Dynamic programming,Fermat's principle and stochastic Eikonal equations

An extension of Fermat's principle to the stochastic case is presented in order to treat ray propagation in random media. The concept of dynamic programming permits one to derive a sequence of stochastic eikonal equations from which a sequence of rays can be traced using Hamilton's equations. The extension is motivated by analogous stochastic control problems in which the concepts and methods are adapted to the present problem. To facilitate understanding of the main ideas the presentation is given in a simple and somewhat naive manner. Two simple examples are presented to demonstrate the ideas and techniques.     

多变量、多分布随机边界元法结构可靠性计算

在对非正态变量正态化的基础上，利用对随机变量求偏导数的方法，了二维弹性随机边界元积分方程，提出了多变量、多分布的随机结构可靠性计算方法，文中主要考虑了材料的随机结构强度，随机材料性能参数以及随机载荷等，数值算例表明，本文的算法是可靠而有效的。     

Modeling of chemotactic steering of bacteria-based microrobot using a population-scale approach

The bacteria-based microrobot (Bacteriobot) is one of the most effective vehicles for drug delivery systems. The bacteriobot consists of a microbead containing therapeutic drugs and bacteria as a sensor and an actuator that can target and guide the bacteriobot to its destination. Many researchers are developing bacteria-based microrobots and establishing the model. In spite of these efforts, a motility model for bacteriobots steered by chemotaxis remains elusive. Because bacterial movement is random and should be described using a stochastic model, bacterial response to the chemo-attractant is difficult to anticipate. In this research, we used a population-scale approach to overcome the main obstacle to the stochastic motion of single bacterium. Also known as Keller-Segel''s equation in chemotaxis research, the population-scale approach is not new. It is a well-designed model derived from transport theory and adaptable to any chemotaxis experiment. In addition, we have considered the self-propelled Brownian motion of the bacteriobot in order to represent its stochastic properties. From this perspective, we have proposed a new numerical modelling method combining chemotaxis and Brownian motion to create a bacteriobot model steered by chemotaxis. To obtain modeling parameters, we executed motility analyses of microbeads and bacteriobots without chemotactic steering as well as chemotactic steering analysis of the bacteriobots. The resulting proposed model shows sound agreement with experimental data with a confidence level <0.01.     

Tracking controller design for random nonlinear benchmark system

This paper considers a benchmark system consisting of a rolling ball and a moving car in the oscillating surroundings. By using the Lagrange law, the dynamic model without disturbance is first constructed, then according to the relative motion principle, random oscillation of surroundings is transformed into the random noises in the constructed Lagrange equation. The special structure of the quasi-lower triangle of Lagrange equation motivates us to pay more attention to the vectorial backstepping technique. By selecting an appropriate Lyapunov-like function, a tracking controller with tunable parameters is designed such that all signals of the closed-loop system are bounded and track error can be made arbitrarily small.     

Nuclear norm-based recursive subspace prediction of time-varying continuous-time stochastic systems via distribution theory

A method for nuclear norm-based recursive subspace prediction of time-varying continuous-time stochastic systems via distribution theory is proposed. The random distribution theory is adopted to describe the time-derivative of stochastic processes, which is the key to obtain the input–output algebraic equation. The low-rank matrix approximation of the input–output projection matrix is established by nuclear norm minimization instead of the singular value decomposition. Moreover, the optimization problem is deduced by the alternating direction method of multipliers. According to the angle rotation between past and present subspaces spanned by the extended observability matrices, the future signal subspace is predicted by the present subspace. Further, the system matrices are predicted and the corresponding system model is obtained. The results of simulation studies show the effectiveness of the presented method.     

Output regulation control for stochastic systems with additive noise

This paper investigates the robust output regulation problem for stochastic systems with additive noises. As is known, for the output regulation control problem, a general method is to regard that the system is disturbed by an autonomous exosystem (which is consisted by external disturbances and reference signals), and for the system disturbed by the white noise, the stochastic differential equations (SDEs) should be utilized in modeling, accordingly, a controller with a feedforward regulator is constructed for the stochastic system with an exosystem, which can not only cancel the external disturbance, but also transform the trajectory tracking problem into the stabilization problem; In consideration of the state variables in stochastic systems cannot be measured completely, we embed an observer to the controller, such that the random interference can be suppressed, and the trajectory tracking can be achieved. Based on the stochastic control theory, the criteria of the exponential practical stability in the mean square is presented for the closed-loop system, finally, through tuning the controller parameters, the mean square of the tracking error can converge to an arbitrarily small neighborhood of the origin.     

Stochastic suppression and stabilization of nonlinear differential systems with general decay rate

In this paper, we investigate stochastic suppression and stabilization for a class of non-autonomous differential systems. Given a deterministic non-autonomous differential system, we introduce two independent Brownian motions and perturb this system into a new stochastic differential system. By using Lyapunov analysis method and some stochastic techniques, we show that a polynomial Brownian noise may guarantee the existence of global solution of the perturbed system while another linear Brownian noise may stabilize this system with general decay rate. An application of stochastic stabilization of differential system in the modeling of population growth is indicated.     

