The object of this thesis is the study of some new financial models. The common feature is that they all involve optimal decisions. Some of the decisions take the form of a control and we enter the theory of stochastic optimal control and of Hamilton-Jacobi-Bellman (HJB) equations. Other decisions are "binary" and we deal with the theory of optimal stopping and free boundary problems. Throughout the thesis we will prefer a heuristic and intuitive approach to a too technical one which could hide the underlying ideas. In the first part we introduce the reader to option pricing, HJB equations and free boundary problems, and we review briefly the use of these mathematical tools in finance. The second part of the thesis deals with passport options. The pricing of these exotic options involves stochastic optimal control and free boundary problems. Finally, in the last part we study the end-of-the-year bonus for traders: how to optimally reward a trader?

Vehicle behaviour near the limit of adhesion is studied using linear optimal . control techniques and relatively simple vehicle models. Both time-invariant and time-varying approaches are used. Controllability is applied as a post-processing tool to analyse the resultant vehicle behaviour. First, a 4WS controller is developed using a linear time-invariant method, with a reference model control structure. Two handling objectives are defined, which are thought to provide predictable dynamics. Advantages of using a reference model control are clearly shown. With a developed control structure, it is shown that the prescribed target dynamics is achieved, provided tyre forces are available. It is also found that the controller is robust to small changes in the various vehicle parameter values. As a next step, time-varying modelling approach was used in order to better represent the vehicle operating conditions through the various dynamic range, including the limit of adhesion. An iterative vehicle path optimisation problem is formulated using a linear time-varying control approach. The validity of the optimisation method is studied against the steady-state simulation result at the limit of adhesion. It is shown that the method is capable of finding a trajectory in the vicinity of the friction limit, where the front tyres are used fully whilst retaining some margin at the rears. However, a couple of Issues are discovered. First, due to the quadratic nature of the road geometry cost function, the trajectory could get locked if the vehicle runs very close to the edge of the road. Hence, the . optimisation needs to be formulated such that the level of "optimality" on the trajectory remains consistent throughout the manoeuvre at each iteration. Secondly, it is found that inappropriate control demands are produced if the system matrix becomes poorly conditioned near the limit. This results in optimisation failure. In order to understand the mechanism of this failure, controllability of linear timevarying system was analysed and its properties were discussed in detail. First, the calculation methods of the controllability gramian matrix are investigated and some practical limitations are found. The gramian matrix is then used to define an open loop control sequence. It is found that the damping of the system has a significant influence on the control strategy. Subsequently, "the moving controllability window of a fixed time period" is found to provide the most relevant information of changing dynamics through the time. The study showed that the failure of the optimisation in the vicinity of the friction limit was indeed due to lack of controllability and the optimisation method itself was functioning correctly. The vehicle path optimisation problem is then extended to include longitudinal dynamics, enabling simulation of more general manoeuvres. The single corner simulation showed that the optimisation converges to an "out-in-out" path, with iterative solution improving continuously in a first order manner. Simulations with various controller settings showed that the strategy is reasonably robust provided that the changes in parameter settings are kept within a reasonable magnitude. It is also confirmed that the optimisation is able to drive a vehicle close to the limit under different types of operations required, i.e. braking, cornering and acceleration. The study was then performed with slightly more complex road geometry in order to investigate if the· optimisation is capable of prioritising certain· part of the manoeuvre in order to achieve better overall result. Unfortunately, this problem could not be solved successfully. The optimisation concentrated on the latter part of the manoeuvre as it had higher sensitivity to the final cost. This resulted in clearly sub-optimal overall performance. Finally, relatively simple study is conducted to investigate the correlation between various vehicle settings and optimisation results. Using the path optimisation problem formulation, iris found that the more oversteer vehicles are able to achieve better· result with more margin left in rear tyre force capacity. The handling objective functions used for the 4 WS controller is also calculated for the resultant trajectories. It is found that the neutral steer cost had a strong correlation, whereas the linearity cost showed no noticeable correlation. The controllability analysis was applied on the various vehicle settings using step steer simulation. It showed that more understeering vehicle retains higher controllability throughout the dynamics range. It is also found that higher inertia gives better controllability near the limit, however, it gives less controllability at more moderate operating conditions.

