Bellman Inequality Research Articles

Existence and k‐Mittag–Leffler–Ulam stabilities of a Volterra integro‐differential equation via (k,ϱ)‐Hilfer fractional derivative

In this paper, we investigate the existence, uniqueness, and analysis of two types of ‐Mittag–Leffler–Ulam stabilities in a Volterra integro‐differential fractional differential equation that involves the ‐Hilfer operator. We utilize the Banach fixed‐point theorem to establish the existence and uniqueness of solutions. We examine the stability properties, including the ‐Mittag–Leffler–Ulam–Hyers ‐ and k‐Mittag–Leffler–Ulam–Hyers–Rassias ‐ stabilities, by employing the Grönwall–Bellman inequality. Additionally, we provide an example to confirm our findings.

Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations

Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in Lp spaces with p≥2 under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end.

Dynamic memory-based event-triggered security control for networked control systems subjected to false-injection-data attack

This study addresses the security control problem for networked control systems that are subjected to false-data-injection attacks. The designed controller rests on a dynamic memory-based event-triggered mechanism, which exploits historically transmitted data and an auxiliary term to improve transmission efficiency and ensure satisfactory control performance. Then, a controller that defends against the false-data-injection attacks is proposed by making full use of the historical information. With the aid of Cauchy–Schwarz and Gronwall–Bellman inequalities, the existence of such a controller is validated through a novel Lyapunov-based method, which allows the Lyapunov function to be nonpositive. Further, a test on a dynamic power system is provided to verify the effectiveness and superiority of the proposed method.

SOME NEW TYPES OF GRONWALL-BELLMAN INEQUALITY ON FRACTAL SET

Gronwall–Bellman-type inequalities provide a very effective way to investigate the qualitative and quantitative properties of solutions of nonlinear integral and differential equations. In recent years, local fractional calculus has attracted the attention of many researchers. In this paper, based on the basic knowledge of local fractional calculus and the method of proving inequality on the set of real numbers, some new Gronwall–Bellman inequalities are discussed and established on the fractal space. Our results are a powerful supplement to the existing literature.

Non-fragile output-feedback control for time-delay neural networks with persistent dwell time switching: A system mode and time scheduler dual-dependent design

This paper is concerned with non-fragile output-feedback control for time-delay neural networks with persistent dwell time (PDT) switching in a continuous-time setting. The main purpose is to design an output-feedback controller subject to gain fluctuations, guaranteeing both asymptotic stability and L2-gain of the closed-loop control system. To achieve reduced conservatism, the controller is formulated to depend not only on the system mode but also on a time scheduler constructed based on the PDT switching rule and minimum time span. A criterion for the asymptotic stability and L2-gain analysis is established through the application of the Gronwall–Bellman inequality and mathematical induction. Then, a numerically tractable design approach for the desired controller is proposed, utilizing a four-section piecewise time-dependent Lyapunov–Krasovskii functional and several nonlinearity decoupling techniques. For comparative purposes, a simple case, independent of the time scheduler, is also investigated, and the corresponding controller design approach is presented. Finally, a simulation example is given to illustrate the effectiveness and superiority of the proposed system mode and time scheduler dual-dependent controller design approach.

On Generalizations of Hölder's and Minkowski's Inequalities

We present the generalizations of Hölder's inequality and Minkowski's inequality along with the generalizations of Aczel's, Popoviciu's, Lyapunov's and Bellman's inequalities. Some applications for the metric spaces, normed spaces, Banach spaces, sequence spaces and integral inequalities are further specified. It is shown that $({\mathbb{R}}^n,d)$ and $\left(l_p,d_{m,p}\right)$ are complete metric spaces and $({\mathbb{R}}^n,{\left\|x\right\|}_m)$ and $\left(l_p,{\left\|x\right\|}_{m,p}\right)$ are $\frac{1}{m}-$Banach spaces. Also, it is deduced that $\left(b^{r,s}_{p,1},{\left\|x\right\|}_{r,s,m}\right)$ is a $\frac{1}{m}-$normed space.

