One-sided Lipschitz Research Articles

One Sided Lipschitz Evolution Inclusions in Banach Spaces

Using the notion of limit solution, we study multivalued perturbations of m-dissipative differential inclusions with nonlocal initial conditions. These solutions enable us to work in general Banach spaces, in particular L1. The commonly used Lipschitz condition on the right-hand side is weakened to a one-sided Lipschitz one. No compactness assumptions are required. We consider the cases of an arbitrary one-sided Lipschitz condition and the case of a negative one-sided Lipschitz constant. Illustrative examples, which can be modifications of real models, are provided.

Existence and pathwise uniqueness of solutions for stochastic differential equations involving the local time at point zero

In this article, we aim to obtain the existence and pathwise uniqueness of the solution to the one-dimensional stochastic differential equations involving the local time (SDELT) at point zero. The existence and pathwise uniqueness theorem for class of SDELT is established under the drift coefficient satisfies a one-sided Lipschitz condition plus the superlinear condition.

State and disturbance simultaneous estimation for a class of nonlinear time-delay fractional-order systems

This article addresses the problem of estimating simultaneously the state and unknown disturbance of one-sided Lipschitz fractional-order systems with time-delay. The nominal models of nonlinearities are assumed to satisfy both the one-sided Lipschitz condition and the quadratically inner-bounded condition. Different from the state observer reported in the literature, which only dealt with one-sided Lipschitz integer-order time-delay systems or nonlinear fractional-order time-delay systems where the nonlinear function satisfying Lipschitz condition, the state observers in this article can be applied to a wide class of nonlinear time-delay systems (one-sided Lipschitz fractional-order time-delay systems and one-sided Lipschitz integer-order time-delay systems). We employ the Razumikhin stability theorem and a recent result on the Caputo fractional derivative of a quadratic function to derive a sufficient condition for the asymptotic stability of the observer error dynamic system. The stability condition is obtained in terms of linear matrix inequalities, which can be effectively solved using the MATLAB LMI Control Toolbox. Two examples are provided to show the effectiveness of the proposed design approach.

Guaranteed‐performance consensus tracking for one‐sided Lipschitz non‐linear multi‐agent systems with switching communication topologies

The guaranteed-performance consensus tracking (GPCT) for non-linear multi-agent systems (MASs) with switching communication topologies is presented. Different from the existing work, the non-linear range of one-sided Lipschitz non-linear MASs is wider than the Lipschitz ones, and the upper bound of the tracking performance is given while achieving the consensus tracking. First, a guaranteed-performance tracking control protocol is proposed, where consensus tracking regulation performance is involved. Then, by the linear matrix inequality (LMI), sufficient conditions are provided to achieve the GPCT under the one-sided Lipschitz and quadratic inner bounded conditions. It is worth noting that a special LMI structure is directly constructed by introducing these two conditions to simplify the complexity of the derivation. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed results.

Observer-Based $$H_\infty $$ Control for One-Sided Lipschitz Nonlinear Systems with Uncertain Input Matrix

Observer-Based $$H_\infty $$ Control for One-Sided Lipschitz Nonlinear Systems with Uncertain Input Matrix

Optimal Control of Mean Field Equations with Monotone Coefficients and Applications in Neuroscience

We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution X=X^alpha of the stochastic mean-field type evolution equation in {mathbb {R}}^ddXt=b(t,Xt,L(Xt),αt)dt+σ(t,Xt,L(Xt),αt)dWt,X0∼μ(μgiven),(1)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} dX_t=b(t,X_t,{\\mathcal {L}}(X_t),\\alpha _t)dt+\\sigma (t,X_t,{\\mathcal {L}}(X_t),\\alpha _t)dW_t\\,, \\quad X_0\\sim \\mu (\\mu \\text { given),}\\qquad (1) \\end{aligned}$$\\end{document}under assumptions that enclose a system of FitzHugh–Nagumo neuron networks, and where for practical purposes the control alpha _t is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipschitz condition, and that the dynamics (2) satisfies an almost sure boundedness property of the form pi (X_t)le 0. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipschitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle for (2), and numerically investigate a gradient algorithm for the approximation of the optimal control.

Selection by vanishing common noise for potential finite state mean field games

The goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the square intensity of a common noise that is inserted in the mean field game or, equivalently, as the diffusivity parameter in the related parabolic version of the master equation. As established in the recent contribution (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98–162), the randomly forced mean field game becomes indeed uniquely solvable for a relevant choice of a Wright-Fisher common noise, the counterpart of which in the master equation is a Kimura operator on the simplex. We here elaborate on (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98–162) to make the mean field game with common noise both uniquely solvable and potential, meaning that its unique solution is in fact equal to the unique minimizer of a suitable stochastic mean field control problem. Taking the limit as the intensity of the common noise vanishes, we obtain a rigorous proof of the aforementioned selection principle. As a byproduct, we get that the classical solution to the viscous master equation associated with the mean field game with common noise converges to the gradient of the value function of the mean field control problem without common noise. We hence select a particular weak solution of the master equation of the original mean field game. Lastly, we establish an intrinsic uniqueness criterion for this solution within a suitable class of weak solutions to the master equation satisfying a weak one-sided Lipschitz inequality.

