We consider problems of optimal motion control for multi-agent systems where assignments as well as motions have costs. In particular, we consider a demand distribution and a distribution of resource agents, which require and provide support respectively. We formulate a time-varying assignment problem which trades off two typically competing costs, namely an assignment cost which depends on distances between resource and demand, and a motion cost associated with moving resource agents to locations with lower assignment costs. We use the formalism of optimal transport theory, and the Wasserstein distance in particular, to capture assignment cost, while motion cost is captured by vehicular velocities over time. Both particle and continuum models for large-scale systems are considered, leading to infinite-dimensional nonlinear optimal control problems in general. We show how in the special case of one spatial dimension, the optimal control problem can be converted into an infinite dimensional Linear-Quadratic tracking problem by reparameterizing in terms of quantile functions, which can then be converted into a family of decoupled scalar Linear-Quadratic tracking problems. An analytic solution is provided in the general case. We investigate further two special cases where the demand distribution is static and where it is periodic in time. Explicit results and simulations are provided in each of these cases as well.

This dissertation consists of three parts centered around the topic of spatially distributed systems.

The first part treats a specific spatially distributed system, the so-called Rijke tube, an experiment illustrating the unstable interplay of heat exchange and gas dynamics. The experiment is described in detail and it is demonstrated how closed-loop system identification tools can be applied to obtain a transfer function model, before a spatially distributed model is developed and analyzed. The model in its most idealized form can be described in the frequency domain by a matrix of non-rational transfer functions, which facilitates analysis with classical methods such as the root locus.

The second part considers the following problem: for a given plant and cost function, could there be a finite-length periodic trajectory that achieves better performance than the optimal steady state? Termed optimal periodic control (OPC), this problem has received attention over several decades, however most available methods employ state-space based methods and hence scale very badly with plant dimension. Here, the problem is approached from a frequency-domain perspective, and methods whose complexity is independent of system dimension are developed by recasting the OPC problem for linear plants with certain memoryless polynomial nonlinearities as the problem of minimizing a polynomial.

Finally, the third part extends results for a special class within spatially distributed systems, that of spatially invariant systems, from systems defined on L_2 (square-integrable) spaces to systems whose state space is an inner-product Sobolev space as they arise when considering systems of higher temporal order. It is shown how standard results on exponential stability, stabilizability and LQ control can be generalized by carefully keeping track of spatial frequency weighting functions related to the Sobolev inner products, and simple recipes for doing so are given.

We consider the optimal controller design problem for linear time-invariant, spatially-distributed systems. The controller to be designed is itself a distributed system; each subcontroller component is restricted to have access to only a local subset of system information, which is shared across the network according to an underlying communication graph. The design problem of interest is to synthesize optimal controllers (with respect to some performance measure) subject to this limited information sharing architecture. In this dissertation, we contribute to two directions of research in this setting: i) analysis of constraints that ensure such localization and can be imposed in a tractable manner, and ii) characterization of settings in which the unconstrained centralized optimal controller has an inherent degree of spatial localization.

For direction (i), we follow the `System Level Synthesis' (SLS) approach and consider directly designing the closed-loops as opposed to the controller or corresponding Youla parameter. Structural constraints on the closed-loop can be imposed in a convex manner, and we demonstrate that the optimal controller design problem subject to closed-loop transfer function sparsity constraints is a convex relaxation of the optimal controller design problem subject to structural constraints on a controller state-space realization (implementation). We provide an implicit parameterization of all achievable closed-loop mappings for

a broad class of systems, including continuous- or discrete-time spatially-invariant systems over an infinite spatial domain.

Under certain assumptions, in the state feedback setting we convert this implicit closed-loop parameterization to an explicit affine linear parameterization.

Our parameterizations allow for conversion of the closed-loop structured H2 optimal controller design problem to a standard model matching problem with finitely many transfer function parameters, allowing for analytic solutions in certain problem settings.

We further take a step toward quantifying the performance gap between structured closed-loop transfer function design and structured controller realization design by studying the setting of relative feedback controllers. To do so, we provide a compact and convex characterization of all relative feedback controllers, and demonstrate that the relative feedback requirement can be imposed as a convex constraint on the closed-loop in certain problem settings. We use this characterization to show that the optimal relative feedback controller design problem subject to closed-loop structural constraints may be infeasible.

In direction (ii), we consider the optimal control of PDEs over a Sobolev space. We demonstrate that the optimal state feedback is a spatial convolution operator given by an exponentially decaying convolution kernel, thus enabling implementation with a localized architecture, extending previous results in the L2 setting. The main tool we utilize is a transformation from a Sobolev to an L2 space, which is constructed from a spectral factorization of the spatial frequency weighting matrix of the Sobolev norm.

