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Topics in Stochastic Stability, Optimal Control and Estimation Theory


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.

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