This dissertation progressively addresses research problems related to the trajectory optimization for fuel cell powered UAVs, from propulsion system model development, to optimal trajectory analyses and optimal trajectory applications. A dynamic model of a fuel cell powered UAV propulsion system is derived by combining a fuel cell system dynamic model, an electric motor dynamic model, and a propeller performance model. The influence of the fuel cell system dynamics on the optimal trajectories of a fuel cell powered UAV is investigated in two phases. In the first phase, the optimal trajectories of a fuel cell powered configuration and that of a conventional gas powered configuration are compared for point-to-point trajectory optimization problems with different performance index functions. In the second phase, the influence of the fuel cell system parameters on the optimal fuel consumption cost of the minimum fuel point-to-point optimal trajectories is investigated. This dissertation also presents two applications for the minimum fuel point-to-point optimal trajectories of a fuel cell powered UAV: three-dimensional minimum fuel route planning and path generation, and fuel cell system size optimization with respect to a UAV mission.

Asymptotic stability is a key notion of system stability for controlled dynamical systems as it guarantees that the system trajectories are bounded in a neighborhood of a given isolated equilibrium point and converge to this equilibrium over the infinite horizon. In some applications, however, asymptotic stability is not the appropriate notion of stability. For example, for systems with a continuum of equilibria, every neighborhood of an equilibrium contains another equilibrium and a nonisolated equilibrium cannot be asymptotically stable. Alternatively, in stabilization of spacecraft dynamics via gimballed gyroscopes, it is desirable to find state- and output-feedback control laws that guarantee partial-state stability of the closed-loop system, that is, stability with respect to part of the system state. Furthermore, we may additionally require finite-time stability of the closed-loop system, that is, convergence of the system's trajectories to a Lyapunov stable equilibrium in finite time. The Hamilton-Jacobi-Bellman optimal control framework provides necessary and sufficient conditions for the existence of state-feedback controllers that minimize a given performance measure and guarantee asymptotic stability of the closed-loop system. In this research, we provide extensions of the Hamilton-Jacobi-Bellman optimal control theory to develop state-feedback control laws that minimize nonlinear-nonquadratic performance criteria and guarantee semistability, partial-state stability, finite-time stability, and finite-time partial state stability of the closed-loop system.

The Earth harbors around 300 000 plant species. The rich and complex environment provided by plants is considered a key factor for the extraordinary diversity of the terrestrial fauna by, for example, providing food and shelter. This thesis contributes to the understanding of these questions by investigating how the interplay of physiology, demography, and evolution gives rise to variation and diversity in fundamental plant traits. This will help us answer questions such as: How has this amazing diversity of plant species emerged? Which mechanisms maintain diversity? How are plant strategies and plant diversity influenced by variations in the environment? A plant faces multiple problems to survive and reproduce successfully. These problems can be modeled by considering traits, trade-offs and a fitness measure. For example: How to maximize growth rate, while maximizing structural stability? I will investigate four plant models in order to understand the function of plants, and mechanisms promoting diversity.  Paper I: We study how annual plants with and without growth constraints should optimize their flowering time when productivity or season length changes. With a dynamic ontogenetic growth model and optimal control theory we prove that a bang-bang reproductive control is optimal under constrained growth and constant mortality rate. We find that growth constraints can flip the direction of optimal phenological response for increasing productivity. The reason is that the growth rate of vegetative mass saturates at high productivity and therefore it is better to flower earlier and take advantage of a longer reproductive period. If season length extends equally both in the beginning and the end of the season, growth constraints control the direction of the optimal response as well. Our theory can help explaining phenological patterns along productivity gradients, and can be linked to empirical observations made on a calendar scale. Paper II: We introduce a new measure of tree crown-rise efficiency based on the loss of biomass of the tree during growth. The more mass the tree looses during growth, the less crown-rise efficient it is. Top-heavy shapes loose more biomass than bottom-heavy shapes. Light-use efficiency is defined as the mean light assimilation of the leaves in the crown times the ratio of leaf mass and total mass. We then study the trade-off between