Lipschitz Coefficient Research Articles

A note on reflected BSDEs in infinite horizon with stochastic Lipschitz coefficients

We consider an infinite horizon, obliquely reflected backward stochastic differential equation (RBSDE). The main contribution of the present work is that we generalize previous results on infinite horizon RBSDEs to the setting where the driver has a stochastic Lipschitz coefficient. As an application, we consider robust optimal stopping problems for functional stochastic differential equations (FSDEs) where the driver has linear growth.

Hyers–Ulam–Rassias Stability of Nonlinear Implicit Higher-Order Volterra Integrodifferential Equations from above on Unbounded Time Scales

In this paper, we present sufficient conditions for Hyers-Ulam-Rassias stability of nonlinear implicit higher-order Volterra-type integrodifferential equations from above on unbounded time scales. These new sufficient conditions result by reducing Volterra-type integrodifferential equations to Volterra-type integral equations, using the Banach fixed point theorem, and by applying an appropriate Bielecki type norm, the Lipschitz type functions, where Lipschitz coefficient is replaced by unbounded rd-continuous function.

Doubly reflected BSDEs driven by RCLL martingales under stochastic Lipschitz coefficient

In this paper, we study a backward stochastic differential equation driven by a Right Continuous with Left Limits (RCLL) martingale with two completely separated RCLL barriers. When the coefficient is stochastically Lipschitz, we demonstrate the existence and uniqueness of a square-integrable adapted solution using the penalization method. Additionally, we provide a fair price for a game contingent claim between two traders with additional exogenous knowledge of the same stock price in a public financial market driven by an Azéma’s martingale. We also determine a saddle point for the game when the obstacles are left upper semi-continuous along stopping times.

An efficient data-driven approximation to the stochastic differential equations with non-global Lipschitz coefficient and multiplicative noise

<abstract><p>This paper studied the numerical approximation of the stochastic differential equations driven by non-global Lipschitz drift coefficient and multiplicative noise. An efficient data-driven method, called extended continuous latent process flow, was proposed for the underlying problem. Compared with the piecewise construction of a variational posterior process used in the classical continuous latent process flow developed by Deng et al. <sup>[<xref ref-type="bibr" rid="b13">13</xref>]</sup>, the principle idea of our method was to derive a variational lower bound by constructing a posterior latent process conditional on all information over the whole time interval to maximize the log-likelihood generated by the observations, which reduces the computational cost and, thus, provides a convenient way to approximate the considered equation. Particularly, our new method showed a better approximation to the underlying equation than the classical drift-$ \theta $ discretization scheme through numerical error comparison. Numerical experiments were finally reported to demonstrate the effectiveness and generalization performance of the proposed method.</p></abstract>

Cooperative output feedback tracking control of stochastic nonlinear heterogeneous multi‐agent systems

SummaryWe study cooperative output feedback tracking control of stochastic nonlinear heterogeneous leader‐following multi‐agent systems. Each agent has a continuous‐time stochastic nonlinear heterogeneous dynamics with an unmeasurable state, and there are additive and multiplicative noises along with information exchange among agents. We propose admissible distributed observation strategies for estimating the leader's and the followers' states, and admissible cooperative output feedback control strategies based on the certainty equivalence principle. By output regulation theory and stochastic analysis, we show if the dynamics of each agent satisfies the Lipschitz condition, and the product of the leader's Lipschitz coefficient, the intensity of multiplicative measurement noises, and the constant related to the leader's Lipschitz coefficient and dimension is less than the minimum nonzero eigenvalue of graph Laplacian, then there exist admissible distributed observation and cooperative control strategies to ensure mean square bounded output tracking. Finally, the effectiveness of our control strategies is demonstrated by a numerical simulation.

Nonlinear Volterra Integrodifferential Equations from above on Unbounded Time Scales

The paper is devoted to studying the existence, uniqueness and certain growth rates of solutions with certain implicit Volterra-type integrodifferential equations on unbounded from above time scales. We consider the case where the integrand is estimated by the Lipschitz type function with respect to the unknown variable. Lipschitz coefficient is an unbounded rd-function and the Banach fixed-point theorem at a functional space endowed with a suitable Bielecki-type norm.

