On a game problem involving systems with time delay

A dynamic programming approach is used to obtain feedback solutions to a game problem involving linear systems having a time delay. As is known, the optimal feedback gains in linear time delay systems with quadratic criterion are determined by partial differential equations [1]. In the particular case studied here, the solution to these equations can be obtained by first separating variables and then solving the resulting simpler equations. The optimal feedback control also has the advantage that it can be realized on-line.     

A Partially Observed Nonzero‐Sum Stochastic Differential Game with Delays and its Application to Finance

In this paper, we deal with a new kind of partially observed nonzero‐sum differential game governed by stochastic differential delay equations. One of the special features is that the controlled system and the utility functionals involve both delays in the state variable and the control variables under different observation equations for each player. We obtain a maximum principle and a verification theorem for the game problem by virtue of Girsanov's theorem and the convex variational method. In addition, based on the theoretical results and Malliavin derivative techniques, we solve a production and consumption choice game problem.     

Non‐Zero Sum Differential Games of Backward Stochastic Differential Delay Equations Under Partial Information

In this paper, we study a new type of differential game problems of backward stochastic differential delay equations under partial information. A class of time‐advanced stochastic differential equations (ASDEs) is introduced as the adjoint process via duality relation. By means of ASDEs, we suggest the necessary and sufficient conditions called maximum principle for an equilibrium point of non‐zero sum games. As an application, an economic problem is putted into our framework to illustrate the theoretical results. In terms of the maximum principle and some auxiliary filtering results, an equilibrium point is obtained.     

Piecewise-linearized and linearized θ-methods for ordinary and partial differential equations

Initial- and boundary-value problems appear frequently in many branches of physics. In this paper, several numerical methods, based on linearization techniques, for solving these problems are reviewed. First, piecewise-linearized methods and linearized θ-methods are considered for the solution of initial-value problems in ordinary differential equations. Second, piecewise-linearized techniques for two-point boundary-value problems in ordinary differential equations are developed and used in conjunction with a shooting method. In order to overcome the lack of convergence associated with shooting, piecewise-linearized methods which provide piecewise analytical solutions and yield nonstandard finite difference schemes are presented. Third, methods of lines in either space or time for the solution of one-dimensional convection-reaction-diffusion problems that transform the original problem into an initial- or boundary-value one are reviewed. Methods of lines in time that result in boundary-value problems at each time step can be solved by means of the techniques described here, whereas methods of lines in space that yield initial-value problems and employ either piecewise-linearized techniques or linearized θ-methods in time are also developed. Finally, for multidimensional problems, approximate factorization methods are first used to transform the multidimensional problem into a sequence of one-dimensional ones which are then solved by means of the linearized and piecewise-linearized methods presented here.     

Necessary and sufficient conditions of risk‐sensitive optimal control and differential games for stochastic differential delayed equations

In this paper, we consider risk‐sensitive optimal control and differential games for stochastic differential delayed equations driven by Brownian motion. The problems are related to robust stochastic optimization with delay due to the inherent feature of the risk‐sensitive objective functional. For both problems, by using the logarithmic transformation of the associated risk‐neutral problem, the necessary and sufficient conditions for the risk‐sensitive maximum principle are obtained. We show that these conditions are characterized in terms of the variational inequality and the coupled anticipated backward stochastic differential equations (ABSDEs). The coupled ABSDEs consist of the first‐order adjoint equation and an additional scalar ABSDE, where the latter is induced due to the nonsmooth nonlinear transformation of the adjoint process of the associated risk‐neutral problem. For applications, we consider the risk‐sensitive linear‐quadratic control and game problems with delay, and the optimal consumption and production game, for which we obtain explicit optimal solutions.     

Asymptotic series solution of optimal systems with small time delay

A fixed final time free end-point optimal problem with a small time delay is considered. This problem leads to a nonlinear boundary-value problem consisting of both advanced- and retarded-type differential-difference equations. An asymptotic power series solution to the problem is constructed in terms of time delay. This asymptotic approximation procedure is aimed at improving a "nominal design" in which the small time delay is neglected. A scalar example illustrates the construction of asymptotic expansions. Numericai results on a coupled-core nuclear-reactor power control problem clearly show the advantages of the method.     

Robust H∞ Filtering and Deconvolution for Continuous Time Delay Systems Based on Game‐Theoretic Approach

Many dynamical systems involve not only process and measurement noise signals but also parameter uncertainty and unknown input signals. This paper aims to estimate the state and unknown input for linear continuous time‐varying systems subject to time delay in state, norm‐bounded parameter uncertainty, and a known input. Such a problem is reformulated into a two‐player differential game whose saddle point solution gives rise to one sufficient solvable condition for the estimation problem. The possible optimal estimators are obtained by solving the two coupled Riccati differential equations. We demonstrate, through two examples, how the proposed estimator is valid for estimating state and unknown input.     

Fractional Differential Mathematical Models of the Dynamics of Nonequilibrium Geomigration Processes and Problems with Nonlocal Boundary Conditions

The analytical solutions of boundary-value problems with nonlocal boundary conditions are presented for two fractional differential mathematical models of the dynamics of a geomigration process non-equilibrium in time. The models based on the equations with the Caputo and Hilfer derivatives of fractional order are considered.     

Zur numerischen Behandlung gestörter Verzweigungsprobleme bei gewöhnlichen Differentialgleichungen

We present a method for the numerical solution of perturbed bifurcation problems of boundary-value problems for ordinary differential equations by standard techniques. This is achieved by reparametrisation by perturbation parameters and definition of appropriate boundary-value problems which have isolated solutions. The efficiency of the method is demonstrated by a numerical example —a special rod buckling problem.     