A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results.

Robust Stabilization for Uncertain Non-Minimum Phase Switched Nonlinear System under Arbitrary Switchings

This paper addresses the problem of stabilization of non-minimum phase switched nonlinear systems where the internal dynamics with symmetries or non-symmetries of each mode may be unstable. The authors initially build a stabilizing Lyapunov controller for each mode in order to stabilize its own unstable internal dynamics. The proposed approach is based on the exact input-output feedback linearization technique and the Lyapunov stability theory. The stability results for non-minimum phase switched nonlinear systems with arbitrary switching rules are then obtained using generalized Gronwall–Bellman inequalities. Finally, numerical examples are provided to demonstrate the efficacy of the achieved results.

Stochastic Sampled-Data Exponential Synchronization of Markovian Jump Neural Networks With Time-Varying Delays.

In this article, the exponential synchronization of Markovian jump neural networks (MJNNs) with time-varying delays is investigated via stochastic sampling and looped-functional (LF) approach. For simplicity, it is assumed that there exist two sampling periods, which satisfies the Bernoulli distribution. To model the synchronization error system, two random variables that, respectively, describe the location of the input delays and the sampling periods are introduced. In order to reduce the conservativeness, a time-dependent looped-functional (TDLF) is designed, which takes full advantage of the available information of the sampling pattern. The Gronwall-Bellman inequalities and the discrete-time Lyapunov stability theory are utilized jointly to analyze the mean-square exponential stability of the error system. A less conservative exponential synchronization criterion is derived, based on which a mode-independent stochastic sampled-data controller (SSDC) is designed. Finally, the effectiveness of the proposed control strategy is demonstrated by a numerical example.

The existence and averaging principle for stochastic fractional differential equations with impulses

In this paper, a class of fractional stochastic differential equations (SFDEs) with impulses is considered. By virtue of Mönch's fixed point theorem and Banach contraction principle, we explore the existence and uniqueness of solutions to the addressed system. Furthermore, with the aid of the Jensen inequality, Hölder inequality, Burkholder–Davis–Gundy inequality, Grönwall–Bellman inequality, and some novel assumptions, the averaging principle of our considered system is obtained. At the end of this paper, an example is provided to illustrate the theoretical results.

Existence Results for Impulsive Fractional Integrodifferential Equations Involving Integral Boundary Conditions

This research paper is devoted to investigating the existence results for impulsive fractional integrodifferential equations in the form of Atangana - Baleanu - Caputo (ABC) fractional derivative, by using Gronwall–Bellman inequality and Krasnoselskii’s fixed point theorem to study the existence and uniqueness of the problem with integral boundary conditions. At the end, the examples are illustrated to verify results.

Finite-time synchronization of fractional-order memristive neural networks via feedback and periodically intermittent control

This paper discusses the finite-time synchronization (FTS) of fractional-order memristive neural networks (FMNNs) with time-varying delays. Firstly, based on Gronwall–Bellman inequality and the fractional-order derivative of the power function, two novel propositions on finite-time fractional functional differential inequality are built. Secondly, the feedback controller and intermittent controller designed in this paper are all delay-independent controllers, which can work even when the prior state cannot be measured or the specific time delay function is unknown. Thirdly, in addition to the traditional Lyapunov function with absolute value form, a more general and flexile Lyapunov function based on p-norm form is constructed to analyze the criteria of FTS. Meanwhile, an improved estimation of settling time of fractional system is given explicitly, which is more accurate and general than the existing results. Eventually, the validity of theoretical analysis is confirmed by numerical examples.

Trajectory controllability of nonlinear fractional Langevin systems

Abstract In this paper, we discuss the trajectory controllability of linear and nonlinear fractional Langevin dynamical systems represented by the Caputo fractional derivative by using the Mittag–Leffler function and Gronwall–Bellman inequality. For the nonlinear system, we assume Lipschitz-type conditions on the nonlinearity. Examples are given to illustrate the theoretical results.