light-use efficiency to crown-rise efficiency for tree crown shapes. We assume that the total tree mass is constant, and a constant vertical light gradient represent the shading from a surrounding forest. We find large differences in crown shapes at intermediate vertical light gradient, when both self-shading and mean-field shading are important, suggesting light-use vs crown-rise efficiency as a new trade-off that can explain tree diversity. Our crown-rise efficiency measure could easily be integrated into existing forest models. Paper III: We extend an evolutionary tree crown model, where trees with different heights compete for light, with drought-induced mortality rates depending on ground-water availability and the depth of an optional taproot. The model does not include competition for ground water. Our model explains how ground-water availability can shape plant communities, when taproot and non-taproot strategies can coexist, and when only one of these strategies can persist. We investigate how emerging plant diversity varies with water table depth, soil water gradient and drought-induced mortality rate. The taproot enables plants to reach deep water, thus reducing mortality, but also carries a construction cost, thus inducing a trade-off. We find that taproots maintain plant diversity under increasing drought mortality, and that taproots evolve when groundwater is accessible at low depths. There are no viable strategies at high drought mortality and deep water table. Red Queen evolutionary dynamics appear at intermediate drought mortality in mixed communities with and without taproots, as the community never reaches a final evolutionarily stable composition. Paper IV: We extend a size-structured plant model, with self-shading and two evolving traits, crown top-heaviness and crown width-to-height ratio. The model allows us to identify salient trade-offs for the crown shape. The most important trade-off for top-heaviness is light-use efficiency vs crownrise efficiency, and the most important trade-off for width-to-height ratio is self-shading vs branch costs. We find that when the two traits coevolve; the outcome is a single common evolutionarily stable strategy (ESS), far away from the highest net primary production (NPP). When only sun angle is decreasing with increasing latitude both the crown width-to-height ratio and crown top-heaviness decrease. However, when light response in addition to the sun angle decreases with increasing latitude, the crown width-to-height ratio is nearly invariant of latitude except at low site productivity when the ratio decreases with latitude. Top-heaviness is always decreasing with increasing latitude. Finally, we find that crown top-heaviness increases with the NPP or leaf-area index (LAI) at ESS, but crown width-to-height ratio is maximal at an intermediate NPP or LAI. / Artikel I: Arters reproduktionsframgång (fitness), till exempel antal avkommor eller frön som produceras under livet, är ofta avgörande för huruvida de är evolutionärt framgångsrika eller inte. Här undersöker vi hur ettåriga växter med eller utan tillväxtbegränsningar ska optimera sin blomningstid när produktivitet eller säsongslängd ändras. Det är optimalt att gå direkt från tillväxt till blomning när tillväxten är begränsad och dödligheten är konstant. Vid ökad produktivitet sker blomningen tidigare med tillväxtbegränsningar men senare utan tillväxtbegränsningar, vilket beror på att med tillväxtbegränsningar ökar den vegetativa massan långsamt. Därför är det bättre att blomma tidigare och ta tillvara på en längre reproduktionsperiod. Vi får samma resultat om säsongslängden ökar lika mycket i början och slutet av säsongen. Vår teori kan bidra till att förutsäga blomningstider vid produktivitetsförändringar och säsongsförändringar. Artikel II: Tillväxten hos träd kan begränsas av brist på ljus, vatten, och näring, men också genom förlust av grenar. Vi introducerar ett nytt mått på tillväxteffektiviteten hos trädkronor baserat på förlust av biomassa under trädets tillväxt. Ju mer massa trädet förlorar under tillväxt, desto mindre tillväxteffektiva är de. Topptunga former förlorar mer biomassa än bottentunga former. Vi studerar avvägningar mellan ljuseffektivitet och tillväxteffektivitet för trädformer, där ljuseffektiviteten definieras som medelljusupptaget för löven i kronan. Vi antar en konstant totalmassa, och en statisk vertikal skuggning som representerar skuggningen från en omgivande skog. Vi hittar stora skillnader i kronformer vid en medelhög skuggning, då både självskuggningen och medelskuggningen har betydelse. Vårt mått för tillväxteffektivitet kan enkelt integreras i existerande skogsmodeller. Studien visar att avvägningar mellan tillväxteffektivitet och ljuseffektivitetet kan vara viktig för mångfalden av trädformer i en skog. En överraskande upptäckt är att konformade eller sfäriska trädkronor aldrig är effektiva, men däremot timglasformade kronor. Artikel III: Växter kan försvara sig på olika sätt mot torka, till exempel genom att rulla ihop bladen eller genom att reproducera tidigare och därigenom undvika uttdragen torka. Här undersöker vi fördelarna med en pålrot vid torka. En pålrot är en rot som växer nedåt för att nå djupliggande