Linear Stochastic Differential Systems Research Articles

Stochastic differential formula and solution of control problem

Exact solution of finite-dimensional linear stochastic differential system with control is derived. Its uniqueness up to stochastic modification is proved. In order to achieve this, detailed grounding of existence for stochastic integral over a process with orthogonal increments is provided. Particularly, density of step-functions set in the set of all integrable functions is proved. Integration by parts formula for stochastic integral is derived. Analogue of Fubini theorem is proved in the case, when one measure is linear and another one is a random orthogonal measure. Stochastic differential formula for finite-dimensional integral functional is established. Formulation of Ito’s stochastic differential formula and its comparison with the result is presented. The solution formula for controlled linear stochastic differential system was conditionally used in applied sciences. Its rigorous proof provides the necessary background for further research.

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an H∞ control problem applicable to an infinite horizon. Finally, we present the conclusions.

Filtering of hidden Markov renewal processes by continuous and counting observations

<p>This paper introduces a subclass of Markov renewal processes (MRPs) and presents a solution to the optimal filtering problem in a stochastic observation system, where the state is modeled by an MRP and observed indirectly through noisy measurements. The MRPs considered here can be interpreted as continuous-time Markov chains (CTMCs) with a finite set of abstract states representing distributions of random vectors. The paper outlines the probabilistic properties of MRPs, emphasizing the ability to express any arbitrary function of the MRP as the solution to a linear stochastic differential system (SDS) with a martingale on the right-hand side (RHS). Using these properties, an optimal filtering problem is formulated in stochastic observation systems, where the hidden state belongs to the class of MRPs, and the observations consist of both diffusion and counting components. The drift terms in all observations depend on the system state. An optimal filtering estimate for a scalar function of the MRP is provided through the solution of an SDS with innovation processes on the RHS. Additionally, the paper presents a version of the Kushner-Stratonovich equation, describing the evolution of the conditional probability density function (PDF). To demonstrate the practical application of the estimation method, the paper presents a communications-related example, focusing on monitoring the qualitative state and numerical characteristics of a network channel using noisy observations of round-trip time (RTT) and packet loss flow. The paper also highlights the robustness of the filtering algorithm in scenarios where the MRP distribution is uncertain.</p>

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

A survey of mean-square destabilization of multidimensional linear stochastic differential systems with non-normal drift

Mean-square stability analysis of linear stochastic differential systems obtained perturbing ordinary systems by linear terms driven by independent Wiener processes is investigated. The so obtained stochastic regions are contractions of the asymptotic stability domain of the linear ordinary system. In this work, the mean-square stability regions exact shape is provided by means of necessary and sufficient conditions in terms of the eigenvalues of the drift and the intensities of the noises. Special attention is paid to how different structures of the perturbation affect the mean-square stability of systems with non-normal drifts. In each case, the obtained explicit stability condition reveals the role played by the parameter that controls the non-normality.

Comparative Study of Markov Chain Filtering Schemas for Stabilization of Stochastic Systems under Incomplete Information

The object under investigation is a controllable linear stochastic differential system affected by some external statistically uncertain piecewise continuous disturbances. They are directly unobservable but assumed to be a continuous-time Markov chain. The problem is to stabilize the system output concerning a quadratic optimality criterion. As is known, the separation theorem holds for the system. The goal of the paper is performance analysis of various numerical schemes applied to the filtering of the external Markov input for system stabilization purposes. The paper briefly presents the theoretical solution to the considered problem of optimal stabilization for systems with the Markov jump external disturbances: the conditions providing the separation theorem, the equations of optimal control, and the ones defining the Wonham filter. It also contains a complex of the stable numerical approximations of the filter, designed for the time-discretized observations, along with their accuracy characteristics. The approximations of orders 12, 1, and 2 along with the classical Euler–Maruyama scheme are chosen for the comparison of the Wonham filter numerical realization. The filtering estimates are used in the practical stabilization of the various linear systems of the second order. The numerical experiments confirm the significant influence of the filtering precision on the stabilization performance and superiority of the proposed stable schemes of numerical filtering.

Stability of Stochastic Delay Differential Systems With Variable Impulses Due to Logic Choice

This paper is concerned with stability problems of stochastic delay differential systems with variable impulses due to logic choice. Firstly, a class of variable impulses due to logic choice is introduced in this paper which is more general than the logic impulses established by (Suo and Sun, 2015) and (Zhang et al., 2018). Then, by establishing a connection between the stochastic delay differential system with logic impulses and a corresponding stochastic delay differential system without logic impulses, some sufficient conditions for stability of the systems are obtained. Finally, the application in a class of linear stochastic delay differential systems with logic impulses is discussed, and several stability criteria together with two numerical examples are given.