Approximation of the invariant distribution for a class of ergodic SDEs with one-sided Lipschitz continuous drift coefficient using an explicit tamed Euler scheme

We study the behavior in a large time regime of an explicit tamed Euler-Maruyama scheme applied to a class of ergodic Ito stochastic differential equations with one-sided Lipschitz continuous drift coefficient and bounded globally Lipschitz diffusion coefficient. Our first main contribution is to prove moments for the numerical scheme, which, on the one hand, are uniform with respect to the time-step size, and which, on the other hand, may not be uniform but have at most polynomial growth with respect to time. Our second main contribution is to apply this result to obtain weak error estimates to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process, as a function of the time-step size and of the time horizon. The explicit tamed Euler scheme is shown to be computationally effective for the approximation of the invariant distribution: even if the moment bounds and error estimates are not proved to be uniform with respect to time, the obtained polynomial growth results in a marginal increase in the upper bound of the computational cost. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for stochastic differential equations with non-globally Lipschitz coefficients using an explicit tamed Euler-Maruyama scheme.

Generalized BSDEs for time inhomogeneous Lévy processes under non-deterministic Lipschitz coefficient

In this paper, we study the generalized backward stochastic differential equations driven by inhomogeneous Lévy processes (GBSDELs in short). We establish the existence and uniqueness of solution by using Picard's iteration setting under non-deterministic Lipschitz and monotone condition.

Adaptive observer design for uncertain nonlinear ODE‐PDE cascades

SummaryWe are revisiting the problem of adaptive observer design for systems that are constituted of an Ordinary Differential Equation (ODE), containing a globally Lipschitz function of the state, and a linear Partial Differential Equation (PDE) of a diffusion–reaction heat type. The ODE and PDE are connected in series and both are subject to parametric uncertainties. In addition to nonlinearity and uncertainty, the system complexity also lies in the fact that no sensor can be implemented at the junction point between the ODE and the PDE. In the absence of parameter uncertainty, nonadaptive state observers are available featuring exponential convergence. However, convergence is guaranteed only under the condition that either the Lipschitz coefficient is sufficiently small or the PDE domain length is sufficiently small. To get around this limitation, and also to account for parameter uncertainty, we develop a design that involves two concatenated adaptive observers, covering the two subintervals of the PDE domain. The proposed design employs one extra sensor, providing the measurement of the PDE state at an inner position close to the ODE‐PDE junction point. Both observers are shown to be exponentially convergent, under ad‐hoc persistent excitation (PE) conditions, with no limitation on the Lipschitz coefficient and domain length.

On the stochastic control-stopping problem

We study the stochastic control-stopping problem when the data are path-dependent and of polynomial growth. The approach is based on backward stochastic differential equations (BSDEs for short). The problem turns into the study of a specific reflected BSDE with a stochastic Lipschitz coefficient for which we show existence and uniqueness of the solution. We then establish its relationship with the value function of the control-stopping problem. The optimal strategy is exhibited. Finally in the Markovian framework we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

Observer Design for ODE-PDE Cascade with Lipschitz ODE Nonlinear Dynamics and Arbitrary Large PDE Domain

This paper deals with observer design for systems constituted of an Ordinary Differential Equation (ODE), containing a Lipschitz function of the state, and a linear Partial Differential Equation (PDE) of diffusion-reaction heat type. In addition to nonlinearity, system complexity also lies in the fact that no sensor can be implemented at the junction point between the ODE and the PDE. Boundary observers providing state estimates of the ODE and PDE do exist but their exponential convergence is only ensured under the condition that either the Lipschitz coefficient is sufficiently small or the PDE domain length is sufficiently small. Presently, we show that limitation is removed by using two connected observers and implementing one extra sensor providing the PDE state at an inner position close to the ODE-PDE junction point. The measurements provided by both sensors are used by the first observer which estimates, not only the PDE state, but also its spatial derivative along the PDE subdomain located between both sensor positions. Using these estimates, the second observer determines estimates of the ODE state and the remaining PDE states. All state estimates are shown to be exponentially convergent to their true values.

