Time-delayed Arguments Research Articles

Numerical and theoretical analysis for fixed points for solving differential equation with retarded argument using an efficient iterative method

This paper includes a new three stage iterative method Aℳ* and uses that method to test some convergence theorems in Banach spaces, together with the example to prove efficiency of Aℳ* is the central focus of this paper, along with explaining, using an example, that Aℳ* is converging to an invariant point faster than all Picards, Mann, Ishikawa, Noor, SP, S, Picard-S, Garodia, K and K*. It must be noted that it is also addressed by the Aℳ* iterative method, the effects of stability and data dependency on contraction mapping. In addition, the Aℳ* iterative method can be used to resolve differential equations with a time-delayed argument.

Solving time delay fractional optimal control problems via a Gudermannian neural network and convergence results

ABSTRACT In this paper, we propose a Gudermannian neural network scheme to solve optimal control problems of fractional-order system with delays in state and control. The fractional derivative is described in the Caputo sense. The problem is first transformed, using a Padé approximation, to one without a time-delayed argument. We try to approximate the solution of the Hamiltonian conditions based on the Pontryagin minimum principle. For this purpose, we use trial solutions for the states, Lagrange multipliers, and control functions where these trial solutions are constructed by using two-layered perceptron. We then minimize the error function using an unconstrained optimization scheme where weight and biases associated with all neurons are unknown. Some numerical examples are given to illustrate the effectiveness of the proposed method.

An enhanced control parameterization technique with variable switching times for constrained optimal control problems with control-dependent time-delayed arguments and discrete time-delayed arguments

In this paper, we consider an optimal control problem whose dynamic system consists of control-dependent time-delayed arguments in the control and discrete time-delayed arguments in the state. Recently, control parameterization methods using hybrid time-scale transformation techniques have been developed to optimize both the control parameters and their switching times when the dynamic system consists of discrete time-delayed arguments only. This transformation maps the variable switching times in the original time horizon to fixed switching times in the new time horizon. Thus, the purpose of this paper is to extend the above technique to solve problems with both control-dependent time-delayed arguments and discrete time-delayed arguments. Using the above technique, we convert both the delayed state vector and the delayed control vector into non-delayed vectors in the new time horizon; hence both the control parameters and their switching times can be optimized. The gradients of the objective functions and the constraint functions with respect to the switching times and the controls are obtained. Two numerical examples, one of which is a practical problem involving the optimal transmission of messages in an internet, are solved to demonstrate the efficiency of our method.

Optimal transmission of messages in computer networks – an optimal control problem involving control-dependent time-delayed arguments

In this paper, we find the optimal transmission of messages in computer networks. This problem has been formulated as a nondelayed optimal control problem in several recent papers on TCP (transmission control protocol). Since the actual transmission of messages from origins to destinations should consist of both forward transmission delays of the buffers and latency of the links, we remodel the problem as a time-delayed optimal control problem consisting of both control-dependent time-delayed arguments and discrete time-delayed arguments. We then develop a modified control parameterization method for solving this time-delayed optimal control problem. The gradients of the new objective function and constraint functions generated by this modified control parametrization method are derived. A numerical example is solved by using the time-delayed version of the problem that we formulate, as well as the nondelayed version of the problem in the literature. Numerical results clearly illustrate the efficiency of the modified control parameterization method for solving both versions of this optimal transmission problem. Comparison of results of the two versions concerning the optimal transmission rates at the origins, the optimal output flow rates at the destination, and the queue sizes at the buffers are obtained. These comparison results clearly reflect how the optimal transmission of messages in computer networks in real life can be affected by both the forward transmission delays of the buffers and the latency of the links.

Solving infinite-horizon optimalcontrol problems of the time-delayedsystems by a feed forward neural network model

ABSTRACT A numerical method using neural network for solving infinite-horizon time-delayed optimal control problems is studied. The problem is first transformed, using a Páde approximation, to one without a time-delayed argument. By a suitable change of variable, the obtained non-delay infinite-horizon optimal control problem is converted to a finite-horizon nonlinear optimal control problem. We try to approximate the solution of Hamiltonian conditions based on the Pontryagin minimum principle (PMP). For this purpose, we introduce an error function that contains all PMP conditions. We then minimize the error function where weights and biases associated with all neurons are unknown. Substituting the optimal values of the weights and biases in the trial solutions, we obtain the optimal solution of the original problem. Several examples are given to show the efficiency of the method.

