Backward Differential Equations Research Articles

Nonlinear Optimal Control Based on FBDEs and its Application to AGV.

This article focuses on solving a finite-horizon nonlinear optimal control problem by using the Pontryagin's maximum principle. In practical applications, linearization is a common approach for solving nonlinear dynamical systems. However, it is not universally applicable due to various reasons, such as instability and low accuracy. In contrast to linearization, the inherent challenge in directly solving the above nonlinear optimal control problem lies in addressing the highly coupled nonlinear forward and backward differential equations. In order to address this problem, an equivalent relationship is established between these equations and a new optimization problem. By exploiting the inherent relationship between supervised learning and an optimization problem from the view of a dynamical system, a deep neural network framework is constructed for describing the new optimization problem. Furthermore, a numerical algorithm for optimal control, which is very powerful for a large variety of nonlinear dynamical systems, is implemented by training a deep residual network. Finally, the effectiveness of the algorithm is demonstrated by solving a trajectory tracking control problem for automatic guided vehicle. The obtained results reveal that the proposed control scheme can achieve high-precision tracking.

Uniqueness Problem for the Backward Differential Equation of a Continuous-State Branching Process

Uniqueness Problem for the Backward Differential Equation of a Continuous-State Branching Process

Optical constants and thickness gradients for light intensities in glass substrate layers and for complex electric fields in indium tin oxide (ITO) transparent conductor thin film layer, using Bode and Nyquist wavelength-dependent diagrams

Abstract The optical constants of a thin film layer of transparent conductive indium tin oxide (ITO), deposited on a thick glass substrate (G), were obtained for an ITO-G sample using a multilayer matrix method by fitting the regular transmittance and specular reflectance measurements to the collimated equations of the four-flux model, once the optical constants of the glass substrate layer had previously been obtained from a G sample of a single glass substrate layer. The accuracy of the results was validated using a feedback system called the three-extinction matching requirement, that is, obtaining the same extinction coefficients from the optical constants and from the forward and backward collimated differential equations of the four-flux model. To calculate the extinction coefficients of the glass from optical constants, the compression of the wavelength of the incident light in the glass substrate and in the thin ITO film layer was demonstrated, with a thickness less than the wavelength of incident light. The thickness gradients of the forward and backward collimated light intensities, on the glass substrate of the ITOG sample, were compared with those obtained for the single-layer G sample. Then, the thickness gradients for the complex forward and backward electric fields of the ITO thin film layer were obtained and plotted using spectral magnitude and phase Bode plots and new Nyquist plots, i.e. imaginary versus real parts, dependent on wavelength.

Finite-horizon and infinite-horizon linear quadratic optimal control problems: A data-driven Euler scheme

In this paper, we devote our focus to studying finite-horizon and infinite-horizon linear quadratic optimal control (LQOC) problems in continuous time. We focus on algorithms for approximating optimal controls of these two problems without the knowledge of all system coefficients. Firstly, we transform the backward differential Riccati equation (DRE) corresponding to the finite-horizon LQOC problem into a forward ordinary differential equation. Then, we discretize the forward equation by the Euler method. Subsequently, we use the collected state and input data to calculate all discretized points of the forward equation. At the same time, we obtain the optimal control of the finite-horizon LQOC problem. Moreover, we approximate the optimal control of the infinite-horizon LQOC problem by virtue of an asymptotic behavior between the DRE and an algebraic Riccati equation (ARE) arising in the infinite-horizon LQOC problem. Finally, we validate the obtained results via two practically motivated examples.

Intrinsic and Extrinsic Scattering and Absorption Coefficients New Equations in Four-flux and Two-flux Models Used for Determining Light Intensity Gradients

