Finite-Dimensional Approximation in Dual Domain

Abstract

This thesis is comprised of two main parts. In the first part of the thesis, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic) approximation of the bounded confidence opinion dynamics, where opinions are influenced by environmental noises and opinions of radicals (stubborn individuals). The distribution of radical opinions serves as an infinite-dimensional exogenous input to the FP equation, visibly influencing the steady opinion profile. We first establish the mathematical properties of the FP equation. In particular, we (i) show the well-posedness of the dynamic equation, (ii) provide existence result accompanied by a quantitative global estimate for the corresponding stationary solution, and, (iii) establish an explicit lower bound on the noise level that guarantees exponential convergence of the dynamics to stationary state. Combining the results in (ii) and (iii) readily yields the input-output stability of the system for sufficiently large noises. Next, using Fourier analysis, the structure of opinion clusters under the uniform initial distribution is examined. Specifically, two numerical schemes for (i) identification of order-disorder transition and (ii) characterization of initial clustering behavior are provided. The results of the analysis are validated through several numerical simulations of the continuum-agent model (partial differential equation) and the corresponding discrete-agent model (interacting stochastic differential equations) for a particular distribution of radicals.

In the second part of the thesis, we focus on the value iteration algorithm for solving optimal control problems. We propose two novel numerical schemes for approximate implementation of the dynamic programming (DP) operation concerned with finite-horizon, optimal control of deterministic, discrete-time systems with input-affine dynamics. The proposed algorithms involve discretization of the state and input spaces and are based on an alternative path that solves the dual problem corresponding to the DP operation. We provide error bounds for the proposed algorithms, along with detailed analyses of their computational complexity. In particular, for a specific class of problems with separable data in the state and input variables, the proposed approach can reduce the typical time complexity of the DP operation from O(XU) to O(X +U), where X and U denote the size of the discrete state and input spaces, respectively. We next discuss the extensions of the proposed conjugate value iteration algorithm for problems with separable data. The extensions are three-fold: We consider (i) infinite-horizon, discounted cost problems with (ii) stochastic dynamics, while (iii) computing the conjugate of input cost numerically. In particular, we analyze the convergence, complexity, and error
of the proposed algorithm under these extensions. The theoretical results are validated through multiple numerical examples.