Higher order Sobolev Spaces and polyharmonic boundary value problems in Carnot Groups

The main task of this work is to present a concise survey on the theory of certain function spaces in the contexts of Hörmander vector fields and Carnot Groups, and to discuss briefly an application to some polyharmonic boundary value problems on Carnot Groups of step 2.

Distributed non-cooperative robust economic predictive control for dynamically coupled linear systems.

In this thesis, a tube-based Distributed Economic Predictive Control (DEPC) scheme is presented for a group of dynamically coupled linear subsystems. These subsystems are components of a large scale system and control inputs are computed based on optimizing a local economic objective. Each subsystem is interacting with its neighbors by sending its future reference trajectory, at each sampling time. It solves a local optimization problem in parallel, based on the received future reference trajectories of the other subsystems. To ensure recursive feasibility and a performance bound, each subsystem is constrained to not deviate too much from its communicated reference trajectory. This difference between the plan trajectory and the communicated one is interpreted as a disturbance on the local level. Then, to ensure the satisfaction of both state and input constraints, they are tightened by considering explicitly the effect of these local disturbances. The proposed approach averages over all possible disturbances, handles tightened state and input constraints, while satisfies the compatibility constraints to guarantee that the actual trajectory lies within a certain bound in the neighborhood of the reference one. Each subsystem is optimizing a local arbitrary economic objective function in parallel while considering a local terminal constraint to guarantee recursive feasibility. In this framework, economic performance guarantees for a tube-based distributed predictive control (DPC) scheme are developed rigorously. It is presented that the closed-loop nominal subsystem has a robust average performance bound locally which is no worse than that of a local robust steady state. Since a robust algorithm is applying on the states of the real (with disturbances) subsystems, this bound can be interpreted as an average performance result for the real closed-loop system. To this end, we present our outcomes on local and global performance, illustrated by a numerical example.

LQG control policy design against agent identity privacy problem

This thesis project studies the agent identity privacy problem in the scalar linear quadratic Gaussian (LQG) control system. For the agent identity privacy problem in the LQG control, privacy models and privacy measures have to be established first. It depends on a trajectory of correlated data rather than a single observation. I propose here privacy models and the corresponding privacy measures by taking into account the two characteristics. The agent identity is a binary hypothesis: Agent A or Agent B. An eavesdropper is assumed to make a hypothesis testing on the agent identity based on the intercepted environment state sequence. The privacy risk is measured by the Kullback-Leibler divergence between the probability distributions of state sequences under two hypotheses. By taking into account both the accumulative control reward and privacy risk, an optimization problem of the policy of Agent B is formulated. The optimal deterministic privacy-preserving LQG policy of Agent B is a linear mapping. A sufficient condition is given to guarantee that the optimal deterministic privacy-preserving policy is time-invariant in the asymptotic regime. An independent Gaussian random variable cannot improve the performance of Agent B. The numerical experiments justify the theoretic results and illustrate the reward-privacy trade-off. Based on the privacy model and the LQG control model, I have formulated the mathematical problems for the agent identity privacy problem in LQG. The formulated problems address the two design objectives: to maximize the control reward and to minimize the privacy risk. I have conducted theoretic analysis on the LQG control policy in the agent identity privacy problem and the trade-off between the control reward and the privacy risk.Finally, the theoretic results are justified by numerical experiments. From the numerical results, I expected to have some interesting observations and insights, which are explained in the last chapter.