Collimated, diffuse and total light intensity gradients, for forward and backward light senses, were determined in a three substrate layers – glass/electrolyte/glass – almost transparent sample, using optical constants and new equations for intrinsic and extrinsic scattering and absorption coefficients. These new equations were obtained for the inner electrolyte layer from the systems of differential equations of the four-flux and two-flux radiative transfer models, used for determining intrinsic and extrinsic coefficients, respectively, once knowing the optical constants of outer glass layers, from a single glass substrate sample measured in advance. Extinction coefficients were determined from optical constants and considering the wavelength compression of light when it enters into a material, decreasing its speed with respect to the vacuum. The same extinction coefficients for glass and electrolyte layers were computed in three different ways. First, from optical constants, determined using collimated transmittance and reflectance solutions of four-flux model. From them and as intermediate parameters for glass and electrolyte layers, collimated interface reflectance and attenuation due to extinction were computed since they were required for determining inner collimated light intensities at the interfaces, which were used at the collimated forward and backward differential equations, solving for the forward and backward extinction coefficients. The three-extinction matching requirement was successfully satisfied for the glass and electrolyte layers. Two average crossing parameters equations, for each sense, and four forward scattering ratios equations, for collimated and for diffuse light intensities for each sense, were used in the system of diffuse differential equations for intrinsic coefficients. For them, intuitive equations were proposed, based on collimated and diffuse light intensities at each interface. New equations for extrinsic parameters were determined by equalizing the system of total differential equations of the two-flux model to the sum of the systems of collimated and diffuse differential equations of the four-flux model.

Adaptive-frequency event-based approach to finite horizon state estimation for complex networks using multi-channel random access protocol

The issue of H∞ state estimation is addressed for complex networks (CNs) with inner coupling disturbance (ICD) while utilizing adaptive-frequency event-triggered mechanism (AFETM) and multi-channel random access protocol (MRAP). To maintain transmission frequency as much as possible while reducing data transmission, the proposed AFETM dynamically modifies the threshold based on the historical number of transmissions within a defined time interval. Data transmission after AFETM is scheduled by the MRAP, which enables the random selection and transmission of multiple signals, thereby avoiding network conflicts and congestion. The primary objective is to devise a estimator that guarantees the estimation errors meet the prescribed H∞ performance. The attainment of sufficient conditions to achieve this objective involves the utilization of stochastic analysis. On this basis, the expected estimator gain is determined through the solution of the backward coupled difference equations. Ultimately, the performance of the devised estimator is validated by simulation.

Open-loop and closed-loop Nash equilibria for the LQ stochastic difference game

This paper investigates the conditions for the existence of open-loop and closed-loop Nash equilibria for the linear quadratic (LQ) two-player stochastic difference game. It is shown that the former is characterized by the solvability of a forward–backward stochastic difference equation (FBSΔE), while the latter is described by the solvability of a class of coupled symmetric difference Riccati equations (ΔREs). Different from the existing works, the relationship between the closed-loop representation of open-loop Nash equilibrium and the closed-loop Nash equilibrium has been studied. It is found that the closed-loop representation of open-loop Nash equilibrium is different from the outcome of closed-loop Nash equilibrium for the general case. However, for the LQ zero-sum stochastic difference game, they are consistent.

Finite-Horizon H∞ State Estimation for Complex Networks With Uncertain Couplings and Packet Losses: Handling Amplify-and-Forward Relays.

This article is concerned with the state estimation problem for a class of complex networks (CNs) with uncertain inner couplings and packet losses over communication networks. The inner couplings are allowed to be uncertain and varying in a specific interval. The amplify-and-forward (AaF) relay protocols are introduced to improve the communication quality and enhance the propagation distance. The Bernoulli random variables are used to characterize the randomly occurring packet losses encountered in communication channels. The focus of this article is on the design of a state estimator for each node of CNs such that a prescribed H∞ performance constraint is satisfied for the dynamical error system over a finite horizon. A sufficient condition is first provided to verify the existence of the desired H∞ state estimator, and the estimator gain is then determined by solving two coupled backward Riccati difference equations (RDEs). Subsequently, a recursive state estimation algorithm is put forward that is suitable for online computation. Finally, a numerical example is given to demonstrate the effectiveness of the proposed estimation method.

First- and Second-Order Maximum Principles for Discrete-Time Stochastic Optimal Control With Recursive Utilities

This paper deals with the discrete-time stochastic optimal control problems with recursive utilities under weakened convexity assumption. A new stochastic maximum principle is established. Moreover, by constructing two new adjoint equations and two new variational equations for the backward stochastic difference equation (BS <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta$</tex-math></inline-formula> E), we obtain the second-order necessary optimality condition of quasi-singular control. Finally, as an illustration, a discrete-time mean-variance portfolio selection mixed with a recursive utility functional optimization problem is solved.