grundvatten. Vi utvidgar en evolutionär modell av trädkronor med grundvatten och en pålrot, där träd med olika höjd konkurrerar om ljus. Det finns ingen konkurrens om vatten. Vi undersöker hur mångfalden hos träden beror på vattendjup, vattengradient och dödlighet orsakad av torka. Med hjälp av pålroten kan träden nå djupt vatten och därigenom minska dödligheten, men den medför också en kostnad, så en avvägning måste göras. Vi ser att pålrötter upprätthåller mångfalden hos växterna vid ökad mortalitet, och att pålrötter uppstår när grundvattnet är grunt. Det finns inga strategier som kan överleva om grundvattnet är djupt och dödligheten är hög. Vår modell kan förklara hur grundvatten kan förändra sammansättningen på trädsamhällen, när träd med och utan pålrot kan samexistera, och under vilka förutsättningar endast en av strategierna förväntas dominera. Artikel IV: Träd som växer upp i en skog måste konkurrera med andra träd om ljus, framförallt större träd. Detta ger upphov till en asymmetrisk ljuskonkurrens, där de små träden hämmas av större träd. Små träd har därmed små chanser att överleva utom då skogen nyligen störts och det öppnas upp en glänta. Vid denna ljuskonkurrens kan man anta att trädkronans form har stor betydelse för trädets framgång. Frågan är hur de evolutionärt fördelaktiga kronformerna beror på latituden och produktiviteten. Vi antar att latituden påverkar solens genomsnittliga vinkel och ljusrespons. Vi utvidgar en storleksstrukturerad trädmodell med självskuggning där två evolverande egenskaper beskriver kronans topptyngd och bredd. Med modellen kan vi undersöka vilka strategiska avvägningar som bestämmer om kronans form blir konkurrenskraftig. En topptung krona har högt ljusupptag eftersom det finns mest ljus högt upp i grenverket. Å andra sidan har den en låg tillväxteffektivitet eftersom topptunga kronor måste tappa mycket grenar för att behålla sin form. En bred krona har en låg självskuggning eftersom bladen är utspridda. Å andra sidan har den höga kostnader för de långa grenar som krävs. Vi finner att när dessa egenskaper evolverar tillsammans så finns endast en evolutionärt stabil strategi (ESS), långt från den högsta nettoproduktionen. När endast solvinkeln minskar med ökande latitud minskar både kronans bredd och topptyngd, men när både solvinkel och ljusrespons minskar med ökande latitud så är bredden nästan oförändrad utom vid låg produktivitet då den minskar med latituden. Kronans topptyngd minskar alltid med latituden. Slutligen ser vi hur kronans topptyngd alltid ökar med nettoproduktionen vid ESS, medan kronans bredd har ett maxium för ett mellanvärde hos nettoproduktionen vid ESS.

Renewable energy is key to a sustainable future. However, the intermittency of most renewable sources and lack of sufficient storage in the current power grid means that reliable integration of significantly more renewables will be a challenging task. Moreover, increased integration of renewables not only increases uncertainty, but also reduces the fraction of traditional controllable generation capacity that is available to cope with supply-demand imbalances and uncertainties. Less traditional generation also means less rotating mass that provides very short term, yet very important, kinetic energy storage to the system and enables mitigation of the frequency drop subsequent to major contingencies but before controllable generation can increase production. Demand, on the other side, has been largely regarded as non-controllable and inelastic in the current setting. However, there is strong evidence that a considerable portion of the current and future demand, such as electric vehicle load, is flexible. That is, the instantaneous power delivered to it needs not to be bound to a specific trajectory. In this thesis, we focus on harnessing demand flexibility as a key to enabling more renewable integration and cost reduction. We start with a data driven analysis of the potential of flexible demands, particularly plug-in electric vehicle (PEV) load. We first show that, if left unmanaged, these loads can jeopardize grid reliability by exacerbating the peaks in the load profile and increasing the negative correlation of demand with wind energy production. Then, we propose a simple local policy with very limited information and minimal coordination that besides avoiding undesired effects, has the positive side-effect of substantially increasing the correlation of flexible demand with wind energy production. Such local policies could be readily implemented as modifications to existing "grid friendly" charging modes of plug-in electric vehicles. We then propose improved localized charging policies that counter balance intermittency by autonomously responding to frequency deviations from the nominal frequency and show that PEV load can offer a substantial amount of such ancillary services. Next, we consider the case where real-time prices are employed to provide incentives for demand response. We consider a flexible load under such a pricing scheme and obtain the optimal policy for responding to stochastic price signals to minimize the expected cost of energy. We show that this optimal policy follows a multi-threshold form and propose a recursive