This thesis touches upon several topics in cochlear modeling. Throughout the literature, mathematical models of the cochlea vary according to the degree of biological realism to be incorporated. This thesis casts the cochlear model as a continuous space-time dynamical system using operator language. This framework encompasses a wider class of cochlear models and makes the dynamics more transparent and easier to analyze before applying any numerical method to discretize space. In fact, several numerical methods are investigated to study the computational efficiency of the finite dimensional realizations in space. Furthermore, we study the effects of the active gain perturbations on the stability of the linearized dynamics. The stability analysis is used to explain possible mechanisms underlying spontaneous otoacoustic emissions and tinnitus. Dynamic Mode Decomposition (DMD) is introduced as a useful tool to analyze the response of nonlinear cochlear models. Cochlear response features are illustrated using DMD which has the advantage of explicitly revealing the spatial modes of vibrations occurring in the Basilar Membrane (BM). Finally, we address the dynamic estimation problem of BM vibrations using Extended Kalman Filters (EKF). Due to the limitations of noninvasive sensing schemes, such algorithms are inevitable to estimate the dynamic behavior of a living cochlea.

This dissertation consists of four parts that revolve around structured stochastic uncertainty and optimal control/estimation theory.

In the first part, we consider the continuous-time setting of linear time-invariant (LTI) systems in feedback with multiplicative stochastic uncertainties. The objective is to characterize the conditions of Mean-Square Stability (MSS) using a purely input-output approach. This approach leads to uncovering new tools such as stochastic block diagrams. Various stochastic interpretations are considered, such as It\=o and Stratonovich, and block diagram conversion schemes between different interpretations are devised. The MSS conditions are given in terms of the spectral radius of a matrix operator that takes different forms when different stochastic interpretations are considered.

The second part applies the developed theory to analyze the mean-square stability and performance of stochastic cochlear models. The analysis is carried out for a generalized class of biomechanical models of the cochlea, that is formulated as a stochastic spatially distributed system, by allowing stochastic spatio-temporal perturbations within the cochlear amplifier. The simulation-free analysis explains the underlying mechanisms that give rise to cochlear instabilities such as spontaneous otoacoustic emissions and/or tinnitus. Furthermore, nonlinear stochastic simulations are carried out to validate the predictions of the theoretical analysis.

The third part revisits the development of numerical methods to solve optimal control problems using a function-space approach. This approach has the advantage of unifying the framework upon which the various (existing) numerical methods are based on. In fact, this approach motivates the definition of various system and projection operators that make the derivations conceptually transparent. Furthermore, the function-space approach builds useful geometric intuitions that inspire the development of new projection-based methods.

In the last part, we propose a methodology of optimal path design for sensors through a distributed environment. We consider time-limited scenarios where the sensors can only make a small number of measurements, but where some portion of a physics-based model is available for the field of interest (such as temperature). We consider both point-wise and tomographic sensors. The main idea is to recast the sensor path planning problem as a deterministic optimal control problem to minimize metrics related to the optimal estimation error covariance.

This work proposes a new Stirling engine design which we refer to as an actuated Stirling engine. The thermodynamic cycle which drives the engine is controlled by the motion of the displacer piston. In traditional designs this motion is a result the design parameters of the engine. In the Beta engine design the motion of the displacer is determined by the design of the flywheel, while in a free piston design the motion is determined by the restorative force applied to the displacer (be that from a gas spring used in some designs or a mechanical spring used in others). This work proposes a new design wherein the displacer motion is directly controlled via an actuator. This allows for more direct control over the thermodynamic process which drives the engine. To answer the question of how well this engine design compares to a more traditional engine, the best possible design of both engines are compared to one another. This requires optimizing not only the proposed design, but the more traditional design as well.

The traditional Beta engine design is chosen as the benchmark with which the actuated design is compared to. Optimizing the design of the Beta engine is a standard parameter optimization problem, as the motion of the pistons is determined by the flywheel parameters. Optimizing the actuated engine is more difficult as it requires finding the optimal trajectory of the displacer, which is an optimal control problem.

This work is divided into two sections. In the first section the problem of optimally designing both the actuated and Beta engine designs is solved using a very simple of the Stirling engine, the isothermal (or Schmidt) model. In the second section the same design problem is solved but with a much higher fidelity engine model. The large state dimension of the high fidelity model makes it ill suited for use in an optimization algorithm. To resolve this, a method of model reduction is proposed and applied to the high fidelity model to yield a reduced model more suitable for use in an optimization algorithm. For both the simple isothermal and high fidelity models it is shown that the actuated design has the potential to significantly outperform the more traditional Beta engine design.