A novel delay-range-dependent observer-based control approach for one-sided Lipschitz systems under measurement delays

This paper presents the observer-based control methodology for the one-sided Lipschitz (OSL) nonlinear systems over measurement delays. A controller design method, based on the estimated states, has been provided by applying the Lyapunov-Krasovskii functional for the delayed dynamics and by inserting the OSL constraint and quadratic inner-boundedness condition. The stability of the resultant delayed dynamics is achieved through the delay-range-dependent approach, and derivative of Lyapunov functional is exploited through the Wirtinger's integral inequality approach to reduce the conservatism of the conventional Jensen's inequality scheme. Further, a necessary and sufficient solution for the main design method has been provided by employing a tedious decoupling technique to render the observer and controller gains, simultaneously, by using the recursive optimization tools. Furthermore, the solution of matrix inequality-oriented results is handled via the cone complementary linearization technique to validate the controller and observer gains through convex optimization. The effectiveness of the resultant observer-oriented control formulation for the OSL nonlinear systems under measurement delays is validated via numerical simulation examples.

Admissibility analysis and passive output feedback control for one-sided Lipschitz nonlinear singular Markovian jump systems with uncertainties

The paper investigates admissibility analysis, the existence and uniqueness (EU) of solutions and robustly passive asynchronous static output feedback (SOF) control for one-sided Lipschitz (OSL) nonlinear singular Markovian jump systems (SMJSs) with uncertainties. Sufficient conditions are derived to guarantee that OSL nonlinear SMJSs are regular, impulse-free, stochastically stable and have a unique solution with passive performance by using implicit function theorem. With these conditions, the design of robust asynchronous SOF controller is presented in form of linear matrix inequality (LMI) constraints. Finally, an example is given to evaluate the operation of our proposed methods.

Reference tracking and transient response shaping of uncertain quasi one-sided Lipschitz nonlinear systems

This work addresses the reference tracking problem for uncertain systems with quasi one-sided Lipschitz nonlinearity. The uncertainty is assumed to be of a norm bound parametric type. Moreover, transient response shaping using the concept of ‘return time’ is also proposed. The controller design relies on the solution of Linear Matrix Inequalities (LMIs) and hence is computationally efficient. The proposed control law is linear in states, and thus the implementation is often straightforward. To illustrate the capability and simplicity of the proposed theory, three design examples are included.

Synchronization of mutual coupled fractional order one-sided lipschitz systems

Up to now, the problem of synchronization of mutual coupled integer order systems has been tackled by several researchers. However, few works have been done to investigate such context on fractional order systems. In this context, this paper presents a complete methodology to solve this problem. On the other hand, to the best of our knowledge, among all the existing works dealing with the synchronization of mutual coupled fractional order systems, no paper has treated the special class of one-sided Lipschitz systems. In order to validate the theoretical results, two numerical examples are studied in the simulation section.

Stability analysis of quasi one-sided Lipschitz non-linear multi-agent system via sampled data control subject to directed switching topology

Abstract This paper is concerned with the problem of stability and consensus of non-linear multi-agent system by utilizing the sampled-data control. The innovative part of this paper is that the nonlinearity of this class of nonlinear systems is considered to satisfy a quasi one-sided Lipschitz condition. Communication among agents are assumed to be a switching directed graph. The principle target of this paper is to design a sampled data controller such that for all permissible uncertainties, the resulting closed-loop system is stable in the sense of mean square. For this reason, through the development of an appropriate Lyapunov–Krasovskii functional with dual integral terms and usage of Kronecker product properties alongside the matrix inequality techniques, a new set of stability and consensus conditions for the prescribed system is obtained in the form of a linear matrix inequality, which can be easily solved by the well-known effective numerical programming. Finally numerical examples are given to show the validity of the proposed hypothetical results.

A novel approach for event-triggered state-estimation of one-sided Lipschitz systems for efficient bandwidth utilization

This paper proposes a robust event-triggered observer design scheme for the systems fulfilling the one-sided Lipschitz and quadratic inner-boundedness conditions. The suggested observer scheme differs from the traditional methods because it deals with a less conservative and a more generalized class of nonlinear systems while utilizing fewer resources and meeting performance requirements. In contrast to the conventional Lipschitz methods, the proposed method considers a broader class of OSL systems, and it can avoid a high-gain observer, which can be sensitive to the measurement noise. The proposed linear matrix inequalities-based observer design guarantees the uniformly ultimately bounded convergence of the observation error with an exponential decay under disturbances. Also, several scenarios selecting a threshold for event-triggering schemes from an observer design perspective are addressed along with their advantages and issues. Simulation results are provided to validate the established results.