A Markovian system approach to distributed H∞ filtering for sensor networks with stochastic sampling

This paper is concerned with the distributed H_∞ filtering problem for a class of sensor networks with stochastic sampling. System measurements are collected through a sensor network stochastically and the phenomena such as random measurement missing and quantization are also considered. Firstly, the stochastic sampling process of the sensor network is modeled as a discrete-time Markovian system. Then, the logarithmic quantization effect is transformed into the parameter uncertainty of the filtering system, and a set of binary variables is introduced to model the random measurement missing phenomenon. Finally, the resulting augmented system is modeled as an uncertain Markovian system with multiple random variables. Based on the Lyapunov stability theory and the stochastic system analysis method, a sufficient condition is obtained such that the augmented system is stochastically stable and achieves an average H_∞ performance level γ; the design procedure of the optimal distributed filter is also provided. A numerical example is given to demonstrate the effectiveness of the proposed results.     

Solution of the Stochastic control problem in unbounded domains

Bellman's dynamic programming equation for the optimal index and control law for stochastic control problems is a parabolic or elliptic partial differential equation frequently defined in an unbounded domain. Existing methods of solution require bounded domain approximations, the application of singular perturbation techniques or Monte Carlo simulation procedures.In this paper, using the fact that Poisson impulse noise tends to a Gaussian process under certain limiting conditions, a method which achieves an arbitrarily good approximate solution to the stochastic control problem is given. The method uses the two iterative techniques of successive approximation and quasi-linearization and is inherently more efficient than existing methods of solution.     

Further stability results for random nonlinear systems with stochastic impulses

In this paper, the global asymptotic stability in probability and the exponential stability in $m$th moment are investigated for random nonlinear systems with stochastic impulses, whose occurrence is determined by a Poisson process. The stochastic disturbances in the impulsive random nonlinear systems are driven by second-order processes, which have bounded mean power. Firstly, the improved Lyapunov approaches for the global asymptotic stability in probability and the exponential stability in $m$th moment are established for impulsive random nonlinear systems based on the uniformly asymptotically stable function. Secondly, the improved results are further extended to the impulsive random nonlinear systems with Markovian switching. Finally, two examples are provided to verify the feasibility and effectiveness of the obtained results.     

Reproducing kernel resolution space and its applications

In some recent work Kailath and Duttweiler used reproducing Kernel Hilbert space techniques, together with a highly non-standard time structure, to formulate a representation theory for stochastic processes. Although non-standard their structure satisfied the axioms for a resolution space; a Hilbert space to which a time structure is axiomatically adjoined which has received considerable interest in the mathematical system theory literature.In the present paper the work of Kailath and Duttweiler is extended via the concept of a reproducing kernel resolution space which is defined for the covariance of an arbitrary resolution space valued random variable. Applications to the representation and approximation of stochastic processes are considered, a new formula for the optimal stochastic predictor is derived, and a canonical form for the causal factors of an arbitrary covariance is obtained.     

Linear quadratic open-loop Stackelberg game for stochastic systems with Poisson jumps

This paper is concerned with the finite horizon linear quadratic (LQ) Stackelberg game for stochastic systems with Poisson jumps under the open-loop information structure. First, the follower solves a LQ stochastic optimal control problem with Poisson jumps. With the aid of an introduced generalized differential Riccati equation with Poisson jumps (GDREP), the sufficient conditions for the optimization of the follower are put forward. Then, the leader faces an optimal control problem for a forward-backward stochastic differential equation with Poisson jumps (FBSDEP). By introducing new state and costate variables, a sufficient condition for the existence and uniqueness of the open-loop Stackelberg strategies is presented in terms of the solvability of two differential Riccati equations and a convexity condition. In addition, the state feedback representation of the open-loop Stackelberg strategies is obtained via the related differential Riccati equation. Finally, two examples shed light on the effectiveness of the obtained results.     

Analyzing noise-induced tracking errors in control systems with saturation: A stochastic linearization approach

Noise Induced Tracking Error (NITE) refers to the tracking error of the mean of the output in feedback control systems with nonlinear instrumentation subject to zero-mean measurement noise. Most of the previous work rely on the stochastic averaging for NITE analysis, the validity of which requires that the bandwidth of the zero mean measurement noise is much higher than that of the system. This is because the results obtained by stochastic averaging are asymptotic with respect to the noise bandwidth. Due to the asymptotic nature of the analysis tool, it is not straightforward to provide a quantitative argument for high bandwidth. An alternative method in the literature that can analyze NITE is stochastic linearization for random input, which is analogous to the well known describing function approach for sinusoidal input. Unlike stochastic averaging, stochastic linearization is not an asymptotic approximation. Therefore, analysis can be carried out for any given noise bandwidth. We carry out NITE analysis using stochastic linearization for a class of LPNI systems that are prone to NITE; identify the system conditions under which the averaging analysis of NITE may yield inaccurate results for a finite noise bandwidth; and prove that the results from the two methods agree as the noise bandwidth approaches infinity. In addition, an existing NITE mitigation strategy is extended based on the proposed method. Numerical examples are given to illustrate the results.     