Hyperbolically Lipschitz Continuity, Area Distortion, and Coefficient Estimates for (K, K′)-Quasiconformal Harmonic Mappings of the Unit Disk

Hyperbolically Lipschitz Continuity, Area Distortion, and Coefficient Estimates for (K, K′)-Quasiconformal Harmonic Mappings of the Unit Disk

Robust [formula omitted] model predictive control for constrained Lipschitz non-linear systems

This paper is concerned with the robust H∞ model predictive control problem for a class of non-linear systems subject to state and input constraints and with norm bounded disturbances. It is well known that the disturbance degrades the control performance remarkably and can cause system instability. As well as, design the H∞ model predictive control based on the non-linear model is much more challenging, because the control problem changes to a non-convex non-linear problem. The main contribution is the introduction of the novel robust controller to stabilize the perturbed Lipschitz non-linear systems. The objective is to minimize the L2 gain between the disturbance input and the controlled output. The non-convex control problem is formulated as a linear matrix inequality optimization problem. The designed controller guarantees the closed-loop asymptotic stability with a prescribed H∞ disturbance attenuation level. In order to reduce conservatism, a sum of squares optimization problem is proposed to obtain the optimal value of Lipschitz coefficient. The proposed algorithm is applied to two non-linear systems, a laboratory tank and a DC/AC converter to evaluate its applicability and effectiveness.

Approximation of BSDE with non Lipschitz coefficient

In this paper we study the discrete approximation of backward stochastic differential equations. Under a kind of non-Lipschitz assumption we prove the convergence of the proposed discrete scheme to the solution of backward stochastic differential equation.

Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for -quasiconformal harmonic mappings of unit disk

UDC 517.51 We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem, and coefficient estimate for the classes of -quasiconformal harmonic mappings from the unit disk onto itself.

Regularity of the Inertial Manifolds for Evolution Equations in Admissible Spaces and Finite-Dimensional Feedback Controllers

Our main aims in this paper are to investigate the regularity of inertial manifolds for non-autonomous semi-linear evolution equations and to give an application of inertial manifolds to a feedback control problem. We first prove that the inertial manifolds are smooth if the nonlinear term is smooth. Then, using the theory of inertial manifolds for non-autonomous semi-linear evolution equations, we construct a feedback controller for a class of control problems for the one-dimensional reaction-diffusion equations with the Lipschitz coefficient of the nonlinear term which may depend on time and belong to an admissible space.

Predictable solution for reflected BSDEs when the obstacle is not right-continuous

AbstractIn the present paper, we consider reflected backward stochastic differential equations when the reflecting obstacle is not necessarily right-continuous in a general filtration that supports a one-dimensional Brownian motion and an independent Poisson random measure. We prove the existence and uniqueness of a predictable solution for such equations under the stochastic Lipschitz coefficient by using the predictable Mertens decomposition.

Reflected BSDEs with jumps and two rcll barriers under stochastic Lipschitz coefficient

In this article, we give a solution to doubly reflected backward stochastic differential equations when the barriers are right continuous with left limits, and the noise is driven by a Brownian motion and an independent Poisson random measure. We show the existence and uniqueness of solution by means of the penalization method when the coefficient is stochastic Lipschitz.

Boundary iterative learning control for a class of first-order hyperbolic system with non-local terms

In this paper, the output tracking problem for a class of first-order hyperbolic equations with non-local terms over repeatable finite time interval is addressed. On the boundary of the system, a kind of iterative learning control scheme composed of a feedback and a feedforward mechanisms is proposed. Through constructing the input-output equivalent form of the original system, the robust convergence of the boundary iterative learning control strategy with respect to the initial states errors, iteration dependent, independent and state dependent disturbances is guaranteed via rigorous analysis. It is shown that the upper bound of the convergence error is continuously depend on the bounds of the initial states errors, iteration dependent or independent disturbances and the Lipschitz coefficient of the state dependent nonlinear disturbance. Finally, several numerical examples and a practical plate heat exchanger process applying the proposed boundary ILC scheme are performed to validate the performance of the algorithms.

BSDEs with right upper-semicontinuous reflecting obstacle and stochastic Lipschitz coefficient

Abstract This paper proves the existence and uniqueness of a solution to reflected backward stochastic differential equations with a lower obstacle, which is assumed to be right upper-semicontinuous. The result is established where the coefficient is stochastic Lipschitz by using some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales and other tools from optimal stopping theory.