Some Stability Results and Existence of Solutions for a Backward Differential Equation with Time Advance via ζ—Caputo Fractional Derivative

In this paper, using a fixed point method, we proved the existence and uniqueness of solutions for a backward differential equation with time advance via ζ−Caputo fractional derivative. Furthermore, the Ulam–Hyers–Rassias and the Ulam–Hyers stabilities of the backward differential equation with time advance via ζ−Caputo fractional derivative are investigated. Finally, some experiments are given to illustrate the theoretical results.

Feedback Nash Equilibrium With Packet Dropouts in Networked Control Systems

This brief is concerned with the feedback Nash equilibrium for networked control systems (NCSs) with packet dropouts. The packet dropouts occur when the state information is transformed via a wireless network, causing the fall that the state information is unavailable to design the feedback Nash equilibrium. The main contribution of this brief is that the necessary and sufficient condition for the existence of feedback Nash equilibrium is presented. The key is that by decoupling solving the forward and backward difference equations (FBDEs), the analytical solutions of the two controllers are given in terms of the two coupled Riccati equations.

Continuous-Time Markov Decision Problems with Binary State

We analyse a binary state continuous-time Markov decision problem. The standard Hamilton–Jacobi– Bellman equation is introduced and, with suitable assumptions on the probability rate and on the cost function, it can be replaced by a simpler backward differential equation. Through a numerical example, we show how to find an optimal feedback control using the results presented in this paper.

A Game—Theoretic Model for a Stochastic Linear Quadratic Tracking Problem

In this paper, we solve a stochastic linear quadratic tracking problem. The controlled dynamical system is modeled by a system of linear Itô differential equations subject to jump Markov perturbations. We consider the case when there are two decision-makers and each of them wants to minimize the deviation of a preferential output of the controlled dynamical system from a given reference signal. We assume that the two decision-makers do not cooperate. Under these conditions, we state the considered tracking problem as a problem of finding a Nash equilibrium strategy for a stochastic differential game. Explicit formulae of a Nash equilibrium strategy are provided. To this end, we use the solutions of two given terminal value problems (TVPs). The first TVP is associated with a hybrid system formed by two backward nonlinear differential equations coupled by two algebraic nonlinear equations. The second TVP is associated with a hybrid system formed by two backward linear differential equations coupled by two algebraic linear equations.

Backward multivalued McKean-Vlasov SDEs and associated variational inequalities

The work concerns a type of backward multivalued McKean-Vlasov stochastic differential equations. First, we prove the existence and uniqueness of solutions for backward multivalued McKean-Vlasov stochastic differential equation. Then, it is presented that the first components of their solutions depend continuously on the terminal values. Finally, we give a probabilistic interpretation for viscosity solutions of nonlocal quasi-linear parabolic variational inequalities.

Existence, uniqueness and stability of solutions to fractional backward stochastic differential equations

ABSTRACT Many types of fractional stochastic differential equation (FrSDE), such as Caputo, fractional Brown motion derivatives, and Mittag-Later functions, exist. In recent decades, FrSDE has been a hot topic and can be applied to many fields of research, such as disease transmission, option pricing, and quantitative finance. FrSDEs have various research and applications in financial markets. After comparing many internationally known articles, the fractional order stochastic differential equation proposed in 2016 is most suitable for European option pricing. Over the years, many scholars have studied fractional Brownian motion, fractional ordinary differential equations, and backward differential equations, and no one has studied the application of fractional backward equations in the financial field. Therefore, in this article, to adapt to the financial market more accurately, we construct a backward equation for this kind of FrSDE and construct a new mapping and use the norm method to prove the existence, uniqueness, and stability of the solution to the backward equation. Finally, considering European call options, the Euler Maruyama simulation example of FrSDE is investigated.

Deep Neural Network Solution for Finite State Mean Field Game with Error Estimation

We discuss the numerical solution to a class of continuous time finite state mean field games. We apply the deep neural network (DNN) approach to solving the fully coupled forward and backward ordinary differential equation system that characterizes the equilibrium value function and probability measure of the finite state mean field game. We prove that the error between the true solution and the approximate solution is linear to the square root of DNN loss function. We give an example of applying the DNN method to solve the optimal market making problem with terminal rank-based trading volume reward.