Nearly all imaginable human activities rest on a context-appropriate dynamic control of the flow of retinal data into the nervous system via eye movements. The brain’s task is to move the eyes so as to exert intelligent predictive control over the informational content of the retinal data stream. An intelligent oculomotor controller would first model future contingent upon each possible next action in the oculomotor repertoire, then rank-order the repertoire by assigning a value v(a,t) to each possible action a at each time t, and execute the oculomotor action with the highest predicted value each time. We present a striking evidence of such an intelligent neural control of human eyes in a laboratory task of visual search for a small target camouflaged by a natural-like stochastic texture, a task in which the value of fixating a given location naturally corresponds to the expected information gain about the unknown location of the target. Human searchers behave as if maintaining a map of beliefs (represented as probabilities) about the target location, updating their beliefs with visual data obtained on each fixation optimally using the Bayes Rule. On average, human eye movement patterns appear remarkably consistent with an intelligent strategy of moving eyes to maximize the expected information gain, but inconsistent with the strategy of always foveating the currently most likely location of the target (a prevalent intuition in the existing theories). We derive principled, simple, accurate, and robust mathematical formulas to compute belief and information value maps across the search area on each fixation (or time step). The formulas are exact expressions in the limiting cases of small amount of information extracted, which occurs when the number of potential target locations is infinite, or when the time step is vanishingly small (used for online control of fixation duration). Under these circumstances, the computation of information value map reduces to a linear filtering of beliefs on each time step, and beliefs can be maintained simply as running weighted averages. A model algorithm employing these simple computations captures many statistical properties of human eye movements in our search task. / text

In racing, it is useful to analyze vehicle performance and driving strategies to achieve the best result possible in competitions. This is often done by simulations and test driving. In this thesis optimal control is used to examine how a racing car should be driven to minimize the lap time. This is achieved by calculating the optimal racing line at various tracks. The tracks can have arbitrary layout and consist of corners with non-constant radius. The road can have variable width. A four wheel vehicle model with lateral and longitudinal weight transfer is used. To increase the performance of the optimization algorithm, a set of additional techniques are used. The most important one is to divide tracks into smaller overlapping segments and find the optimal line for each segment independently. This turned out to be useful when the track is long. The optimal racing line is found for various tracks and cars. The solutions have several similarities to real driving techniques. The result is presented as driving instructions in Racer, a car simulator.

More strict control quality requirements are raised to the synthesis of modern algorithmic control systems which can not be satisfied by using classical methods of systems synthesis. Also, the usage of them sometimes becomes impossible, e.g. in cases when a mathematical model of the control object is described by means of complex differential equations or in cases when the model itself is unknown. By applying the suggested synthesis methods of quasi-optimal and variable structure systems as well as algorithms, it is possible to avoid disadvantages common with classical synthesis methods.

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The motor control of anthropomorphic robotic systems is a challenging computational task mainly because of the high levels of redundancies such systems exhibit. Optimality principles provide a general strategy to resolve such redundancies in a task driven fashion. In particular closed loop optimisation, i.e., optimal feedback control (OFC), has served as a successful motor control model as it unifies important concepts such as costs, noise, sensory feedback and internal models into a coherent mathematical framework. Realising OFC on realistic anthropomorphic systems however is non-trivial: Firstly, such systems have typically large dimensionality and nonlinear dynamics, in which case the optimisation problem becomes computationally intractable. Approximative methods, like the iterative linear quadratic gaussian (ILQG), have been proposed to avoid this, however the transfer of solutions from idealised simulations to real hardware systems has proved to be challenging. Secondly, OFC relies on an accurate description of the system dynamics, which for many realistic control systems may be unknown, difficult to estimate, or subject to frequent systematic changes. Thirdly, many (especially biologically inspired) systems suffer from significant state or control dependent sources of noise, which are difficult to model in a generally valid fashion. This thesis addresses these issues with the aim to realise efficient OFC for anthropomorphic manipulators. First we investigate the implementation of OFC laws on anthropomorphic hardware. Using ILQG we optimally control a high-dimensional anthropomorphic manipulator without having to specify an explicit inverse kinematics, inverse dynamics or feedback control law. We achieve this by introducing a novel cost function that accounts for the physical constraints of the robot and a dynamics formulation that resolves discontinuities in the dynamics. The experimental hardware results reveal the benefits of OFC over traditional (open loop) optimal controllers in terms of energy efficiency and compliance, properties that are crucial for the control of modern anthropomorphic manipulators. We then propose a new framework of OFC with learned dynamics (OFC-LD) that, unlike classic approaches, does not rely on analytic dynamics functions but rather updates the internal dynamics model continuously from sensorimotor plant feedback. We demonstrate how this approach can compensate for unknown dynamics and for complex dynamic perturbations in an online fashion. A specific advantage of a learned dynamics model is that it contains the stochastic information (i.e., noise) from the plant data, which corresponds to the uncertainty in the system. Consequently one can exploit this information within OFC-LD in order to produce control laws that minimise the uncertainty in the system. In the domain of antagonistically actuated systems this approach leads to improved motor performance, which is achieved by co-contracting antagonistic actuators in order to reduce the negative effects of the noise. Most importantly the shape and source of the noise is unknown a priory and is solely learned from plant data. The model is successfully tested on an antagonistic series elastic actuator (SEA) that we have built for this purpose. The proposed OFC-LD model is not only applicable to robotic systems but also proves to be very useful in the modelling of biological motor control phenomena and we show how our model can be used to predict a wide range of human impedance control patterns during both, stationary and adaptation tasks.