method to obtain these thresholds. We then extend our results to obtain optimal policies for simultaneous energy consumption and ancillary service provision by flexible loads as well as optimal policies for operation of storage assets under similar real-time stochastic prices. We prove that the optimal policy in all these cases admits a computationally efficient form. Moreover, we show that while optimal response to prices reduces energy costs, it will result in increased volatility in the aggregate demand which is undesirable. We then discuss how aggregation of flexible loads can take us a step further by transforming the loads to controllable assets that help maintain grid reliability by counterbalancing the intermittency due to renewables. We explore the value of load flexibility in the context of a restructured electricity market. To this end, we introduce a model that economically incentivizes the load to reveal its flexibility and provides cost-comfort trade-offs to the consumers. We establish the performance of our proposed model through evaluation of the price reductions that can be provided to the users compared to uncontrolled and uncoordinated consumption. We show that a key advantage of aggregation and coordination is provision of "regulation" to the system by load, which can account for a considerable price reduction. The proposed scheme is also capable of preventing distribution network overloads. Finally, we extend our flexible load coordination problem to a multi-settlement market setup and propose a stochastic programming approach in obtaining day-ahead market energy purchases and ancillary service sales. Our work demonstrates the potential of flexible loads in harnessing renewables by affecting the load patterns and providing mechanisms to mitigate the inherent intermittency of renewables in an economically efficient manner. / text

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本文以最適控制理論證明，獎勵再生能源產出之政策，應以再生能源淨能源產出做為獎勵的基礎，而非現行以再生能源總能源產出做為獎勵基礎之模式。這裡的淨能源產出，指的是再生能源廠商生產出之再生能源，減去生產再生能源時所用的能源投入。本文首先將社會最適情況下的總能源產出分別與以淨能源產出和總能源產出為獎勵政策基礎之價格與數量政策下之總能源產出作比較，提出獎勵再生能源產出的政策，需以淨能源產出做為獎勵的基礎，而非現行以總能源產出作為基礎的政策，接著，在以淨能源產出為基礎的政策下，探討環境外部性與防治成本以及研究發展的議題。 / By the optimal control theory, this paper proves that policies on encouraging the production of renewable energy should be based on its net output or net energy instead of on its gross output or gross energy. Here net energy is defined as the surplus of renewable energy output minus energy input from its production. This paper first compares the optimal gross output of the renewable energy under the social optimal condition with the gross outputs under the price-based policy instrument and the quantity-based policy instrument based on net energy output and gross energy output, respectively, suggesting that policy instruments used to encourage the production of renewable energy should be based on its net output instead of on its gross output. Finally, it probes the cases of environmental externality and R&D based on the net output of renewable energy.

The problem addressed in this research is the in-flight generation of trajectories for autonomous unmanned aircraft, which requires a method of generating pseudo-optimal trajectories in near-real-time, on-board the aircraft, and without external intervention. The focus of this research is the enhancement of a particular inverse dynamics direct method that is a candidate solution to the problem. This research introduces the following contributions to the method. A quaternion-based inverse dynamics model is introduced that represents all orientations without singularities, permits smooth interpolation of orientations, and generates more accurate controls than the previous Euler-angle model. Algorithmic modifications are introduced that: overcome singularities arising from parameterization and discretization; combine analytic and finite difference expressions to improve the accuracy of controls and constraints; remove roll ill-conditioning when the normal load factor is near zero, and extend the method to handle negative-g orientations. It is also shown in this research that quadratic interpolation improves the accuracy and speed of constraint evaluation. The method is known to lead to a multimodal constrained nonlinear optimization problem. The performance of the method with four nonlinear programming algorithms was investigated: a differential evolution algorithm was found to be capable of over 99% successful convergence, to generate solutions with better optimality than the quasi- Newton and derivative-free algorithms against which it was tested, but to be up to an order of magnitude slower than those algorithms. The effects of the degree and form of polynomial airspeed parameterization on optimization performance were investigated, and results were obtained that quantify the achievable optimality as a function of the parameterization degree. Overall, it was found that the method is a potentially viable method of on-board near- real-time trajectory generation for unmanned aircraft but for this potential to be realized in practice further improvements in computational speed are desirable. Candidate optimization strategies are identified for future research.

