Control Of Stochastic Systems Research Articles

Existence of a mild solution and approximate controllability for fractional random integro-differential inclusions with non-instantaneous impulses

This paper investigates the existence and approximate controllability (ACA) of fractional neutral-type stochastic differential inclusions (NTSDIs) characterized by non-instantaneous impulses within a separable Hilbert space (HS) framework. Employing the Atangana–Baleanu–Caputo (ABC) derivative, we transform the system into an equivalent fixed-point (FP) problem through an integral operator. Subsequently, the Bohnenblust–Karlin FP theorem is leveraged to establish existence results. By assuming ACA of the corresponding linear system, we derive sufficient conditions for the ACA of the nonlinear stochastic impulsive control system. Our analysis relies on concepts from stochastic analysis, fractional calculus, FP theory, semigroup theory, and the theory of multivalued maps (MVMs). The theoretical findings are illustrated through a concrete example.

Fault Isolation and Fault‐Tolerant Control Design for Non‐Gaussian Stochastic Distribution Control Systems With Multiple Sensor Faults

ABSTRACTA fault isolation, estimation and fault‐tolerant control algorithm is proposed for non‐Gaussian stochastic distribution control systems with disturbance and multiple sensor faults. Sensor faults are represented as actuator faults virtually, and an observer is devised to detect the sensor fault occurrence time. Then two subsystems are separated by the expanded system through introducing the coordinate transformation matrices. One subsystem contains only sensor faults and does not contain disturbance and the other contains sensor faults and disturbance, which provides convenience for fault isolation. The faults are estimated respectively by the multiple fault isolation observers with the same number of sensors. A fault‐tolerant control scheme is proposed after getting the fault information to compensated sensor faults and track the desired probability density function. Finally, a MATLAB simulation example is used to verify the feasibility of the algorithm.

A Lyapunov–Razumikhin control strategy for stochastic nonlinear delayed systems with polynomial conditions

Numerous control schemes for stochastic systems involving time-varying delays need a strict slow time-delay condition, which does hold for some practical models. Also, some stochastic systems will have different structures as working conditions change. In this article, we will investigate the control of stochastic systems which contain time-varying delays and polynomial conditions. By providing the dynamic-gain based homogeneous domination approach, and selecting a new Lyapunov–Razumikhin (L–R) function, a universal dynamic controller for systems with different growing conditions is presented, and the slow time-delay condition is removed successfully. The algorithm is finally verified with a practical example.

Pth moment asymptotic stability for stochastic complex networked control systems with Lévy noise

Pth moment asymptotic stability for stochastic complex networked control systems with Lévy noise

Finite time prescribed performance control for stochastic systems with asymmetric error constraint and actuator faults

This paper investigates the problem of finite time prescribed performance control (PPC) for a number of nonlinear stochastic systems with asymmetric error constraint, unknown control directions, and actuator faults. Firstly, instead of introducing the performance constraint function in the Lyapunov function, a new asymmetric error conversion function (AECF) is presented, which can successfully constrain the tracking errors into the specified asymmetric boundaries and eliminate the feasibility condition of requiring tracking errors to be bounded. Then, in iterative process, the unknown intermediate virtual controller is approximated by the fuzzy logic system (FLS), which makes each actual virtual controller to be reduced to only one item. Furthermore, the investigated finite time PPC strategy can fully compensate the faults impact on the systems and have only one adaptive parameter. In the end, the efficacy of investigated strategy is verified by inverted pendulum systems with stochastic disturbances.

A Novel Accelerated Multistage Learning Control Mechanism via Virtual Performance Reduction.

This study uses a multistage learning mechanism concept to investigate the accelerated learning control for stochastic systems. In this mechanism, the learning iterations are divided into successive stages, with each stage comprising several iterations. The learning gain is constant in each stage to accelerate the learning process and decreases it from one stage to another to eliminate the noise effect asymptotically. The critical issue is determining the switching iteration when a new stage starts. This study resolves this issue by calculating a virtual performance index of the mean-squared input error and its estimated upper bound. Specifically, the ideal, practical, and improved multistage learning control schemes are proposed to determine the switching iteration and generate the learning gain sequence. The ideal scheme achieves the best performance at the cost of a large computation burden, and the practical scheme saves computation cost, but the performance is not excellent. The improved scheme significantly approximates the best performance by introducing additional stretching parameters to the performance index. Illustrative simulations are provided to verify the theoretical results.