Calculating the Lyapunov exponents of a piecewise-smooth soft impacting system with a time-delayed feedback controller

Lyapunov exponent is a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise-smooth systems with time-delayed arguments one faces a lack of continuity in the variational problem. This paper studies how to build a variational equation for the efficient construction of Jacobians along trajectories of a delayed nonsmooth system. Trajectories of a piecewise-smooth system may encounter the so-called grazing event where the trajectory approaches a discontinuity surface in the state space in a non-transversal manner. For this event we develop a grazing point estimation algorithm to ensure the accuracy of trajectories for the nonlinear and the variational equations. We show that the eigenvalues of the Jacobian matrix computed by the algorithm converge with an order consistent with the order of the numerical integration method, therefore guaranteeing the reliability of the proposed numerical method. Finally, the method is demonstrated on a periodically forced impacting oscillator under the time-delayed feedback control.

Solving optimal control problems of the time-delayed systems by a neural network framework

ABSTRACTA numerical method using neural networks for solving time-delayed optimal control problems is studied. The problem is first transformed into one without a time-delayed argument, using a Páde approximation. We try to approximate the solution of the Hamiltonian conditions based on the Pontryagin minimum principle (PMP). For this purpose, we introduce an error function that contains all PMP conditions. We then minimise the error function where weights and biases associated with all neurons are unknown. Substituting the optimal values of the weights and biases in the trial solutions, we obtain the optimal solution of the original problem. Several examples are given to show the efficiency of the method.

Optimal production schedule in a single-supplier multi-manufacturer supply chain involving time delays in both levels

This paper considers an optimal production scheduling problem in a single-supplier-multi-manufacturer supply chain involving production and delivery time-delays, where the time-delays for the supplier and the manufacturers can have different values. The objective of both levels is to find an optimal production schedule so that their production rates and their inventory levels are close to the ideal values as much as possible in the whole planning horizon. Each manufacturer's problem, which involves one time-delayed argument, can be solved analytically by using the necessary condition of optimality. To tackle the supplier's problem involving $n+1$ different time-delayed arguments (where $n$ is the number of manufacturers) by the above approach, we need to introduce a model transformation technique which converts the original system of combined algebraic/differential equations with $n+1$ time-delayed arguments into a sum of $n$ sub-systems, each of which consists of only two time-delayed arguments. Thus, the supplier's problem can also be solved analytically. Numerical examples consisting of a single supplier and four manufacturers are solved to provide insight of the optimal strategies of both levels.

Solving infinite-horizon optimal control problems of the time-delayed systems by Haar wavelet collocation method

A numerical method using Haar wavelets for solving infinite-horizon time-delayed optimal control problems is studied. The problem is first transformed, using a Pade approximation, to one without a time-delayed argument. By a suitable change of variable, the obtained non-delay infinite-horizon optimal control problem is converted to a finite-horizon non-linear optimal control problem. An approximation scheme based on Haar wavelets in the time domain is then proposed for solving the obtained optimal control problem. Haar wavelets integral operational matrix and direct collocation method are used to find the approximated optimal trajectory and the optimal control law of the original problem. Two illustrative examples are provided to show the feasibility and the efficiency of the proposed method.

Solving optimal control problems of the time-delayed systems by Haar wavelet

We consider an approximation scheme using Haar wavelets for solving time-delayed optimal control problems with terminal inequality constraints. The problem is first transformed, using a Páde approximation, to one without a time-delayed argument. Terminal inequality constraints, if they exist, are converted to equality constraints via Valentine-type unknown parameters. A computational method based on Haar wavelets in the time domain is then proposed for solving the obtained nondelay optimal control problem. The Haar wavelets integral operational matrix and direct collocation method are utilized to find the approximated optimal trajectory and the optimal control law of the original problem. Numerical results are also given for several test examples to demonstrate the applicability and the efficiency of the method.

The control parametrization enhancing transform for constrained time--delayed optimal control problems

The Control Parametrization Enhancing Technique (CPET), is extended to a general class of constrained time-delayed optimal control problems. A model transformation approach is used to convert the time-delayed problem to an optimal control problem involving mixed boundary conditions, but without time-delayed arguments. The CPET is then used to solve this non delayed problem. Two test examples have been solved to illustrate the efficiencies of the CPET for time delayed problems.