Properties of value function and existence of viscosity solution of HJB equation for stochastic boundary control problems

In the present paper, we study stochastic boundary control problems where the system dynamics is a controlled stochastic parabolic equation with Neumann boundary control and boundary noise. Under some assumptions, the continuity and differentiability of the value function are proved. We also define a new type of Hamilton–Jacobi–Bellman (HJB) equation and prove that the value function is a viscosity solution of this HJB equation.     

Stability of delayed Hopfield neural networks under a sublinear expectation framework

In this paper, we consider the stability of a class of stochastic delay Hopfield neural networks driven by G-Brownian motion. Under a sublinear expectation framework, we give the definition of exponential stability in mean square and construct some conditions such that the stochastic system is exponentially stable in mean square. Moreover, we also consider the stability of the Euler numerical solution of such equation. Finally, we give an example and its numerical simulation to illustrate our results.     

A time-varying stochastic framework for modeling wireless channels

Wireless communication channels can be complex and exhibit unpredictable behavior. This behavior can be successfully represented by stochastic processes undergoing random perturbation due to shifts in the underlying physical state of the channel. These state transitions manifest themselves as changes in channel power level measurements taken over time from a remote reference beacon. In this paper, we expand the application of switched process theory to develop models of maritime and land satellite communications using generalized switched Fokker-Planck equation techniques. This application yields a broad class of models that may be treated analytically and numerically. These models are ideal for system-level simulation due to high numerical efficiency and limited maintenance calculation requirements.     

Unraveling the links between dimensions of innovation and organizational performance

Economists typically define innovation as a process or practice that is new to an industry; hence they emphasize a firm's speed of innovation relative to other firms in the industry. Organizational theorists, on the other hand, usually focus on the number of products or processes that are new to the firm; hence, they emphasize innovation magnitude. This study builds a bridge between these two approaches by exploring the link between two dimensions of innovation—speed and magnitude—and two measures of a firm's performance—objective financial reports and executive ratings of perceived effectiveness. We propose that each dimension of innovation will be associated with a different measure of firm performance. Using data from the commercial banking industry, we find interesting results that partially support our predictions based on the theory that different dimensions are indeed linked to different measures of performance. Implications for future research and practice are discussed.     

Infinite horizon linear quadratic Pareto game of the stochastic singular systems

This paper is concerned with the linear quadratic (LQ) Pareto game of the stochastic singular systems in infinite horizon. Firstly, the optimal control problem of the weighted sum cost functional is discussed. Utilizing the equivalent transformation method, the weighted sum LQ optimal control problem is transformed into a stochastic LQ optimization problem. Based on the classical stochastic LQ optimal control theory, the necessary and sufficient condition for the solvability of the indefinite weighted sum LQ optimal control is put forward. Then, the LQ Pareto game of the stochastic singular systems is studied. By the discussion of the convexity of the cost functionals, a sufficient condition for the existence of the Pareto solutions is obtained via the solvability of the corresponding generalized algebraic Riccati equation (GARE). Moreover, we derive all Pareto solutions based on the solution of a Lyapunov equation. Finally, an example is given to show the effectiveness of the proposed results.     

Stochastic input-to-state stability of random impulsive nonlinear systems

In this paper, the stochastic input-to-state stability is investigated for random impulsive nonlinear systems, in which impulses happen at random moments. Employing Lyapunov-based approach, sufficient conditions for the stochastic input-to-state stability are established based on the connection between the properties of system and impulsive intervals. Two classes of impulsive systems are considered: (1) the systems with single jump map; (2) the systems with multiple jump maps. Finally, some examples are provided to illustrate the effectiveness of the proposed results.     

Predictive control of aircraft control systems for maneuverable target

A new aircraft guidance law utilizing predictive control algorithm is proposed. Compared with extended proportional navigation guidance law, this guidance law is designed specifically for using against a maneuverable target. It can track the target maneuver effectively. Uncertainty in the predicted target's future maneuvering is handled by constructing a stochastic predictive cost function and utilizing the stochastic minimum principle to deduce a predictive control algorithm. The guidance law is established by adopting the continuous approximation method into the continuous predictive control algorithm. The simulation results demonstrate that the stochastic predictive guidance law has the ability of trajectory prediction. Compared with the PID proportional navigation guidance law, the stochastic predictive guidance law has smaller overload and can intercept the maneuvering target more effectively.