Qualitative Theory of Stochastic Non-Linear Systems

A data-driven approach for discovering stochastic dynamical systems with non-Gaussian Lévy noise

The stability of stochastic dynamic systems have widespread application prospects and greater theoretical significance. Its theory are mainly applied to engineering control, communication equipment, military technology, biology, finance, etc. In this article, we consider the stability of a Class of nonlinear impulsive neutral stochastic differential dynamic systems. A new set of conditions proving the mean square stability of this nonlinear impulsive neutral stochastic dynamic systems are derived by means of the Banach fixed point theorem. Some well-known results are improved and generalized.

A stochastic dynamical system is a system composed of many interacting components which includes stochastic behavior. Feedback control theory is designed to use information from the current state of the system in order to temper the final state toward the desired outcome. In this dissertation, I perform studies on two choices of classic and quantum dynamical systems to cover vast variety of statistical tools and methods. For classical model, I perform a study on finite-size effects at the phase transition in the Nagel-Schreckenberg traffic model as an example of non-equilibrium many body stochastic dynamical system. For quantum mechanical model, I introduce an optimal measurement-based feedback control protocol for cooling a single qubit as an example of complex system with one cell and many degrees of freedom. I examine the Nagel-Schreckenberg traffic model for a variety of maximum speeds. I show that the low density limit can be described as a dilute gas of vehicles with a repulsive core. At the transition to jamming, I observe finite-size effects in a variety of quantities describing the flow and the density correlations, but only if the maximum speed Vmax is larger than a certain value. A finite-size scaling analysis of several order parameters shows universal behavior, with scaling exponents that depend on Vmax The jamming transition at large Vmax can be viewed as the nucleation of jams in a background of freely flowing vehicles. Feedback control of quantum systems via continuous measurements involves complex nonlinear dynamics. As a result, even for a single qubit the optimal measurement for feedback control is known only in very special cases. I show here that for a broad class of noise processes, a series of compelling arguments can be applied to greatly simplify the problem of steady-state preparation of the ground-state, while loosing little in the way of optimality. Using numerical optimization to solve this simplified control problem, I obtain for the first time a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. The protocol can be described relatively simply, and contains a discontinuity as a function of feedback strength.

In order to accelerate computations and improve long time accuracy of numerical simulations, this thesis develops multiscale geometric integrators. For general multiscale stiff ODEs, SDEs, and PDEs, FLow AVeraging integratORs (FLAVORs) have been proposed for the coarse time-stepping without any identification of the slow or the fast variables. In the special case of deterministic and stochastic mechanical systems, symplectic, multisymplectic, and quasi-symplectic multiscale integrators are easily obtained using this strategy. For highly oscillatory mechanical systems (with quasi-quadratic stiff potentials and possibly high-dimensional), a specialized symplectic method has been devised to provide improved efficiency and accuracy. This method is based on the introduction of two highly nontrivial matrix exponentiation algorithms, which are generic, efficient, and symplectic (if the exact exponential is symplectic). For multiscale systems with Dirac-distributed fast processes, a family of symplectic, linearly-implicit and stable integrators has been designed for coarse step simulations. An application is the fast and accurate integration of constrained dynamics. In addition, if one cares about statistical properties of an ensemble of trajectories, but not the numerical accuracy of a single trajectory, we suggest tuning friction and annealing temperature in a Langevin process to accelerate its convergence. Other works include variational integration of circuits, efficient simulation of a nonlinear wave, and finding optimal transition pathways in stochastic dynamical systems (with a demonstration of mass effects in molecular dynamics).

Systems and Control is now a mature subject with a deep mathematical theory and a powerful engineering methodology; its roots lie in antiquity but it emerged in the form of the Watt governor more than two centuries ago as an enabling technology of the Industrial Revolution, and its development since then involves critical contributions in a broad range of domains stretching from the Space Race to the economically vital area of industrial process control. However, the current natural and designed environment presents a vast array of control problems generated by systems of immense complexity which are often populated by subsystems possessing both discrete and continuous states and which, moreover, display so-called intelligent behaviour. A sample would have to include (i) the need to stabilize and control in a distributed manner the huge communication, information, transportation and economic networks of the modern world, (ii) the control of systems such as agricultural, fisheries and ecological systems with the objective of achieving sustainability, (iii) the management of the current integration of distributed and intrinsically stochastic renewable energy sources into deregulated and immensely expanded transmission networks serving users of all scales, and, finally, the most difficult problem area of all, (iv) the effective stabilization of the planet's climate through political and economic intervention based on system theoretic concepts and grounded in the physical sciences. Clearly the problem areas above demand a range of responses from various disciplines, but they raise at least two specific, interlinked challenges of a basic nature for Systems and Control: 1. To create fundamental theory and effective methodology to analyze and control systems with both discrete and continuous states and dynamics. Despite two decades of advances, Hybrid System theory is still in an early stage of development, for instance continuous-to-discrete state aggregation still lacks an adequate theory and so hierarchical control is in its infancy. Furthermore, work taking inspiration from the life sciences, such as gene controlled and self-healing systems, is limited to say the least. 2. To create adequate methods for the analysis and decentralized control of networked systems of huge complexity with social characteristics. While there has been progress in the development of game theory for stochastic dynamical systems, in particular Mean Field Game (MFG) theory, many challenges lie ahead, two examples being (i) relating MFG methods to the currently emerging information based control theory wherein the capacity of each of the feedback channels in a network plays a critical role, and (ii) the development of a dynamical game theory of the formation and stability of coalitions. In addition, the state aggregation problem introduced in the first challenge area also naturally falls under this second heading.

Event-triggered [formula omitted] stabilization for networked stochastic systems with multiplicative noise and network-induced delays

Approximation identification for the stochastic time-delayed dynamical system

Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map

Limits of invariant measures of stochastic Burgers equations driven by two kinds of [formula omitted]-stable processes

Mean-square practical stability for stochastic large-scale dynamical systems is studied. The vector Lyapunov functions and the basic comparison principle of stochastic systems as well as the decomposition-aggregation method for large-scale systems were used to obtain results for various types of mean-square practical stability of nonlinear stochastic large-scale dynamical systems. The objective is to analyze stochastic large-scale systems in terms of their lower-order subsystems and in terms of their interconnecting structure. These results make it possible to determine the mean-square practical stability of stochastic large-scale dynamical systems by testing the corresponding practical stability of the auxiliary deterministic systems. >

In this chapter, an introduction will be given on how to use the theory and methods of stochastic dynamical systems theory to understand phenomena of climate variability. We will restrict the application of this theory to very basic aspects of the El Nino/Southern Oscillation and the Atlantic Multidecadal Oscillation.

A solution method for linear and geometrically nonlinear MDOF systems with random properties subject to random excitation

Tau leaping of stiff stochastic chemical systems via local central limit approximation

In this paper, we study the Wong-Zakai approximations given by a smoothed approximation of the white noise and their associated long term pathwise behavior for the stochastic lattice dynamical systems. To be exactly, we first establish the existence of the random attractor for the random lattice dynamical system driven by the smoothed noise and then show the convergence of solutions and random attractors to these of stochastic lattice dynamical systems driven by a multiplicative noise and an additive white noise, respectively, when the perturbation parameters tend to zero.

Dissipativity theory and applications of nonlinear stochastic systems with Markov jump and Lévy noise