An Asymptotic Radius of Convergence for the Loewner Equation and Simulation of $$SLE_{\kappa }$$ Traces via Splitting

In this paper, we shall study the convergence of Taylor approximations for the backward Loewner differential equation (driven by Brownian motion) near the origin. More concretely, whenever the initial condition of the backward Loewner equation (which lies in the upper half plane) is small and has the form $Z_{0} = \varepsilon i$, we show these approximations exhibit an $O(\varepsilon)$ error provided the time horizon is $\varepsilon^{2+\delta}$ for $\delta > 0$. Statements of this theorem will be given using both rough path and $L^{2}(\mathbb{P})$ estimates. Furthermore, over the time horizon of $\varepsilon^{2-\delta}$, we shall see that "higher degree" terms within the Taylor expansion become larger than "lower degree" terms for small $\varepsilon$. In this sense, the time horizon on which approximations are accurate scales like $\varepsilon^{2}$. This scaling comes naturally from the Loewner equation when growing vector field derivatives are balanced against decaying iterated integrals of the Brownian motion. As well as being of theoretical interest, this scaling may be used as a guiding principle for developing adaptive step size strategies which perform efficiently near the origin. In addition, this result highlights the limitations of using stochastic Taylor methods (such as the Euler-Maruyama and Milstein methods) for approximating $SLE_{\kappa}$ traces. Due to the analytically tractable vector fields of the Loewner equation, we will show Ninomiya-Victoir (or Strang) splitting is particularly well suited for SLE simulation. As the singularity at the origin can lead to large numerical errors, we shall employ the adaptive step size proposed by Tom Kennedy to discretize $SLE_{\kappa}$ traces using this splitting. We believe that the Ninomiya-Victoir scheme is the first high order numerical method that has been successfully applied to $SLE_{\kappa}$ traces.

Robust Discrete-Time Pontryagin Maximum Principle on Matrix Lie Groups

This article considers a discrete-time robust optimal control problem on matrix Lie groups. The underlying system is assumed to be perturbed by exogenous unmeasured bounded disturbances, and the control problem is posed as a min–max optimal control wherein the disturbance is the adversary and tries to maximize a cost that the control tries to minimize. Assuming the existence of a saddle point in the problem, we present a version of the Pontryagin maximum principle (PMP) that encapsulates first-order necessary conditions that the optimal control and disturbance trajectories must satisfy. This PMP features a saddle point condition on the Hamiltonian and a set of backward difference equations for the adjoint dynamics. We also present a special case of our result on Euclidean spaces. We conclude with applying the PMP to robust version of single axis rotation of a rigid body.

Encoding-decoding-based finite-horizon recursive secure state estimation for dynamic coupled networks with random coupling strength☆

The recursive secure state estimator design issue is investigated in this article for a class of dynamic coupled networks (DCNs) through the encoding-decoding scheme. The random variables distributed over specified intervals are utilized to depict the stochastically varying coupling strengths. For the sake of facilitating the data transmission, saving network resources and cutting down security risks, the two-description coding scheme (TDCS) is employed in the communication channels. Based on the TDCS, the encoders are installed at two independent communication channels, and the packet loss phenomena are considered in two encoder-to-decoder channels. The random denial of service (DoS) attack is taken into account when the decoded data are transmitted to the estimator through the shared communication networks. By recursively dealing with two coupled backward Riccati difference equations (CBRDEs), the desired finite-horizon H∞ state estimator gain matrix (EGM) can be determined. Finally, a numerical example is exhibited to illustrate the efficiency of the designed finite-horizon secure state estimator.

Optimal control of stochastic singular affine systems with Markovian jumps

We consider an optimal control problem for a class of stochastic singular affine systems with Markovian jumps. We establish the existence and uniqueness of the solution to stochastic singular affine systems with Markovian jumps for the first time. Via square completion technique and the generalized Itô’s formula, we derive new kinds of generalized differential Riccati equations (GDREs) and generalized backward differential equations (GBDEs), which give sufficient conditions for the well-posedness of the optimal control problem, and present an explicit representation of optimal control. Also, we discuss the solvability of the GDREs in two cases. As an application, we present a leader-follower differential game to demonstrate the practicability of our results.