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Analysis of Closed-loop Dynamical Systems Using ReLU NNs
No full textThis thesis develops a framework for analyzing the stability and invariance properties of closed-loop dynamical systems whose dynamics are derived from neural networks with rectified linear units (ReLUs). The ReLU neural network defines continuous piecewise affine (PWA) functions, which makes them ideal for analysis using polyhedral geometry and optimization techniques. To begin, we introduce two exact algorithms for enumerating polytopic regions of affine behavior induced by ReLU networks: a half-space-based method that checks the feasibility of activation patterns and a geometric vertex-based method. Using these methods, we are able to perform precise region-wise analysis, and these methods serve as the foundation for the rest of the thesis. Following this, we present an automated method for verifying Lyapunov stability in systems with closed-loop dynamics derived from ReLU networks. By converting Lyapunov conditions into quantifier-free linear constraints over each region of the domain, we enable efficient verification and synthesis of Lyapunov functions based on ReLU neural networks. We propose two refinement strategies to enhance the expressiveness of the candidate Lyapunov function: one based on the sign variation of the Lyapunov derivative, and the other based on directional variations in the vector field across region boundaries. To analyze invariance, we develop a barrier-function-based framework aligned with the PWA structure of ReLU-modeled systems. We introduce the use of a leaky ReLU function as the K-infinity function in the barrier condition to improve flexibility while retaining tractability. This facilitates the synthesis of valid barrier certificates by enabling compatibility with non-smooth formulations. To expand the certified invariant set, we propose the Union of Invariant Sets (UIS) method, in which multiple ReLU-based barrier functions are merged into a single certificate. To reduce conservatism from full-domain analysis, we introduced SEROAISE (Sequential Estimation of Region of Attraction via Invariant Set Estimation), a framework that first certifies an invariant set using a barrier function and then limits Lyapunov verification to that region. Our method of expanding the certified invariant set is Non-Uniform Growth of Invariant Sets (NUGIS), which selectively includes boundary points that satisfy the invariance condition. Lastly, the framework integrates a control policy refinement step that modifies ReLU-based closed-loop dynamics to obtain a larger invariant set. Together, these contributions provide a unified, automation-friendly methodology for certifying stability and forward invariance in systems with closed-loop dynamics derived from ReLU. The framework is validated through numerical experiments across a range of nonlinear control scenarios
Unreasonable Behavior or Unreasonable Results: The Seventh Circuit’s Approach to Deliberate Indifference and Its Implications
No full textNo One Mourns the Mid-Majors: Can Mid-Major Schools Survive Under the House v. NCAA Settlement Agreement, or is College Athletics Destined to Downsize?
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Mosaic Grazing - Animal Selectivity in Tropical Savannas of Northern Australia
No full textIn set stocked, heavily grazed, unburnt, native grass/sown legume pastures in the tropical savannas of northern Australia, animals were found to graze very selectively. This contrasts with the more uniform light grazing usually occurring in the extensive, lightly grazed native grasslands which are normally burnt annually. Once heavy selective grazing of an area was initiated, this grazing pressure was maintained, while ungrazed areas grew rank and unacceptable. Although three different sized paddocks were used to give different stocking rates, the area heavily grazed was the same at all stocking rates. In unfertilised paddocks, plants died within two years when selectively grazed, resulting in bare soil. The area affected by heavy grazing then extended each year. In fertilised pastures, although the perennial grasses died out in the heavily grazed patches, these areas were then colonised by the sown legume (Stylosanthes spp.). Given the economic and legislative pressures for intensification of grazing in these northern savanna pastures, preferential grazing may exacerbate the instability of the grazing-intolerant native pastures of the region and future management may have to be designed to minimize selective grazing
Preliminary Report on Experiment of Winter Rotational Grazing of Local Cattle
No full textIn winter, grass growth declines sharply and animal\u27s demand increases with the passage of time. In Dec., affected by frostbite and low temperatures, pasture output drops greatly and cannot meet the needs of cattle. For this reason, in Nanning, late Dec. is considered the winter grazing in a critical period (WGCP). Huts and shelters should be put up in the areas where animals should continue to graze throughout the winter. It is relatively economical to give animal supplementary feeding started in late Dec
Light Interception, Productive Structure and Production of the Phragmites comminis Grassland in the Delta of Nakdong River in Korea
No full textInterception and absorption of sunlight, seasonal changes of productive structure, productivity, and standing crops of the Phragmites communisgrassland which was the pure stand in the delta of Nakdong river in Korea, were investigated in 1977,1982 and 1983.
The lighfpenetrating through the stratified canopies is decreased in sigmoid pattern. The cumulative leaf area of a definite space in a certain plant is the same as the growth of leaf area of the canopy at that time. When the leaf area grows according to the logistic equation, vertical accumulation of leaf area in a grassland is given by the general dynamic equation. Light interception at the maximum growth rate of leaf area is given by this equation(l). The developed productive structure of live materials was occupied by the upper part of photosynthetic system and the under part of nonphotosynthetic system. Therefore, the pattern of the light interception was changed from a sigmoid pattern to an exponential pattern. P. communis contributed to the standing crops of live materials of above ground parts was in excess of 3,660g/m2 throughout the growing season. The peak standing crop ,3 ,399. 0g/m2, was reached in September. The net productivity of this grassland was in excess of 142g/m2 for one growing season except for Octber. Annual net productions of the reed grassland in 1977, 1982 and 1983 were 4.10, 4.25 and 4.47kg/m2/year, respectively. The maximum leaf area indices were 10.3,11.0 and 10.4, respectively. These data were very high in the natural grassland
Studies on the Mountain Pastures Classification and Class Evaluation of Guichou, China
No full textThe pastures of the mountain range in Guichou are a secondary succession type, which has been shaped since the forest was destroyed. They were formed by forest regressive succession; from forest to bushwood, and with further degeneration from bushes to pastures. Therefore, in the author\u27s opinion, it is appropriate to use the secondary succession series method of geobotany to classify the mountain range pastures in southern China. This method can objectively describe the origin and developing stages of this range type, the features of the vegetation succession series in subtropical areas, and express their economic value and utilization, which form an important basis for determining the mountain range pastures. The class evaluation of pastures resources is based on the grass quality, grass food value, pasture productivity, mountain slope and water-source condition etc., and consequently, the mountain pastures are divided into 3 grades, grade 1 the pasture type, grade 2 the pasture class, and grade 3 the pasture group
Quickening Snow Melting in Early Spring with Black Powder in Snowy District
No full textExperiments were conducted during 1977-1980 to quiken snow melting in early spring with three kinds of black powders at 450 m a.s.l.,Tsunan-machi, Niigata Prefecture. Snow melting materials involved blast furnace slag, di-amido lime and calcium silicate and their effects on snow melting were compared with each other at various application rates on snow surface in late March every year. Snow melting was earlier with 600 kg/ha di-amido lime resulting from lower albedo of 25 % and it could quicken the snow-off by 10-13 days as compared to the control plot. This early snow melting induced the higher soil temperature and increased the herbage production at the first cut of the grass mixture and rye twofold over the control plot. Using a broadcaster o.n a sledge pulled by snow mobile was efficient in applying the di-amido lime