Adaptive output feedback control of stochastic systems with mismatched uncertainties input–output quantization

Inspired by the fact that remote control systems always demand to be quantified, and considering that there are always stochastic disturbances in the actual situation, this study discusses a sort of systems with unknown constants due to the existence of parameters that are untoward to measure. Therefore, in this research, a sort of stochastic nonlinear systems with input–output quantization mismatched uncertainties is studied by utilizing backstepping method. The difficulty is that the output signal becomes non-differential after being quantized. In addition, only the quantized output signal in the system can be regarded as known during the analysis. Therefore, the update of unknown parameters and the design of controller will grow arduous. In order to solve the above dilemma, a state observer and a projection operator are introduced. Moreover, in the design of the controller, the power form of Lyapunov function and differential method is employed to avoid the derivation of non-differentiable functions. Since the controller proposed in this study contains a small number of parameters that need to be designed, it is practical and can eliminate the problem of computational explosion. Simulation results demonstrate the effectiveness of the proposed scheme.

Controllability of partially observed stochastic semilinear fractional control systems

Controllability of partially observed stochastic semilinear fractional control systems

Design stabilisers for multi-input affine control stochastic systems via stochastic control Lyapunov functions

This paper studies the problem of state feedback stabilisation, in the sense of a weak solution, for a class of multi-input affine control stochastic systems whose drift and diffusion terms are dependent on the control and for which classical methods are not applicable. Based on the generalised stochastic Lyapunov theorem and on the technique of stochastic control Lyapunov functions, sufficient conditions for the existence of a continuous stabilising feedback control are proposed, and a state feedback controller can be developed to guarantee the closed-loop system globally asymptotically stable in probability. This work generalises previous results on the stabilisation of single-input affine control stochastic systems. The obtained results are illustrated by a numerical example.

Disturbance observer based fixed-time control of stochastic systems

In this paper, the novel fixed-time anti-disturbance control scheme is proposed for a class of stochastic systems subjected to multiple disturbances and faults, and the disturbances consist of derivative-bounded disturbances and multiply noise. Based on the pole placement method, the fixed-time disturbance observer (Fixed-time DO) is devised to estimate derivative-bounded disturbances and an adaptive law is employed to approach the time-varying fault. Then a composite fixed-time controller is constructed to make the system converge to equilibrium position within a pre-specified time. At the same time, simulation examples illustrate that the proposed controller is effective and the convergence time is not related to the initial states of the system.

On convergence of occupational measures sets of a discrete-time stochastic control system, with applications to averaging of hybrid systems

We establish that, under certain conditions, the set of random occupational measures generated by the state-control trajectories of a discrete-time stochastic system as well as the set of their mathematical expectations converge to a non-random, convex and compact set. We apply these results to the averaging a hybrid system with a slow continuous-time component and a fast discrete-time component. It is shown that the solutions of the hybrid system are approximated by the solutions of a differential inclusion. The novelty of our results is that we allow the state-control space of the fast component to be non-denumerable.

Learning to Order for Inventory Systems with Lost Sales and Uncertain Supplies

We consider a stochastic lost-sales inventory control system with lead time L over a planning horizon T. Supply is uncertain, and it is a function of the order quantity (because of random yield/capacity, etc.). We aim to minimize the T-period cost, a problem that is known to be computationally intractable even under known distributions of demand and supply. In this paper, we assume that both the demand and supply distributions are unknown and develop a computationally efficient online learning algorithm. We show that our algorithm achieves a regret (i.e., the performance gap between the cost of our algorithm and that of an optimal policy over T periods) of [Formula: see text] when [Formula: see text]. We do so by (1) showing that our algorithm’s cost is higher by at most [Formula: see text] for any [Formula: see text] compared with an optimal constant-order policy under complete information (a widely used algorithm) and (2) leveraging the latter’s known performance guarantee from the existing literature. To the best of our knowledge, a finite sample [Formula: see text] (and polynomial in L) regret bound when benchmarked against an optimal policy is not known before in the online inventory control literature. A key challenge in this learning problem is that both demand and supply data can be censored; hence, only truncated values are observable. We circumvent this challenge by showing that the data generated under an order quantity q2 allow us to simulate the performance of not only q2 but also, q1 for all [Formula: see text], a key observation to obtain sufficient information even under data censoring. By establishing a high-probability coupling argument, we are able to evaluate and compare the performance of different order policies at their steady state within a finite time horizon. Because the problem lacks convexity, commonly used learning algorithms, such as stochastic gradient decent and bisection, cannot be applied, and instead, we develop an active elimination method that adaptively rules out suboptimal solutions. This paper was accepted by Victor Martínez-de-Albéniz, operations management. Funding: This work is supported by the National Science Foundation [Grant CCF-2312205]. Z. Zhou also acknowledges the New York University’s 2024 Center for Global Economy and Business [Faculty Research Grant] and New York University [Research Catalyst Prize]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2022.02476 .