Recent developments in quantum information processing have presented new and interesting ways to perform advanced algorithms and improve signal to noise ratios. Examples of these include optimal control theory pulse generation algorithms and the usage of high Q-factor resonators. However, these developments are blocked by current spectrometer designs. This thesis details the design and testing of a new spectrometer with sufficient accuracy, bandwidth, and control to implement these advances. The proposed solution is to use an FPGA-based development board together with custom computer software. This gives access to high-speed analogue inputs and outputs, as well as digital output pins. The spectrometer is then used in two X-band electron spin resonance experiments, showing how the advantages of the system allow for superior results to that possible with the previous equipment. In addition, the setup is used in a Nitrogen Vacancy (NV) system where a rabi experiment is performed.

Quantum information is a growing field showing exciting possibilities for computational speed-up and communications. For the successful implementation of quantum computers, high-precision control is required to reach fault-tolerant thresholds. Control of quantum systems pertains to the manipulation of states and their evolution. In order to minimize the effects of the environment on the control operations of the qubits, control pulses should be made time-optimal. In addition, control pulses should be made robust to noise in the system, dispersion in energies and coupling elements, and uncertain parameters. In this thesis, we examine a robust time-optimal gradient ascent technique which is used to develop controls of the motional degrees of freedom for an ensemble of neutral atoms in a one-dimensional optical lattice in the high dispersion regime with shallow trapping potentials. As such, the system is analyzed in the delocalized basis. The system is treated as an ensemble of atoms with a range of possible quasimomenta across the first Brillouin zone. This gives the ensemble of Hamiltonians, indexed by the quasimomenta, a distinct spectra in their motional states and highly inhomogeneous control Hamiltonians. Thus, the optical lattice is seen as a model system for robust control. We find optimized control pulses designed using an ensemble modification of gradient-ascent pulse engineering robust to any range of quasimomentum. We show that it is possible to produce rotation controls with fidelities above 90\% for half of the first Brillouin zone with gate times in the order of several free oscillations. This is possible for a spectrum that shows upwards of 75\% dispersion in the energies of the band structure. We also show that NOT controls for qubit rotations on the entire Brillouin zone fidelities above 99\% were possible for 0.6\% dispersion in energies. The gate times were also in the order of several free oscillations. It is shown that these solutions are palindromic in time due to phase differences in some of the energy couplings when comparing one half of the Brillouin zone to another. We explore the limits of discretized sampling of a continuous ensemble for control.

Research Doctorate - Doctor of Philosophy (PhD) / This thesis studies control systems with communication constraints. Such constraints arise due to the fact that practical control systems often use non-transparent communication links, i.e., links subject to data-rate constraints, random data-dropouts or random delays. Traditional control theory cannot deal with such constraints and the need for new tools and insights arises. We study two problems: control with average data-rate constraints and control over analog erasure channels with i.i.d. dropout profiles. When focusing on average data-rate constraints, it is natural to ask whether information theoretic ideas may assist the study of networked control systems. In this thesis we show that it is possible to use fundamental information theoretic concepts to arrive at a framework that allows one to tackle performance related control problems. In doing so, we show that there exists an exact link between control systems subject to average data-rate limits, and control systems which are closed over additive i.i.d. noise channels subject to a signal-to-noise ratio constraint. On the other hand, in the case of control systems subject to i.i.d. data-dropouts, we show that there exists a second-order moments equivalence between a linear feedback system which is interconnected over an analog erasure channel, and the same system when it is interconnected over an additive i.i.d. noise channel subject to a signal-to-noise ratio constraint. From the results foreshadowed above, it follows that the study of control systems closed over signal-to-noise ratio constrained additive i.i.d. noise channels is a task of relevance to many networked control problems. Moreover, the interplay between signal-to-noise ratio constraints and control objectives is an interesting issue in its own right. This thesis starts with such a study. Then, we use the resultant insights to address performance issues in control systems subject to either average data-rate constraints or i.i.d. data-dropouts. Our approach shows that, once key equivalences are exposed, standard control intuition and synthesis machinery can be used to tackle networked control problems in an exact manner. It also sheds light into fundamental results in the literature and gives (partial) answers to several previously open questions. We believe that the insights in this thesis are of fundamental importance and, to the best of the author's knowledge, novel.