimensional systems described by an abstract differential equation ẋ(t) = (A + BΔC)x(t), x(0) = x0, on a Hilbert space, where A, B, C are linear, possibly unbounded operators and Δ is an unknown, linear, bounded perturbation. The two classes of systems are defined in terms of properties imposed on the triple {A, B, C}. It is proved that for every Δ the perturbed system {A + EΔF, B, C} inherits all the properties of the unperturbed system {A, B, C} if {A, E, F} and {A, B, C} are in the same class. © 1996 Kluwer Academic Publishers.

A problem of universal adaptive stabilization in Rℝn is approached in a qualitative manner to relate the form of the trajectories in the extended state space ℝn+1 to the root locus of the associated fixed-parameter linear system. Relationships are derived between the values of the limit gain and the initial conditions. Numerical studies are used to illustrate and support these ideas by computation of generic trajectories and attempted computation of certain non-generic possibilities. The implications of the study for more general dynamic situations are outlined.

It is well-known that exponential stabilization of a neutral system with unstable difference operator is only possible by allowing for control laws containing derivative feedback. We show that closed-loop stability of a neutral system with unstable open-loop difference operator obtained by applying a derivative feedback scheme is extremely sensitive to arbitrarily small time delays in the feedback loop.

In this paper we consider two problems in "non-identifier-based," universal adaptive control within the framework of Mårtensson [Adaptive Stabilization, Ph.D. thesis, Lund Institute of Technology, 1986]. In this framework, any linear system stabilizable by constant linear output feedback is adaptively stabilized by an adaptive piecewise-linear output feedback control law. The essential feature we exploit is that of a piecewise-linear output feedback which switches through a set of feedback matrices, with switching controlled by an output-driven differential equation. For each initial condition the state of the system converges to zero and the time-varying gain matrix converges to a "limit gain." in this setting we consider two related problems. The first concerns the sensitivity of closed-loop solutions under small perturbations of the initial data. The second concerns generic properties, with respect to the set of initial conditions, of stabilization by the limit gain. We adopt a topological approach, based on a decomposition of the dynamics of the resultant nonlinear, closed-loop system into a sequence of homeo/diffeomorphisms derived from the switching nature of the dynamics. Using this decomposition we show that the set of initial conditions for which solutions are stable under small perturbations and the limiting gain is stabilizing has full Lebesgue measure and dense interior. This latter result has been conjectured in the literature. The results are illustrated by examples of planar control systems where the sets of initial conditions yielding nonstabilizing limit gians are computed.

We consider the dynamics of closed loop systems arising from universal adaptive stabilization, with scalar feedback gain, of a class of linear systems with n-dimensional state vector. The state and gain dynamics evolve in Rn × R. We analyze qualitatively the dynamics of the closed loop nonlinear system and of an associated linear system - the limit system - obtained by implementing the linear feedback determined by the limit value of the gain. We focus on cases when this limit system can have imaginary axis eigenvalues. We give a classification of the dynamics on the corresponding center manifold, for which we derive analytical and computational results. The analysis is always made complicated by the degeneracy of the vector field defining the adaptation law. Further degenerate behavior, arising when the feedback law involves switching in the sign of the gain, is also considered.

The problem of adaptive stabilization is an important problem in applied control theory and is often approached from an input-output point of view with emphasis on the proof of stability of the states and outputs and convergence of the adaptive gain. However, the parameterization of the solutions in terms of initial state conditions and limit control gains is not fully understood. This paper investigates the dynamics of the basic nonlinear closed-loop adaptive scheme proposed by Byrnes and Willems using a detailed series expansion solution of the special case of closed loop dynamics in ℝ3 projected on to ℝ2. It is demonstrated that the series expansion approach can be used to calculate explicitly initial conditions where the adaptive controllers converge to destabilizing limit gains and (hence to) divide the (resulting) state space into two regions describing two different approaches to the origin. The partition is parameterized by the limiting values of certain control gain parameters. It is verified that parameters leading to unstable ‘limit systems’ lead to projected trajectories (or paths) that lie precisely on the dividing curve. © 1995, Taylor & Francis Group, LLC. All rights reserved.

We analyze the spectrum of a class of abstract partial differential equations with boundary feedback control. If we interpret the feedback as a perturbation then we obtain a robustness radius for spectrum placement. We obtain continuity of the spectrum using this radius and Rouche's theorem. We exploit extensively recent developments in the representation theory for abstract linear systems, especially regular systems. We also analyze ill-posed systems by interchanging the roles of input and output. We apply these results to a wave equation and an ill-posed beam equation. © 1995.

