No abstract

We propose two novel numerical schemes for approximate implementation of the Dynamic Programming (DP) operation concerned with finite-horizon optimal control of discrete-time, stochastic 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 a 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. In a broader perspective, the key contribution here can be viewed as an algorithmic transformation of the minimization in DP operation to addition via discrete conjugation. This bridge enables us to utilize any complexity reduction on the discrete conjugation front within the proposed algorithms. In particular, motivated by the recent development of quantum algorithms for computing the discrete conjugate transform, we discuss the possibility of a quantum mechanical implementation of the proposed algorithms.

In this article, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic) approximation of 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 establish 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. The results of 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.Index Terms-Opinion dynamics, distributed parameter systems, stochastic systems, nonlinear systems, stability of NL systems. I. INTRODUCTIONR ECENT decades have witnessed enormous progress in study of complex systems and their system-theoretic properties [1], [2]. The main effort has been invested into the study of "self-organization" and "spontaneous order" phenomena [3] that have inspired the development of synchronization and consensus theory [4], [5]. Paradoxically, these regular behaviors arising from local interactions between subsystems (agents, nodes) of a complex system are studied much better than various "irregular" dynamic effects such as persistent disagreement and clustering, exhibited by many real-world systems. Although some culprits of this asynchrony and dissent (e.g. symmetries and other special structures in the coupling mechanisms, exogenous forces acting on some nodes, heterogeneous dynamics of nodes, etc.) have been revealed in the literature [6]-[10], only a few mathematical models have been proposed that are sufficiently "rich" to capture the

No abstract