Data-Driven Optimal Control of Affine Systems: A Linear Programming Perspective

In this letter, we discuss the problem of optimal control for affine systems in the context of data-driven linear programming. First, we introduce a unified framework for the fixed point characterization of the value function, Q-function and relaxed Bellman operators. Then, in a model-free setting, we show how to synthesize and estimate Bellman inequalities from a small but sufficiently rich dataset. To guarantee exploration richness, we complete the extension of Willems’ fundamental lemma to affine systems.

Ulam Stability of n-th Order Delay Integro-Differential Equations

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

Delay-independent stability criteria for fractional order time delayed gene regulatory networks in terms of Mittag-Leffler function

This paper investigates the finite-time delay-free stability criteria for fractional-order gene regulatory networks (FOGRNs) with feedback regulation time delays. The existence of unique equilibrium points of considered systems is established based on the Banach fixed point theorem and Cauchy–Schwarz inequality. Further, when the order γ satisfies 0<γ<1 and 1<γ<2, some novel delay-free sufficient conditions are derived to ensure the finite-time stability for addressing fractional order systems by using the ideas of generalized Gronwall–Bellman inequality, equivalent norm techniques, and Laplace transform. In the end, the article comes up with two numerical cases to manifest the applicability and superiority of the obtained theoretical results.

A novel result on averaging principle of stochastic Hilfer-type fractional system involving non-Lipschitz coefficients

In this paper, we investigate a class of stochastic Hilfer-type fractional differential equations (SHFDEs) involving non-Lipschitz coefficients. With the aid of the Laplace transform and its inverse, we derive an implicit solution to the SHFDEs. Subsequently, we present the averaging principle result using Jensen’s inequality, the Cauchy–Schwarz inequality, Doob’s martingale inequality, and the Grönwall–Bellman inequality. Finally, we solve an example problem numerically to prove the correctness of the theoretical solution.

The Time-fractional Airy Equation on the Metric Graph

Initial boundary value problem for the time-fractional Airy equation on a graph with finite bonds is considered in the paper. Properties of potentials for this equation are studied. Using these properties the solutions of the considered problem were found. The uniqueness theorem is proved using the analogue of Gr¨onwall-Bellman inequality and a-priory estimate

New results on the uniform exponential stability of nonautonomous perturbed dynamical systems

AbstractIn this article, we investigate the asymptotic behaviors of the solutions of nonlinear dynamic systems nearby an equilibrium point, when the nominal parts are subject to non‐necessarily small perturbations. We show that, under some estimates on the perturbation term, the equilibrium point remains (globally) uniformly exponentially stable. The obtained stability results can easily be applied in practice since they are based on the Gronwall–Bellman inequalities rather than the classical Lyapunov methods that require the knowledge of a Lyapunov function. Several numerical examples, as well as an application to control and mechanical systems, are given in illustration.

Online Allocation and Pricing: Constant Regret via Bellman Inequalities

We develop a framework for designing simple and efficient policies for a family of online allocation and pricing problems that includes online packing, budget-constrained probing, dynamic pricing, and online contextual bandits with knapsacks. In each case, we evaluate the performance of our policies in terms of their regret (i.e., additive gap) relative to an offline controller that is endowed with more information than the online controller. Our framework is based on Bellman inequalities, which decompose the loss of an algorithm into two distinct sources of error: (1) arising from computational tractability issues, and (2) arising from estimation/prediction of random trajectories. Balancing these errors guides the choice of benchmarks, and leads to policies that are both tractable and have strong performance guarantees. In particular, in all our examples, we demonstrate constant-regret policies that only require resolving a linear program in each period, followed by a simple greedy action-selection rule; thus, our policies are practical as well as provably near optimal.