With the increase of world population and industrial growth of developing countries, demand for energy, in particular electric power, has gone up at an unprecedented rate over the last decades. To meet the demand, electric power generation by use of fossil fuel has increased enormously thereby producing increased quantity of greenhouse gases. This contributes more and more to atmospheric pollution, which climate scientists believe can adversly affect the global climate, as well as health and the welfare of the world population. In view of these issues, there is global awareness to look for alternate sources of energy such as natural gas, hydropower, wind, solar, geothermal and biomass. It is recognized that this requires replacement of existing infrastructure with new systems, which cannot be achieved overnight. Optimal control theory has been widely used in diverse areas of physical sciences, medicine, engineering and economics. The main motivation of this thesis is to use this theory to find the optimum strategy for integration of all currently available renewable energy sources with the existing electric power generating systems. The ultimate goal is to eliminate fossil fuels. Eight main energy sources namely, Coal, Petroleum, Natural Gas, Conventional Hydro, Wind, Solar, Geothermal and Biomass are considered in a dynamic model. The state of the dynamic model represents the level of energy generation from each of the sources. Different objective functions are proposed in this thesis. These range from meeting the desired target level of power generation from each of the available sources at the end of a given plan period, to reducing the implementation and investment costs; from minimizing the production from polluted energy sources to meeting the electricity demand during a whole plan period. Official released data from the U.S. Energy Information Administration have been used as a case study. Based on real life data and the mathematics of optimal control theory, we present an optimal policy for integration of renewable energy sources to the national power grid.

Three problems are being investigated in this thesis. The first two relate to the modelling and analysis of martensitic phase transitions, while the third is concerned with some mathematical tools used in this setting. After a short introduction (Chapter 1) and overviews of the calculus of variations and martensitic phase transformations (Chapter 2), the research part of this thesis is divided into three chapters. We show in Chapter 3 that for the two wells $\mathrm{SO}(3)U$ and $\mathrm{SO}(3)V$ to be rank-one connected, where the $3\times 3$ symmetric positive definite $U$ and $V$ have the same eigenvalues, it is necessary and sufficient that $\mathrm{det}(U-V)=0$, a result that does not hold in higher dimensions. Using this criterion and a result of Gurtin, formulae for the twinning plane and the shearing vector are obtained, which yield an extremely simple condition for the occurrence of so-called compound twins. Our results also provide a simple classification of the twinning mode of the two wells by looking at the crystallographic properties of the eigenvectors of the difference $U-V$. As an illustration, we apply our results to cubic-to-tetra gonal,tetragonal-to-monoclinic and cubic-to-monoclinic transitions. Chapter 4 focuses on the mathematical analysis of biaxial loading experiments in martensite, more particularly on how hysteresis relates to metastability. These experiments were carried out by Chu and James and their mathematical treatment was initiated by Ball, Chu and James. Experimentally it is observed that a homogeneous deformation $y_1(x)= U_1x$ is the stable state for `small' loads while $y_2(x)=U_2x$ is stable for `large' loads. A model was proposed by Ball, Chu and James which, for a certain intermediate range of loads, predicts crucially that $y_1(x)=U_1x$ remains metastable i.e., a local - as opposed to global - minimiser of the energy). This result explains convincingly the hysteresis that is observed experimentally. It is easy to get an upper bound for when metastability finishes. However, it was also noticed that this bound (the Schmid Law) may not be sharp, though this required some geometric conditions on the sample. In this chapter, we rigorously justify the Ball-Chu-James model by means of De Giorgi's $\Gamma$-convergence, establish some properties of local minimisers of the (limiting) energy and prove the metastability result mentioned above. An important part of the chapter is then devoted to establishing which geometric conditions are necessary and sufficient for the counter-example to the Schmid Law to apply. Finally, Chapter 5 investigates the structure of the solutions to the two-well problem. Restricting ourselves to the subset $K=\{H\}\cup \mathrm{SO}(2)V \subset\mathrm{SO}(2)U\cup\mathrm{SO}(2)V$ and assuming the two wells to be compatible, we let $T_1$ and $T_2$ denote the two (not necessarily distinct) twins of $H$ on $\mathrm{SO}(2)V$ and ask the following question: if $\nu_x$ is a non-trivial gradient Young measure almost everywhere supported on $K$, does its support necessarily contain a pair of rank-one connected matrices on a set of positive measure? Although we do not provide a solution for the general case, we show that this is true whenever (a) $\nu_x\equiv \nu$ is homogeneous and $\mathrm{supp}\nu\cap \mathrm{SO}(2)V$ is connected, (b) $\nu_x\equiv \nu$ is homogeneous and $T_1=T_2$ i.e., when the two wells are trivially rank-one connected) or (c) $\mathrm{supp}\nu_x \subset F$ a.e., for some finite set $F$. We also establish a more general case provided a strong `rigidity' conjecture holds.