This thesis considers applications of gradient-based optimization algorithms to the design and control of some mechanics systems. The material distribution approach to topology optimization is applied to design two different acoustic devices, a reactive muffler and an acoustic horn, and optimization is used to control a ball pitching robot. Reactive mufflers are widely used to attenuate the exhaust noise of internal combustion engines by reflecting the acoustic energy back to the source. A material distribution optimization method is developed to design the layout of sound-hard material inside the expansion chamber of a reactive muffler. The objective is to minimize the acoustic energy at the muffler outlet. The presence or absence of material is represented by design variables that are mapped to varying coefficients in the governing equation. An anisotropic design filter is used to control the minimum thickness of materials separately in different directions. Numerical results demonstrate that the approach can produce mufflers with high transmission loss for a broad range of frequencies. For acoustic devices, it is possible to improve their performance, without adding extended volumes of materials, by an appropriate placement of thin structures with suitable material properties. We apply layout optimization of thin sound-hard material in the interior of an acoustic horn to improve its far-field directivity properties. Absence or presence of thin sound-hard material is modeled by a surface transmission impedance, and the optimization determines the distribution of materials along a “ground structure” in the form of a grid inside the horn. Horns provided with the optimized scatterers show a much improved angular coverage, compared to the initial configuration. The surface impedance is handled by a new finite element method developed for Helmholtz equation in the situation where an interface is embedded in the computational domain. A Nitschetype method, different from the standard one, weakly enforces the impedance conditions for transmission through the interface. As opposed to a standard finite-element discretization of the problem, our method seamlessly handles both vanishing and non-vanishing interface conditions. We show the stability of the method for a quite general class of surface impedance functions, provided that possible surface waves are sufficiently resolved by the mesh. The thesis also presents a method for optimal control of a two-link ball pitching robot with the aim of throwing a ball as far as possible. The pitching robot is connected to a motor via a non-linear torsional spring at the shoulder joint. Constraints on the motor torque, power, and angular velocity of the motor shaft are included in the model. The control problem is solved by an interior point method to determine the optimal motor torque profile and release position. Numerical experiments show the effectiveness of the method and the effect of the constraints on the performance.

Les premiers résultats en analyse convexe ne nécessitant aucune condition de qualification datent à peu près d'une quinzaine d'années et constituent le début de l'analyse convexe séquentielle. Ils concernaient essentiellement: la somme d'un nombre fini de fonctions convexes, la composition avec une application vectorielle convexe, et les problèmes de programmation mathématique convexe. Cette thèse apporte un ensemble de contributions à l'analyse convexe séquentielle. La première partie de la thèse est consacrée à l'obtention sans condition de qualification de règles de calcul sous-differentiel exprimées séquentiellement. On considère les cas suivants:l'enveloppe supérieure d'une famille quelconque de fonctions convexes semi-continues inférieurement définies sur un espace de Banach; une fonctionnelle intégrale convexe générale définie sur un espace de fonctions intégrales;la somme continue (ou intégrale) de fonctions convexes semi-continues inférieurement définies sur un espace de Banach séparable. Dans la deuxième partie on établit sans hypothèse de qualification sur les données du problème, des conditions nécessaires et suffisantes d'optimalité séquentielle pour divers types de problèmes d'optimisation et de contrôle optimal discret ou continu. / The first results in convex analysis without any qualificationcondition have been established fifteen years ago, and one may say thatsequential convex analysis began with those results. They essentially concerned:The finite sum of convex functions, the composition with a vectorvaluedconvex mapping, and convex mathematical programming. The firstpart of this dissertation provides several contibutions to sequential convexanalysis. The following cases are considered: the upper envelop of a familyof lower semicontinuous convex functions; the integral functional overan integral space; the continuous sum of lower semicontinuous convex functions.In the second part, necessary and sufficient optimality conditions areestablished in sequential form for many types of programming problems anddicrete or continuous optimal control problems.

Approved for public release; distribution is unlimited / Reissued 30 May 2017 with correction to student's affiliation on title page. / Autonomous vehicle teams have great potential in a wide range of maritime sensing applications, including mine countermeasures (MCM). A key enabler for successfully employing autonomous vehicles in MCM missions is motion planning, a collection of algo-rithms for designing trajectories that vehicles must follow. For maximum utility, these algorithms must consider the capabilities and limitations of each team member. At a minimum, they should incorporate dynamic and operational constraints to ensure trajectories are feasible. Another goal is maximizing sensor performance in the presence of uncertainty. Optimal control provides a useful frame-work for solving these types of motion planning problems with dynamic constraints and di_x000B_erent performance objectives, but they usually require numerical solutions. Recent advances in numerical methods have produced a general mathematical and computational framework for numerically solving optimal control problems with parameter uncertainty—generalized optimal control (GenOC)— thus making it possible to numerically solve optimal search problems with multiple searcher, sensor, and target models. In this dissertation, we use the GenOC framework to solve motion planning problems for di_x000B_erentMCMsearch missions conducted by autonomous surface and underwater vehicles. Physics-based sonar detection models are developed for operationally relevant MCM sensors, and the resulting optimal search trajectories improve mine detection performance over conventional lawnmower survey patterns—especially under time or resource constraints. Simulation results highlight the flexibility of this approach for optimal mo-tion planning and pre-mission analysis. Finally, a novel application of this framework is presented to address inverse problems relating search performance to sensor design, team composition, and mission planning for MCM CONOPS development.