Observer Design for One-sided Lipschitz Uncertain Descriptor Systems with Time-varying Delay and Nonlinear Uncertainties

This paper investigates observer design for a class of one-sided Lipschitz descriptor systems with time-varying delay and uncertain parameters. In order to provide a general framework for large-scale systems, the paper considers uncertainties, nonlinearities, disturbance and time-varying delay at both output and state. By constructing Lyapunov–Krasovskii functional, and using the one-sided Lipschitz condition and the quadratic inner-boundedness inequality, we establish the sufficient condition which guarantees that the observer error dynamics is asymptotically stable, and the proposed observer ensures the $$L_2$$ gain bounded by a scalar $$\gamma .$$ Then, we change the condition into a strict matrix inequality condition. Furthermore, based on the obtained results, we establish the linear matrix inequality-based condition to ensure the asymptotically convergence of state estimation error and to accomplish robustness against $$L_2$$ norm bounded disturbances by utilizing change of variables. We propose the computing method of observer gain. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed method.

Uniqueness and energy balance for isentropic Euler equation with stochastic forcing

In this article, we prove uniqueness and energy balance for isentropic Euler system driven by a cylindrical Wiener process. Pathwise uniqueness result is obtained for weak solutions having Hölder regularity Cα,α>1∕2 in space and satisfying one-sided Lipschitz bound on velocity. We prove Onsager’s conjecture for isentropic Euler system with stochastic forcing, that is, energy balance equation for solutions enjoying Hölder regularity Cα,α>1∕3. Both the results have been obtained in a more general setting by considering regularity in Besov space.

Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

AbstractThis paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. The considered systems are described by second-order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit, drift-truncated Euler scheme that fulfills the constraint exactly. Pathwise uniform $L_p$-convergence is established. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semistability and moment growth properties of the scheme.

Uniqueness of Dissipative Solutions to the Complete Euler System

Dissipative solutions have recently been studied as a generalized concept for weak solutions of the complete Euler system. Apparently, these are expectations of suitable measure valued solutions. Motivated from Feireisl et al. (Commun Partial Differ Equ 44(12):1285–1298, 2019), we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space $$B^{\alpha ,\infty }_{p}$$ with $$\alpha >1/2$$ . We prove that the Besov solution satisfying the above mentioned condition is unique in the class of dissipative solutions. In the later part of this article, we prove that the one sided Lipschitz condition gives uniqueness among weak solutions with the Besov regularity, $$B^{\alpha ,\infty }_{3}$$ for $$\alpha >1/3$$ . Our proof relies on commutator estimates for Besov functions and the relative entropy method.

Impulsive consensus of one-sided Lipschitz nonlinear multi-agent systems with Semi-Markov switching topologies

Many real systems involve not only parameter changes but also sudden variations in environmental conditions, which often causes unpredictable topologies switching. This paper investigates the impulsive consensus problem of the one-sided Lipschitz nonlinear multi-agent systems (MASs) with Semi-Markov switching topologies. Different from the existing modeling methods of the Markov chain, the Semi-Markov chain is adopted to describe this kind of randomly occurring changes reasonably. To cope with the communication and control cost constraints in the multi-agent systems, the distributed impulsive control method is applied to address the leader–follower consensus problem. Beyond that, to obtain a wider nonlinear application range, the one-sided condition is delicately developed to the controller design, and the results are different from the ones obtained in the traditional method with the Lipschitz condition (note that the existing results are usually only applicable to the case with small Lipschitz constant). Based on the characteristics of cumulative distribution functions, the theory of Lyapunov-like function and impulsive differential equation, the asymptotically mean square consensus of multi-agent systems is maintained with the proposed impulsive control protocol. Finally, an explanatory simulation is presented to validate the correctness of the proposed approach conclusively.

First-order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems.

In this paper, we derive fully implementable first-order time-stepping schemes for McKean-Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean-Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.

Finite-time event-triggered sliding mode control for one-sided Lipschitz nonlinear systems with uncertainties

This paper investigates the problem of finite-time event-triggered sliding mode control for one-sided Lipschitz nonlinear systems with uncertainties. The system is subjected to event-triggered scheme, which can automatically monitor the data transmission. Firstly, a zero-order-hold is employed to guarantee the control signal to be continuous, and an observer is proposed to estimate the system states. Then, some sufficient conditions are given to ensure the FTB of the resulting closed-loop system under consideration. Based on these, an observer and an observer-based sliding mode controller can be simultaneously involved in the design at one step in terms of linear matrix inequalities (LMIs). Two convex optimization problems subjected to LMIs are formulated to optimize the desired performance indices of interest to us. Finally, two practical examples are provided to verify the effectiveness of the proposed method.