Switching controls for linear stochastic differential systems

We analyze the exact controllability problem of switching controls for stochastic control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to the other such that, in each instant of time, there are as few active actuators as possible. We prove that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. The proof is based on building a new functional by the adjoint system whose minimizers are the switching controls.

Estimation of non-integral and integral quadratic functions in linear stochastic differential systems

This paper focuses on estimation of an non-integral quadratic function (NIQF) and integral quadratic function (IQF) of a random signal in dynamic system described by a linear stochastic differential equation. The quadratic form of an unobservable signal indicates useful information of a signal for control. The optimal (in mean square sense) and suboptimal estimates of NIQF and IQF represent a function of the Kalman estimate and its error covariance. The proposed estimation algorithms have a closed-form estimation procedure. The obtained estimates are studied in detail, including derivation of the exact formulas and differential equations for mean square errors. The results we demonstrate on practical example of a power of signal , and comparison analysis between optimal and suboptimal estimators is presented.

Optimal Stochastic Control Problem for General Linear Dynamical Systems in Neuroscience

This paper considers a d-dimensional stochastic optimization problem in neuroscience. Suppose the arm’s movement trajectory is modeled by high-order linear stochastic differential dynamic system in d-dimensional space, the optimal trajectory, velocity, and variance are explicitly obtained by using stochastic control method, which allows us to analytically establish exact relationships between various quantities. Moreover, the optimal trajectory is almost a straight line for a reaching movement; the optimal velocity bell-shaped and the optimal variance are consistent with the experimental Fitts law; that is, the longer the time of a reaching movement, the higher the accuracy of arriving at the target position, and the results can be directly applied to designing a reaching movement performed by a robotic arm in a more general environment.

On almost sure stability conditions of linear switching stochastic differential systems

In modeling practical systems, it can be efficient to apply Poisson process and Wiener process to represent the abrupt changes and the environmental noise, respectively. Therefore, we consider the systems affected by these random processes and investigate their joint effects on stability. In order to apply Lyapunov stability method, we formulate the action of the infinitesimal generator corresponding to such a system. Then, we derive the almost sure stability conditions by using some fundamental convergence theorem. To illustrate the theoretical results, we construct an example to show that it is possible to achieve stabilization by using random perturbations.

An analysis of a three-factor model proposed by the Danish Society of Actuaries for forecasting and risk analysis

This paper provides the explicit solution to the three-factor diffusion model recently proposed by the Danish Society of Actuaries to the Danish industry of life insurance and pensions. The solution is obtained by use of the known general solution to multidimensional linear stochastic differential equation systems. With offset in the explicit solution, we establish the conditional distribution of the future state variables which allows for exact simulation. Using exact simulation, we illustrate how simulation of the system can be improved compared to a standard Euler scheme. In order to analyze the effect of choosing the exact simulation scheme over the traditional Euler approximation scheme frequently applied by practitioners, we carry out a simulation study. We show that due to its recursive nature, the Euler scheme becomes computationally expensive as it requires a small step size in order to minimize discretization errors. Using our exact simulation scheme, one is able to cut these computational costs significantly and obtain even better forecasts. As probability density tail behavior is key to expected investment portfolio performance, we further conduct a risk analysis in which we compare well-known risk measures under both schemes. Finally, we conduct a sensitivity analysis and find that the relative performance of the two schemes depends on the chosen model parameter estimates.

Linear backward stochastic differential systems of descriptor type with structure and applications to engineering

In this paper, a class of linear stochastic differential systems of descriptor type with symmetric and skew-symmetric coefficients is considered. These kinds of systems have numerous applications in several areas of engineering, systems and control theory with potential applications to multibody systems (constrained mechanical systems with singular mass matrices), power systems, robotics and elsewhere. Thus, in our approach, using the Thompson canonical form for regular pencils, necessary and sufficient conditions for the solvability of a general class of such systems are obtained. In addition, as interesting theoretical applications, the solvability for any terminal condition and the problem of exact controllability are completely settled. An application to controlled mechanical translation systems illustrates the main findings of the paper.