<p>This thesis presents a method for solving partial differential equations (PDEs) using articial neural networks. The method uses a constrained backpropagation (CPROP) approach for preserving prior knowledge during incremental training for solving nonlinear elliptic and parabolic PDEs adaptively, in non-stationary environments. Compared to previous methods that use penalty functions or Lagrange multipliers,</p><p>CPROP reduces the dimensionality of the optimization problem by using direct elimination, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic</p><p>and parabolic PDEs with changing parameters and non-homogeneous terms. The computational complexity analysis shows that CPROP compares favorably to existing methods of solution, and that it leads to considerable computational savings when subject to non-stationary environments.</p><p>The CPROP based approach is extended to a constrained integration (CINT) method for solving initial boundary value partial differential equations (PDEs). The CINT method combines classical Galerkin methods with CPROP in order to constrain the ANN to approximately satisfy the boundary condition at each stage of integration. The advantage of the CINT method is that it is readily applicable to PDEs in irregular domains and requires no special modification for domains with complex geometries. Furthermore, the CINT method provides a semi-analytical solution that is infinitely differentiable. The CINT method is demonstrated on two hyperbolic and one parabolic initial boundary value problems (IBVPs). These IBVPs are widely used and have known analytical solutions. When compared with Matlab's nite element (FE) method, the CINT method is shown to achieve significant improvements both in terms of computational time and accuracy. The CINT method is applied to a distributed optimal control (DOC) problem of computing optimal state and control trajectories for a multiscale dynamical system comprised of many interacting dynamical systems, or agents. A generalized reduced gradient (GRG) approach is presented in which the agent dynamics are described by a small system of stochastic dierential equations (SDEs). A set of optimality conditions is derived using calculus of variations, and used to compute the optimal macroscopic state and microscopic control laws. An indirect GRG approach is used to solve the optimality conditions numerically for large systems of agents. By assuming a parametric control law obtained from the superposition of linear basis functions, the agent control laws can be determined via set-point regulation, such</p><p>that the macroscopic behavior of the agents is optimized over time, based on multiple, interactive navigation objectives.</p><p>Lastly, the CINT method is used to identify optimal root profiles in water limited ecosystems. Knowledge of root depths and distributions is vital in order to accurately model and predict hydrological ecosystem dynamics. Therefore, there is interest in accurately predicting distributions for various vegetation types, soils, and climates. Numerical experiments were were performed that identify root profiles that maximize transpiration over a 10 year period across a transect of the Kalahari. Storm types were varied to show the dependence of the optimal profile on storm frequency and intensity. It is shown that more deeply distributed roots are optimal for regions where</p><p>storms are more intense and less frequent, and shallower roots are advantageous in regions where storms are less intense and more frequent.</p> / Dissertation

This thesis presents an online path planning algorithm developed for unmanned vehicles in charge of autonomous border patrol. In this Pursuit-Evasion game, the unmanned vehicle is required to capture multiple trespassers on its own before any of them reach a target safe house where they are safe from capture. The problem formulation is based on Isaacs’ Target Guarding problem, but extended to the case of multiple evaders. The proposed path planning method is based on Rapidly-exploring random trees (RRT) and is capable of producing trajectories within several seconds to capture 2 or 3 evaders. Simulations are carried out to demonstrate that the resulting trajectories approach the optimal solution produced by a nonlinear programming-based numerical optimal control solver. Experiments are also conducted on unmanned ground vehicles to show the feasibility of implementing the proposed online path planning algorithm on physical applications.

This thesis presents an online path planning algorithm developed for unmanned vehicles in charge of autonomous border patrol. In this Pursuit-Evasion game, the unmanned vehicle is required to capture multiple trespassers on its own before any of them reach a target safe house where they are safe from capture. The problem formulation is based on Isaacs’ Target Guarding problem, but extended to the case of multiple evaders. The proposed path planning method is based on Rapidly-exploring random trees (RRT) and is capable of producing trajectories within several seconds to capture 2 or 3 evaders. Simulations are carried out to demonstrate that the resulting trajectories approach the optimal solution produced by a nonlinear programming-based numerical optimal control solver. Experiments are also conducted on unmanned ground vehicles to show the feasibility of implementing the proposed online path planning algorithm on physical applications.