Sufficiently general framework for simple models of the net immune response

A general framework for modelling the immune response is proposed and analyzed. The models have simple formulation in terms of only few measurable quantities and the complexity of the immune response is accounted for by using nonlinear functions with time-delayed arguments. It is illustrated that such models are capable of describing various typical cases of the immune response. Standard analytical tools are used to show how such models can assist in the problem of parameter estimation.

Numerical solution of time-delayed optimal control problems by iterative dynamic programming

This work presents a numerical method to solve the optimal control problem with time-delayed arguments and a fixed terminal time. A series of auxiliary states obtained from the linearly truncated Taylor series expansion are used to represent the status of a time-delayed state at different time intervals. The backward iterative dynamic programming (IDP) technique can thus be directly employed to solve the delay-free optimal control problem with augmented states. Five numerical examples are provided, demonstrating the proposed method's effectiveness in solving the time-delayed optimal control problems. Copyright © 2000 John Wiley & Sons, Ltd.

Neighbouring optimal feedback law for linear time-delayed dynamical systems

One way to obtain a neighbouring feedback law for a time-delayed optimal control problem is to first transform it into a ‘standard’ optimisation problem. That is one without terms having a time-delayed argument. To this end, I use a Padé approximation to determine a differential relation for y(t), an augmented state that represents x(t−τ). The time-delayed optimisation problem can then be rewritten in terms of an augmented state vector consisting of both the physical state x(t) and the delayed state y(t). Once reformulated, one may use to advantage existing well-developed techniques such as the backward sweep method to obtain the neighbouring feedback law. Results obtained from two examples show good agreement between the exact results and those predicted by the feedback law for small variations in both the initial condition and a system parameter.

Hereditary optimal control problems: Numerical method based upon a pad� approximation

In this paper, we consider a particular approximation scheme which can be used to solve hereditary optimal control problems. These problems are characterized by variables with a time-delayed argumentx(t − τ). In our approximation scheme, we first replace the variable with an augmented statey(t) ≜x(t - τ). The two-sided Laplace transform ofy(t) is a product of the Laplace transform ofx(t) and an exponential factor. This factor is approximated by a first-order Pade approximation, and a differential relation fory(t) can be found. The transformed problem, without any time-delayed argument, can then be solved using a gradient algorithm in the usual way. Four problems are solved to illustrate the validity and usefulness of this technique.

Optimal control computation for parabolic systems with boundary conditions involving time delays

In this paper, we consider a class of optimal control problems involving a second-order, linear parabolic partial differential equation with Neumann boundary conditions. The time-delayed arguments are assumed to appear in the boundary conditions. A necessary and sufficient condition for optimality is derived, and an iterative method for solving this optimal control problem is proposed. The convergence property of this iterative method is also investigated.

On the computational algorithms for time-lag optimal control problems

In this thesis we study the following two types of hereditary optimal control problems: (i) problems governed by systems of ordinary differential equations with discrete time-delayed arguments appearing in both the state and the control variables; (ii) problems governed by parabolic partial differential equations with Neumann boundary conditions involving time-delays.

On a computational algorithm for a class of optimal control problems involving discrete time delayed arguments

AbstractIn this paper, a computational algorithm for solving a class of optimal control problems involving discrete time-delayed arguments is presented. By way of example, a simple model of a production firm is devised for which the algorithm is used to solve a decision-making problem.

Asymptotic Stability of a Time-Delayed Diffusion System

This paper discusses the asymptotic stability of the equilibrium states of a nonlinear diffusion system with time delays. It is assumed that the system is describable by a partial differential-difference equation of the form: ∂u(t,X)∂t=ℒu(t,X)+f(t,X,u(t,X),u(t−T,X),…,∂u(t,X)∂xi,…,∂u(t−T,X)∂xi,…)i=1,…,M where ℒ is a linear operator uniformly elliptic in X defined for all XεΩ … a bounded, open M-dimensional spatial domain; f is a specified function of its arguments. In the development of this paper, the physical origin of the foregoing equation is discussed briefly. Then, conditions for asymptotic stability of the trivial solution are derived via an extended Lyapunov’s direct method. Specific results are given for a simple one-dimensional linear heat equation with time-delayed arguments.