Library of Congress Cataloging-in-Publication Data HUll. David G.Optimal control theory for applications / David G. Hull. p. cm. -(Mechanical engineering series) Includes bibliographical references and index.

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This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).

An integrated sequence stratigraphic study based on outcrop, core and wireline log data documents the combined impact of Cretaceous eustacy and oceanic anoxic events on carbonate shelf morphology and facies distributions in the northern Gulf of Mexico. The diverse facies and abundant data of the Comanche platform serve as a nearly complete global reference section and provide a sensitive record of external processes affecting Cretaceous platform development. Regional cross-sections across the shoreline to shelf-margin profile provide a detailed record of mixed carbonate-siliciclastic strata for the Hauterivian to lower Campanian stages (ca 136 to 80 Ma). The study window on the slowly subsiding passive margin allows the stratigraphic response to external forcing mechanisms to be isolated from regional structural processes. Three second-order supersequences comprised of eight composite sequences are recognized in the Valanginian-Barremian, the Aptian-Albian and the Cenomanian-Campanian. The Valanginian-Barremian supersequence transitioned from a siliciclastic ramp to carbonate rimmed shelf and is a product of glacial ice accumulation and melting, as well as variable rates of mid-ocean ridge volcanism. The Aptian-Albian supersequence chronicles the drowning and recovery of the platform surrounding oceanic anoxic events 1a and 1b. The Cenomanian-Campanian supersequence similarly documents shelf drowning following oceanic anoxic event 1d, after which the platform evolved to a deep-subtidal system consisting of anoxic/dysoxic shale and chalk in the time surrounding oceanic anoxic event 2. Each period of oceanic anoxia is associated with composite sequence maximum flooding, termination of carbonate shelf sedimentation and deposition of condensed shale units in distally steepened ramp profiles. Composite sequences unaffected by oceanic anoxic events consist of aggradational to progradational shelves with an abundance of grain-dominated facies and shallow-subtidal to intertidal environments. Because they are products of eustacy and global oceanographic processes, the three supersequences and most composite sequences defined in the south Texas passive margin are recognizable in other carbonate platforms and published eustatic sea-level curves. Maraca Lisure A g u a rd ie n te Apon Rio Negro M o rr o n e d i P a c e n tr o Va lle de ll' In fe rn o T r e G r o t t e Cima Dell Murelle

a b s t r a c tThe ubiquity of carbonate platforms throughout the Cretaceous Period is recognized as a product of high eustatic sea-level and a distinct climatic optimum induced by rapid sea-floor spreading and elevated levels of atmospheric carbon-dioxide. Notably, a series of global oceanic anoxic events (OAEs) punctuate this time-interval and mark periods of significantly reduced free oxygen in the world's oceans. The best records of these events are often from one-dimensional shelf or basin sections where only abrupt shifts between oxygenated carbonates and anoxic shales are recorded. The Comanche Platform of central Texas provides a unique opportunity to study these events within a well-constrained stratigraphic framework in which their up-dip and down-dip sedimentologic effects can be observed and the recovery of the platform to equilibrium states can be timed and understood. Stable isotope data from whole cores in middle Hauterivian through lower Campanian mixed carbonate-siliciclastic strata are used to construct a 52-myr carbon isotope reference profile for the northern Gulf of Mexico. Correlation of this composite curve to numerous global reference profiles permits identification of several anoxic events and allows their impact on platform architecture and facies distribution to be documented. Oceanic anoxic events 1a, 1b, 1d, and 2 occurred immediately before, after, or during shale deposition in the Pine Island Member, Bexar Member, Del Rio Formation, and Eagle Ford Group, respectively. Oceanic anoxic event 3 corresponds to deposition of the Austin Chalk Group. Platform drowning on three occasions more closely coincided with globally recognized anoxic sub-events such as the Fallot, Albian-Cenomanian, and MidCenomanian events. This illustrates that the specific anoxic event most affecting a given carbonate platform varied globally as a function of regional oceanographic circumstances.Using chemo-and sequence-stratigraphic observations, a four-stage model is proposed to describe the changing facies patterns, fauna, sedimentation accumulation rates, platform architectures, and relative sea-level trends of transgressive-regressive composite sequences that developed in response to global carbon-cycle perturbations. The four phases of platform evolution include the equilibrium, crisis, anoxic, and recovery stages. The equilibrium stage is characterized by progradational shelf geometries and coralrudist phototrophic faunal assemblages. Similar phototrophic fauna typify the crisis stage; however, incipient biocalcification crises of this phase led to retrogradational shelf morphologies, transgressive facies patterns, and increased clay mineral proportions. Anoxic stages of the Comanche Platform were coincident with back-ground deposition of organic-rich shale on drowned shelves and heterotrophic fauna dominated by oysters or coccolithophorids. Eustatic peaks of this stage were of moderate amplitude (~30 m), yet relative sea-level rises were greatly enhanced by reduced sedimentation rates. In the recovery sta...