As technological advances allow us to peer into and beyond microscopic phenomena, new theoretical developments are necessary to facilitate this exploration. In particular, the potential afforded by utilizing quantum resources promises to dramatically affect scientific research, communications, computation, and material science. Control theory is the field dedicated to the manipulation of systems, and quantum control theory pertains to the manoeuvring of quantum systems. Due to the inherent sensitivity of quantum ensembles to their environment, time-optimal solutions should be found in order to minimize exposure to external sources. Furthermore, the complexity of the Schr\"odinger equation in describing the evolution of a unitary operator makes the analytical discovery of time-optimal solutions rare, motivating the development of numerical algorithms. The seminal result of classical control is the Pontryagin Maximum Principle, which implies that a restriction to bounded control amplitudes admits two classifications of trajectories: bang-bang and singular. Extensions of this result to generalized control problems yield the same classification and hence arise in the study of quantum dynamics. While singular trajectories are often avoided in both classical and quantum literature, a full optimal synthesis requires that we analyze the possibility of their existence. With this in mind, this treatise will examine the issue of time-optimal quantum control. In particular, we examine the results of existing numerical algorithms, including Gradient Ascent Pulse Engineering and the Kaya-Huneault method. We elaborate on the ideas of the Kaya-Huneault algorithm and present an alternative algorithm that we title the Real-Embedding algorithm. These methods are then compared when used to simulate unitary evolution. This is followed by a brief examination on the conditions for the existence of singular controls, as well possible ideas and developments in creating geometry based numerical algorithms.

Covariance assignment theory, introduced in the late 1980s, provided the only means to directly control the steady-state error properties of a linear system subject to Gaussian white noise and parameter uncertainty. This theory, however, does not extend to control of the transient uncertainties and to date there exist no practical engineering solutions to the problem of directly and optimally controlling the uncertainty in a linear system from one Gaussian distribution to another. In this thesis I design a dual-mode Receding Horizon Controller (RHC) that takes a controllable, deterministic linear system from an arbitrary initial covariance to near a desired stationary covariance in finite time. The RHC solves a sequence of free-time Optimal Control Problems (OCP) that directly controls the fundamental solution matrices of the linear system; each problem is a right-invariant OCP on the matrix Lie group GLn of invertible matrices. A terminal constraint ensures that each OCP takes the system to the desired covariance. I show that, by reducing the Hamiltonian system of each OCP from T?GLn to gln? x GLn, the transversality condition corresponding to the terminal constraint simplifies the two-point Boundary Value Problem (BVP) to a single unknown in the initial or final value of the costate in gln?. These results are applied in the design of a dual-mode RHC. The first mode repeatedly solves the OCPs until the optimal time for the system to reach the de- sired covariance is less than the RHC update time. This triggers the second mode, which applies covariance assignment theory to stabilize the system near the desired covariance. The dual-mode controller is illustrated on a planar system. The BVPs are solved using an indirect shooting method that numerically integrates the fundamental solutions on R4 using an adaptive Runge-Kutta method. I contend that extension of the results of this thesis to higher-dimensional systems using either in- direct or direct methods will require numerical integrators that account for the Lie group structure. I conclude with some remarks on the possible extension of a classic result called Lie?s method of reduction to receding horizon control.

Συστήματα μαγνητικών ταινιών χρησιμοποιούνται ευρέως ως μέσα αποθήκευσης και αποκατάστασης δεδομένων. Λόγω του χαμηλού τους κόστους αναλογικά των σκληρών δίσκων συνεχίζουν να προτιμούνται σε μεγάλο πλήθος εφαρμογών όπως δευτερεύουσες συσκευές αποθήκευσης, εγγραφής και επεξεργασίας ηχητικών δεδομένων,βιντεοκάμερες κ.ά. Σε περιπτώσεις όπου θέλουμε να επεξεργαστούμε μια μαγνητοταινία για γρήγορη μεταφορά ή εγγραφή δεδομένων θα πρέπει -η ταινία- να μεταφέρεται σε υψηλή ταχύτητα από το μηχανισμό του συστήματος. Η ταχύτητα αυτή να διατηρείται σταθερή και να βρίσκεται σε συγχρονισμό ε την κεφαλή ανάγνωσης/εγγραφής προς αποφυγή σφαλμάτων μεταφοράς και αναγνώρισης δεδομένων, αναπηδήσεων και χρονικών σφαλμάτων. Σε πιθανά σφάλματα η επαναφορά της ταινίας στο επιθυμητό σημείο αποτελεί τη λύση για την αποκατάσταση του σφάλματος. Η επιτάχυνση και επιβράδυνση που δέχεται η ταινία σε κάθε γρήγορη αναζήτηση δεδομένων προκαλούν τη μεγαλύτερη επιβάρυνση σε αυτή. Ειδικά η συχνή επανατοποθέτηση της ταινίας στο ίδιο ση είο λόγω της βαθμιαίας εξασθένισης του σήματος ανάγνωσης. Σκοπός είναι ο σχεδιασμός ενός μηχανισμού εταφοράς της ταινίας ο οποίος προσφέρει μια ικανοποιητική τάση στην ταινία όπως και ταχύτητα μεταφοράς αυτής. / Optimal control and estimation of a magnetic tape-drive system.

