Data based identification and prediction of nonlinear and complex dynamical systems

We thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. Su (formerly at ASU), and Mr. Zhesi Shen for their contributions to a number of original papers on which this Review is partly based. This work was supported by ARO under Grant No. W911NF-14-1-0504. W.-X. Wang was also supported by NSFC under Grants No. 61573064 and No. 61074116, as well as by the Fundamental Research Funds for the Central Universities, Beijing Nova Programme.Peer reviewedPostprin

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton-Jacobi-Isaacs Equations

In this paper, we consider state and control path-dependent stochastic zero-sum differential games, where the dynamics and the running cost include both state and control paths of the players. Using the notion of nonanticipative strategies, we define lower and upper value functionals, which are functions of the initial state and control paths of the players. We prove that the value functionals satisfy the dynamic programming principle. The associated lower and upper Hamilton-Jacobi-Isaacs (HJI) equations from the dynamic programming principle are state and control path-dependent nonlinear second-order partial differential equations. We apply the functional It\^o calculus to prove that the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact subset of an $\kappa$-H\"older space introduced in \cite{Tang_DCD_2015}. Moreover, we show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. For the state path-dependent case, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations.Comment: 29 page

International Conference on Dynamic Control and Optimization - DCO 2021: book of abstracts

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Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition

In the present work, we consider 2-person zero-sum stochastic differential games with a nonlinear pay-off functional which is defined through a backward stochastic differential equation. Our main objective is to study for such a game the problem of the existence of a value without Isaacs condition. Not surprising, this requires a suitable concept of mixed strategies which, to the authors' best knowledge, was not known in the context of stochastic differential games. For this, we consider nonanticipative strategies with a delay defined through a partition $\pi$ of the time interval $[0,T]$. The underlying stochastic controls for the both players are randomized along $\pi$ by a hazard which is independent of the governing Brownian motion, and knowing the information available at the left time point $t_{j-1}$ of the subintervals generated by $\pi$, the controls of Players 1 and 2 are conditionally independent over $[t_{j-1},t_j)$. It is shown that the associated lower and upper value functions $W^{\pi}$ and $U^{\pi}$ converge uniformly on compacts to a function $V$, the so-called value in mixed strategies, as the mesh of $\pi$ tends to zero. This function $V$ is characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation.Comment: Published in at http://dx.doi.org/10.1214/13-AOP849 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multi-Layer Cyber-Physical Security and Resilience for Smart Grid

The smart grid is a large-scale complex system that integrates communication technologies with the physical layer operation of the energy systems. Security and resilience mechanisms by design are important to provide guarantee operations for the system. This chapter provides a layered perspective of the smart grid security and discusses game and decision theory as a tool to model the interactions among system components and the interaction between attackers and the system. We discuss game-theoretic applications and challenges in the design of cross-layer robust and resilient controller, secure network routing protocol at the data communication and networking layers, and the challenges of the information security at the management layer of the grid. The chapter will discuss the future directions of using game-theoretic tools in addressing multi-layer security issues in the smart grid.Comment: 16 page

Markovian Dynamics on Complex Reaction Networks

Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multi-agent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples.Comment: 52 pages, 11 figures, for freely available MATLAB software, see http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.htm

Information theory in biochemical regulatory networks: a theoretical study

In this Thesis we consider the optimization of information transmission as a viable design principle for biochemical networks. We apply this principle to a simple model regulatory circuit, given by an input and a delayed output that switch randomly between two states in continuous time. First we maximize the transmitted information in the network at a given output delay, when the system has no external constraints and it is in steady state or can optimize its initial condition. We find that optimal network topologies correspond to common biological circuits linked to stress response and that circuits functioning out of steady state may exploit absorbing states to be more informative than in steady state. We then take into account that biological regulatory networks need to dissipate energy in order to transmit biochemical signals and that such signaling often happens in challenging environmental conditions. Hence we explore the system's trade-offs between information transmission and energetic efficiency. At fixed delay and dissipated energy, we determine the most informative networks both in the absence and in the presence of feedback. We find that negative feedback loops are optimal at high dissipation, whereas positive feedback loops become more informative close to equilibrium conditions. Moreover, feedback allows the system to transmit almost the maximum available information at a given delay, even in the absence of dissipation. Finally, within a game-theoretic maximin approach, we ask how a biochemical network should be constructed to be most informative in the worst possible initial condition set by the environment. We find that, in the limit of large energy dissipation, the system tunes the ratio of the input and output timescales so that the environmental disturbance is marginalized as much as possible

A sufficient maximum principle for backward stochastic systems with mixed delays

In this paper, we study the problem of optimal control of backward stochastic differential equations with three delays (discrete delay, moving-average delay and noisy memory). We establish the sufficient optimality condition for the stochastic system. We introduce two kinds of time-advanced stochastic differential equations as the adjoint equations, which involve the partial derivatives of the function $f$ and its Malliavin derivatives. We also show that these two kinds of adjoint equations are equivalent. Finally, as applications, we discuss a linear-quadratic backward stochastic system and give an explicit optimal control. In particular, the stochastic differential equations with time delay are simulated by means of discretization techniques, and the effect of time delay on the optimal control result is explained