We show that the simple universal adaptive control law u(t)=N(k(t))y(t)=|y(t)|2, with N(k)=(log k)γ cos((log k)σ) and 3γ+σ

In this paper we analyse non-linear, Willems-Byrnes adaptive stabilization of first order, linear systems x ̇(t) = ax(t) + bu(t), x(0) ε{lunate} R. The Willems-Byrnes control law is u = - h(k)x, k ̇ = x2, k(0) ε{lunate} R and for each x(0) = x0, k(0) = k0 ε{lunate} R, x(t) → 0, k(t) → k(∞, x0) < ∞ as t → ∞. There are three cases according to whether h is smooth, piecewise smooth or piecewise constant. For each x0 ε{lunate} R, k(∞, x0) defines a limit pole (a -bh(k(∞, x0))) ε{lunate} R. We find that in all cases {x0 ε{lunate} R|a -bh(k(∞, x0)) < 0} is open and dense. In the piecewise smooth/constant case we solve explicitly the closed loop dynamical equation for x(t) (and k(t)). We find that a -bh(k(∞, x0)) = 0 corresponds to solutions which decay only algebraically. © 1993.

It is shown that the simple adaptive feedback strategy u(t)=1ln k(t) cos√ln(t), k ̇(t)=y(y)2 is a universal adaptive stabilizer for the class of single-input, single-output, finite-dimensional, linear systems which are stabilizable by either negative or positive high-gain feedback. © 1993.

In this note we consider the problem of adaptive tracking for multivariable relative degree one, minimum phase systems. Our approach is in the spirit of Helmke et al. (1990) who considered adaptive tracking of sinusoidal reference signals for single-input, single-output systems. The case of relative degree r, single-input, single-output, minimum-phase systems has been considered in Mareels (1984). The main contribution we make is in establishing that the sufficient conditions for universal adaptive tracking, namely minimum phase, relative degree one and the internal model principle, hold for the suitably precompensated plant. This enables us to apply the techniques of Helmke et al. (1990). © 1991 Oxford University Press.

The main result of this paper is the identification of a class of real systems for which the supremal Hinrichsen-Pritchard complex stability radius is equal to the supremal Hinrichsen-Pritchard real stability radius. For this class (which includes all planar linear systems with non-zero input operator) the remarkable differences, which are known to exist between these two measures of robustness in the absence of control, can be eliminated by state feedback. The respective differences and equivalences are illustrated by examples of single-input uncertain planar control systems. © 1991.

In this paper, we consider the problem of robustness optimization for infinite-dimensional systems with respect to a class of bounded feedback-control laws. Both unbounded inputs and perturbations are allowed, where the well-posedness of the abstract controlled perturbed system is guaranteed by specifying Pritchard-Salamon-type conditions on the system parameters. The solution of the problem is determined via a doubly parameterized weak algebraic Riccati equation. © 1991 Oxford University Press.

The subject of this paper is a geometric approach to the robustness optimization problem for uncertain finite dimensional linear systems. For the structurally perturbed stabilizable pair (A, B), we show that optimally robust feedback stabilization is possible via the class of feedbacks parameterizing (A, B) feedback invariant subspaces of codimension Rk B. A direct consequence of this geometric approach is a reduction in dimension of the optimization problem. © 1990.

In this paper we consider linear systems which are stable and examine the robustness of this property. The perturbations are assumed to have unbounded structure and we determine the smallest perturbation which destroys stability. We begin with an abstract analysis of the problem, but in later sections specialize to three types of systems: those whose mild solutions are given in terms of 1. (a) strongly continuous semigroups, 2. (b) resolvent operators-integrodifferential systems, 3. (c) evolution operators-time varying systems. For B a Banach space, L2(t0, T; B) denotes the space of all square integrable functions defined on [t0, T] with values in B. For B1, B2 Banach spaces, [t0, T] denotes the space of strongly measurable functions f(·):[t0, T]→L(B1, B2) with ess sup ∥f(·)∥≤∞.B-1(t0, ∞;L(B1, B2)) denotes the space of strongly measurable functions f(·): [t0, ∞) → L(B1, B2) with ∥f(t)∥. 0 and k(·) ε{lunate} L1(t0, ∞). © 1989.

We outline a new approach to the design of robust low order compensators for high order systems. Our approach is formulated in the state space and is based on associating a structured perturbation problem with the compensator design. Analysis of this structured perturbation problem via stability radii (Hinrichsen and Pretchard [5], Hinrichsen et al. [4] and Pritchard and Townley [6]) results in criteria for our compensator design. © 1988.

This paper considers the problem of extending the standard linear time-invariant state space model from a finite dimensional to an infinite dimensional state space setting. Emphasis is placed on constructing a framework wide enough to include models with boundary control and point sensing. The concept of a 'solution' to the model is examined and an analysis of linear state variable feedback is performed. A simple one-dimensional heat diffusion example with point measurement of temperature and point influx of heat is used to illustrate and motivate the general theory.

We introduce the distance of a stable system from the set of unstable systems and determine a formula by which it can be calculated. Examples of delay equations and partial differential equations are used to illustrate the result.