A collection of computational techniques for solving singular boundary-value problems

The objective of this paper is a wide survey and classification of various computational techniques which are used for solving singular boundary-value problems. Here, we have considered various research papers within last 10 years to get a better the know-how of present scenario. This paper contains summary of linear, non-linear second-order two-point singular boundary-value problems (BVPs), two-point singularly perturbed BVPs in ordinary differential equations and singularly elliptic BVPs in partial differential equations.     

Studying Stability of the Econometric Model of Nonantagonistic Game of the Wholesale Market Subjects

Organization of the paper scheme-based wholesale fertilizer supply was suggested as one of the effective means of supporting the agricultural branch which is in prolonged depression. The strategies of the main subjects of economic interaction were based on solving a nonantagonistic game. The differential procedure of choosing the strategies of the market subjects was a system of ordinary differential delay equations. Its stability was considered, and a method of choosing stabilizing strategies of the wholesale market subjects was constructed.     

Discretized LKF method for stability of coupled differential‐difference equations with multiple discrete and distributed delays

Time‐delay systems described by coupled differential‐functional equations include as special cases many types of time‐delay systems and coupled differential‐difference systems with time delays. This article discusses the discretized Lyapunov–Krasovskii functional (LKF) method for the stability problem of coupled differential‐difference equations with multiple discrete and distributed delays. Through independently dividing every delay region that the plane regions consists in two delays to discretize LKF, the exponential stability conditions for coupled systems with multiple discrete and distributed delays are established based on a linear matrix inequality (LMI). The numerical examples show that the analysis limit of delay bound in which the systems are stable may be approached by our result. Copyright © 2011 John Wiley & Sons, Ltd.     

The standard H∞ problem in systems with a singleinput delay

The standard H_∞ problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration     

An efficient approach to approximate solutions of eighth-order boundary-value problems

In this paper, the approximate solutions to the eighth-order boundary-value differential equations are solved by using the Adomian decomposition method (ADM). The numerical solutions of the problem are calculated in the form of a series with easily computable components. The numerical illustrations show that this technique is more reliable, efficient and accurate than the traditional schemes.     

The Search of Equilibrium Strategies for Controlled Boundary Value Problem

The present paper is focused on the finite‐time differential game of m players with nonzero sum. The principal difference from regular cases consists in the fact that the states of players are subordinated to boundary value system of ordinary differential equations (rather than the initial value system). By the end of the game we understand the equilibrium situation and our purpose is to design a well‐founded suitable method for the equilibrium control search.     

Two-Wave Interactions in the Fermi–Pasta–Ulam Model

This work is devoted to investigating the dynamic properties of the solutions to the boundaryvalue problems associated with the classic Fermi–Pasta–Ulam (FPU) system. We analyze these problems for an infinite-dimensional case where a countable number of roots of characteristic equations tend to an imaginary axis. Under these conditions, we build a special nonlinear partial differential equation that acts as a quasi-normal form, i.e., determines the dynamics of the original boundary-value problem with the initial conditions in a sufficiently small neighborhood of the equilibrium state. The modified Korteweg–deVries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation act as quasi-normal forms depending on the parameter values. Under some additional assumptions, we apply the renormalization procedure to the boundary-value problems obtained. This procedure leads to an infinite-dimensional system of ordinary differential equations. We describe a method for folding this system into a special boundary- value problem, which is an analog of the normal form. The main contribution of this work is investigating the interaction of the waves moving in different directions in the FPU problem by using analytical methods of nonlinear dynamics. It is shown that the mutual influence of the waves is asymptotically small, does not affect their shape, and contributes only to a shift in their speeds, which does not change over time.     

Minimum energy control of time delay systems via piecewise linear polynomial functions

The piecewise linear polynomial function approach to the minimum energy control of linear systems with time delay, is presented in this paper. The concepts of a delay shift matrix and an operational matrix for integration are employed in solving the related state and costate equations containing terms with advanced and delayed arguments. An attractive feature of the present method is its ultimate simplicity and convenience. The differential equations with delay and advance terms are converted into a set of linear algebraic equations using a recurrence algorithm. An example demonstrates the accuracy of the method.     

Robust stabilization for uncertain time-delay systems containingsaturating actuators

A class of uncertain time-delay systems containing a saturating actuator is considered. These systems are characterized by delayed state equations (including a saturating actuator) with norm-bounded parameter uncertainty (possibly time varying) in the state and input matrices. The delay is assumed to be constant bounded but unknown. Using a Razumikhin approach for the stability of functional differential equations, upper bounds on the time delay are given such that the considered uncertain system is robustly stabilizable, in the case of constrained input, via memoryless state feedback control laws. These bounds are given in terms of solutions of appropriate finite dimensional Riccati equations     

Optimal control of linear time-delay systems

A method is presented whereby an optimal control may be obtained for a linear time-varying system with time delay. The performance criterion is quadratic with a fixed, finite upper limit, and results in a set of differential equations with boundary conditions whose solution yields an optimal feedback control. A numerical technique is developed for the solution of the differential equations, and two examples are worked.     

Computation of optimal control for linear systems with delay

A numerical algorithm is presented to calculate an optimal control recursively for linear multivariable systems with delay. The algorithm is based on the method of steepest descent in Hilbert space. The optimal control of a multivariable system with delay for a quadratic criterion function is given by the Riccati partial differential equations. These are simultaneous partial differential equations which are difficult to solve numerically. In most computational algorithms, errors are inevitable, a vast memory is required, and a lot of computational time is needed. This makes it impractical to use available computers for these algorithms. The algorithm presented here gives optimal control effectively without a large memory requirement. It also provides a practical computational method for obtaining the optimal control of multivariable systems with delay. Several numerical computations are performed to show how the effectiveness of the algorithm compares with other methods.