Made available in DSpace on 2014-06-11T19:30:27Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-06-10Bitstream added on 2014-06-13T20:00:34Z : No. of bitstreams: 1 santos_ild_dr_sjrp.pdf: 578124 bytes, checksum: f4eef9516c9a15e6505cc44be029c3a7 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Consideramos nesta tese inclusões dinâmicas vetoriais em escalas de tempo e estendemos para esta classe o resultado de compacidade das trajetórias que, p or sua vez, foi combinado com soluções de Euler, tamb ém intro duzidas nesta tese, para garantir a existência de trajetória qua ndo o camp o vetorial da inclusão dinâmica é semicontínuo sup erior. Porém, quando o camp o vetorial da inclusão dinâmica é semicontínuo inferior, é possível obter uma solução da inclusão dinâmica por meio de uma equação dinâmica cujo campo vetorial é contí nuo. Este campo é um a seleção da multifunção que de ne o camp o vetorial. Consideramos também um problema de controle ó ti mo e mostramos que este possui tra jetória admissível ótima sempre que o conjunto de soluções admissíveis é não-vazio e o campo satisfaz as condições de mensurabilidade, convexidade, compacidade e crescimento linear. Além disso, estendemos o Lema de Filippov para a classe de inclusões dinâmicas para mostrar que é possível fazer uma equivalência total do problema de controle no paradigma de inclusão dinâmica com o problema de controle padrão / We consider in this t hes is vectors dynamic inclusions on time scales and extended for this class the result of compactness of the trajectories which, in turn, was combined with Euler solutions, also introduced i n this thesis, t o ensure t he existence of trajectory when the vector eld of t he dynamic inclusi on is upper semicontinuous. However, when the vector eld of the dynamic inclusion is lower semicontinuous, it is possible to obtain a solution of the dynamic inclusion through a dynamic equation whose vector eld is continuous. This eld is a selection of the multifunction de ning the vector eld. We also consider an optimal control problem and we showed that it has an optimal admissible trajectory whenever the admissible solutions set is nonempty and the eld sati s es measurability conditions, conve-xity, compactness and linear growth. Furthermore, we extend the Filippov's Lemma for the class of dynamic inclusions to show that it is possible to do a ful l equivalence of the control problem in the paradigm of dynamic inclusion with the standard control problem

Many applications in engineering, computer science and economics involve mixed-integer optimal control problems. Solving these problems in real-time is a challenging task because of the explosion of integer combinations to evaluate. This thesis focuses on the development of new algorithms for mixed-integer programming with an emphasis on optimal control problems of fast dynamical systems with discrete controls. The first part proposes two reformulations to reduce the computational complexity. The first reformulation avoids integer variables altogether. By considering a sequence of switched dynamics, we analyze the switching time optimization problem. Even though it is a continuous smooth problem, it is non-convex and the cost function and derivatives are hard to compute. We develop a new efficient method to compute the cost function and its derivatives. Our technique brings up to two orders of magnitude speedups with respect to state-of-the-art tools. The second approach reduces the number of integer decisions. In hybrid model predictive control (MPC) the computational complexity grows exponentially with the horizon length. Using approximate dynamic programming (ADP) we reduce the horizon length while maintaining good control performance by approximating the tail cost offline. This approach allows, for the first time, the application of such control techniques to fast dynamical systems with sampling times of only a few microseconds. The second part investigates embedded branch-and-bound algorithms for mixed-integer quadratic programs (MIQPs). A core component of these methods is the solution of continuous quadratic programs (QPs). We develop OSQP, a new robust and efficient general-purpose QP solver based on the alternating direction method of multipliers (ADMM) and able, for the first time, to detect infeasible problems. We include OSQP into a custom branch-and-bound algorithm suitable for embedded systems. Our extension requires only a single matrix factorization and exploits warm-starting, thereby greatly reducing the number of ADMM iterations required. Numerical examples show that our algorithm solves small to medium scale MIQPs more quickly than commercial solvers.