Robust Stabilization Design for Stochastic Partial Differential Systems Under Spatio-temporal Disturbances and Sensor Measurement Noises

We investigate an H∞ robust observer-based stabilization design problem for linear stochastic partial differential systems (LSPDSs) under spatio-temporal disturbances and sensor measurement noises. A general theoretical H∞ robust observer-based stabilization method is introduced at the beginning for LSPDSs under intrinsic fluctuation, external disturbance and sensor measurement noise in the spatio-temporal domain. A complex Hamilton Jacobi integral inequality (HJII) needs to be solved when designing the robust H∞ observer-based stabilization for LSPDSs. For simplifying the design procedure, a stochastic state space model is first developed via the semi-discretization finite difference scheme to represent the stochastic partial differential system at each grid node. Then the stochastic state space models at all grid nodes are merged together into a stochastic spatial state space model. Based on this stochastic spatial state space model, an implementable H∞ robust stabilization design is proposed for LSPDSs via an iterative linear matrix inequality (ILMI) method. The proposed robust H∞ stabilization design can efficiently attenuate the effect of spatio-temporal external disturbances and measurement noises upon LSPDSs from the area energy point of view. Finally, a robust H∞ stabilization example simulation is shown to illustrate the design procedure and to confirm the performance of the proposed robust stabilization design method.

A note on [formula omitted]-theory of linear stochastic parabolic partial differential systems

In this article we construct a Wpγ-theory of linear stochastic parabolic partial differential systems. Here, p∈[2,∞) and γ∈(−∞,∞). We also provide an example to show that for stochastic systems we need more restriction than the algebraic condition which ensures that diffusion survives against wild convection.

Robust Filter for Linear Stochastic Partial Differential Systems via a Set of Sensor Measurements

This study addresses a robust H∞ filtering design problem for linear stochastic partial differential systems (LSPDSs) with external disturbance and measurement noise in the spatio-temporal domain. For LSPDSs, the robust H∞ filter design via a set of sensor measurements needs to solve a complex Hamilton Jacobi integral inequality (HJII) for robust state estimation despite external disturbance and measurement noise. In order to simplify the design procedure, a stochastic spatial state space model is developed to represent the stochastic partial differential system via the semi-discretization finite difference scheme and the Kronecker product. Then based on this model a robust H∞ filter design is proposed to achieve the robust state estimation via solving the linear matrix inequality (LMI). The proposed robust H∞ filter has an efficient ability to attenuate the effect of spatio-temporal external disturbance and measurement noise on the state estimation of LSPDSs from the area energy point of view. Finally, a robust H∞ state estimation example is given for the illustration of design procedure and the performance confirmation of the proposed robust filter design method.

On exponential stability conditions of linear neutral stochastic differential systems with time‐varying delay

SUMMARYWe consider a class of neutral stochastic systems with time‐varying delay and study the exponential stability in the mean square sense. We derive sufficient stability conditions via applying Lyapunov functional method along with some practical techniques. Firstly, in computing the constructed Lyapunov functional, we make use of some basic rules of Itô calculus to reduce the conservatism produced by noise because it, in principle, plays a negative role for preserving stability in the mean square sense. Also, it is an important observation that, using some slack matrices, we can create convex conditions to accommodate the computation to time‐varying delay. In the sequel, we use a perturbation approach to estimate the decay rate of state and come to the conclusion of stability. Finally, we include an example to demonstrate the effectiveness of the method. Copyright © 2012 John Wiley & Sons, Ltd.

A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems

We are concerned with a linear mean-square stability analysis of numerical methods applied to systems of stochastic differential equations (SDEs) and, in particular, consider the θ-Maruyama and the θ-Milstein method in this context. We propose an approach, based on the vectorisation of matrices and the Kronecker product, that allows us to deal efficiently with the matrix expressions arising in this analysis and that provides the explicit structure of the stability matrices in the general case of linear systems of SDEs. For a set of simple test SDE systems, incorporating different noise structures but only a few parameters, we apply the general results and provide visual and numerical comparisons of the stability properties of the two methods.

Stability of linear stochastic differential delay systems under impulsive control

In this study, the authors investigate stability of stochastic differential delay systems under impulsive control. First, an impulsive control for a kind of linear stochastic differential delay system is constructed. Then, the authors point out that different kinds of stability of linear stochastic differential delay systems under impulsive control is equivalent respectively, to the stability of the corresponding linear stochastic differential delay systems without impulses. Finally, several stability criteria of a class of linear stochastic differential delay systems under impulsive control are presented. Moreover, an example is also discussed to illustrate the effectiveness of the proposed results.

On Generalized Regular Stochastic Differential Delay Systems with Time Invariant Coefficients

In this article, we consider the generalized linear regular stochastic differential delay system with constant coefficients and two simultaneous external differentiable and non differentiable perturbations. These kinds of systems are inherent in many application fields; among them we mention fluid dynamics, the modeling of multi body mechanisms, finance and the problem of protein folding. Using the regular Matrix Pencil theory, we decompose it into two subsystems, whose solutions are obtained as generalized processes (in the sense of distributions). Moreover, the form of the initial function is given, so the corresponding initial value problem is uniquely solvable. Finally, two illustrative applications are presented using white noise and fractional white noise, respectively.