Multi-objective continuous review inventory policy using MOPSO and TOPSIS methods

In the dynamic business environment of today, decision-making involves considering various conflicting aspects. Inventory planning problems aim to determine how much and when to order products to satisfy customer demand at the lowest possible cost while maintaining a desirable service level. These problems can be formulated as a MOPSO algorithm, which is used to handle multiple objectives in a continuous review stochastic inventory control system (r, Q). Unfortunately, most multi-objective inventory models have been solved by aggregating objectives using specific weights or by optimizing only one objective and treating the others as constraints. Considering the complexity of real-world inventory control problems, which involves conflicting objectives such as minimizing cost and maximizing service level, the need arises to employ more precise optimizers that can generate better and more diverse non-dominated solutions of reorder point and order size system. In this paper multi-criteria decision-making framework that combines MOPSO algorithm and TOPSIS method to generate a Pareto front of non-dominated solutions and rank them based on decision makers' preferences. Initially, the original MOPSO is applied to the multi-objective inventory control problem, and then the mutation operator is integrated into the MOPSO to maintain diversity in the swarm and explore the entire search space. Next, the leader selection strategy called the geographically-based system (Grids) is replaced by the crowding distance factor to choose the global optimal particle as a leader. Additionally, the ε-dominance concept is employed to limit the archive size and maintain more diversity and convergence in the MOPSO for optimizing the inventory control problem. In conclusion, this work not only pioneers a cutting-edge approach to multi objective inventory control but also underscores its practical value. By facilitating the generation of superior solutions that cater to diverse decision-maker preferences, our methodology resonates deeply with real-world challenges and sets a new benchmark for effective inventory planning.

Optimal Impulse Control for Systems Deriven by Stochastic Delayed Differential Equations

In this paper we study the problem of optimal impulse control for stochastic systems with delay in the case when the value function of the impulse problem depends only on the initial data of the given process through its initial value (value at zero) and some weighted averages. A verification theorem for such impulse control problem is given. As an example the optimal stream of dividends with transaction costs is solved.

A model of the radar situation of the coastal vessel traffic control system

Since the elements of the ship traffic control system at sea (VTS) are geographically distributed, and the observed spaces have well-defined zones, the radar facilities (radars) that make up their technological basis are evaluated by the ability to detect, classify (recognize), and ensure tracking of ships at a certain range in the spatial zone. The solution of these problems is probabilistic in nature, and the indicators are represented by random functions. However, due to the complexity of accounting for random factors in such stochastic control systems, calculations of their efficiency indicators are usually carried out deterministically for the "average" values of random parameters. The development of a mathematical model for evaluating the effectiveness of VTS in its ability to detect marine targets is a scientific task, the solution of which is possible using elements of the theory of random functions and processes.

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper.

The second-order maximum principle for partially observed optimal controls

In this paper, the stochastic control system under partial observation for singular optimal controls in the classical sense is studied. Note that both the system and the observation contain noise, in which the control variable is allowed to enter into all the coefficient terms. By adding some regularity assumptions, the pointwise second-order maximum principle under partial observation can be obtained by Malliavin calculus. Eventually, an example is given to demonstrate how to find the optimal control.

Risk-Aware MPC for Stochastic Systems with Runtime Temporal Logics

This paper concerns the risk-aware control of stochastic systems with temporal logic specifications dynamically assigned during runtime. Conventional risk-aware control typically assumes that all specifications are predefined and remain unchanged during runtime. In this paper, we propose a novel, provably correct model predictive control scheme for linear systems with additive unbounded stochastic disturbances that dynamically evaluates the feasibility of runtime signal temporal logic specifications and automatically reschedules the control inputs accordingly. The control method guarantees the probabilistic satisfaction of newly accepted specifications without sacrificing the satisfaction of the previously accepted ones. The proposed control method is validated by a robotic motion planning case study.

A General Maximum Principle for Discrete Fractional Stochastic Control System of Mean‐Field Type

A General Maximum Principle for Discrete Fractional Stochastic Control System of Mean‐Field Type

A stochastic maximum principle for forward–backward stochastic control systems with quadratic generators and sample-wise constraints

This paper examines the stochastic maximum principle (SMP) for a forward–backward stochastic control system where the backward state equation is characterized by the backward stochastic differential equation (BSDE) with quadratic growth and the forward state at the terminal time is constrained in a convex set with probability one. With the help of the theory of BSDEs with quadratic growth and the bounded mean oscillation (BMO) martingales, we employ the terminal perturbation approach and Ekeland’s variational principle to obtain a dynamic stochastic maximum principle. The main result has a wide range of applications in mathematical finance and we investigate a robust recursive utility maximization problem with bankruptcy prohibition as an example.