Necati Özdemir

Analysis of an axis-symmetric fractional diffusion-wave problem

This paper presents an axis-symmetric fractional diffusion-wave problem which is considered in polar coordinates. The dynamic characteristics of the system are described with a partial fractional differential equation in terms of the Riemann–Liouville fractional derivative. This continuum problem is reduced to a countable infinite problem by using the method of separation of variables. In this way, the closed form solution of the problem is obtained. The Grunwald–Letnikov approach is applied to take a numerical evaluation. The compatibility and effectiveness of this approach are realized by some simulation results which are obtained by a MATLAB program. It can be seen that the analytical and numerical solutions overlap.

Brief Integral control by variable sampling based on steady-state data

In this paper we consider integral control algorithms with convergent adaptive sampling for multivariable infinite-dimensional systems. Steady-state gain information is used in choosing suitable integrator gains and we also consider robustness with respect to error in measuring the steady-state gain.

Fractional optimal control problem of an axis-symmetric diffusion-wave propagation

This paper presents the formulation of an axis-symmetric fractional optimal control problem (FOCP). Dynamic characteristics of the system are defined in terms of the left and right Riemann‐Liouville fractional derivatives (RLFDs). The performance index of a FOCP is described with a state and a control function. Furthermore, dynamic constraints of the system are given by a fractional diffusion-wave equation. It is preferred to use the method of separation of variables for finding the analytical solution of the problem. In this way, the closed form solution of the problem is obtained by a linear combination of eigenfunctions and eigencoordinates. For numerical evaluation, the Grunwald‐Letnikov approximation is applied to the problem. Consequently, some simulation results show that analytical and numerical solutions overlap for = 1. This numerical approach is applicable and effective for such a kind of FOCP. In addition, the changing of some variables related to the problem formulation is analyzed.

Multistage Adomian Decomposition Method for Solving NLP Problems Over a Nonlinear Fractional Dynamical System

This paper deals with implementation of the multistage Adomian decomposition method (MADM) to solve a class of nonlinear programming (NLP) problems, which are reformulated with a nonlinear system of fractional differential equations. The multistage strategy is used to investigate the relation between an equilibrium point of the fractional order dynamical system and an optimal solution of the NLP problem. The preference of the method lies in the fact that the multistage strategy gives this relation in an arbitrary longtime interval, while the Adomian decomposition method (ADM) gives the optimal solution just only in the neighborhood of the initial time. The numerical results taken by the fractional order MADM show that these results are compatible with the solution of NLP problem rather than the ADM. Furthermore, in some cases the fractional order MADM can perform more rapid convergency to the optimal solution of optimization problem than the integer order ones.

Fractional Order Control of Fractional Diffusion Systems Subject to Input Hysteresis

This paper concerns the control of a time fractional diffusion system defined in the Riemann-Liouville sense. It is assumed that the system is subject to hysteresis nonlinearity at its input, where the hysteresis is mathematically modeled with the Duhem operator. To compensate the effects of hysteresis nonlinearity, a fractional order Proportional +Integral +Derivative (PID) controller is designed by minimizing integral square error. For numerical computation, the Riemann-Liouville fractional derivative is approximated by the Grunwald-Letnikov approach. A set of algebraic equations arises from this approximation, which can be solved numerically. Performance of the fractional order PID controllers are analyzed in comparison with integer order PID controllers by simulation results, and it is shown that the fractional order controllers are more advantageous than the integer ones.

Optimal control of a linear time-invariant space–time fractional diffusion process

This paper presents a formulation and numerical solutions of an optimal control problem of a linear time-invariant space–time fractional diffusion equation. The main aim of this formulation is minimization of a performance index, which is a functional of both state and control functions of the diffusion system. The dynamics of the system are defined by the space–time fractional diffusion equation in the sense of Caputo and fractional Laplacian operators. The separation of variables technique and a spectral representation of a fractional Laplacian operator are applied to determine the eigenfunctions that represent the space parameters. Therefore, the state and control functions are defined by linear infinite combinations of eigenfunctions. Optimality conditions described by Euler–Lagrange equations are found by using a Lagrange multiplier technique. The Grünwald–Letnikov definition is used to approximate to the time fractional derivative. The applicapability and effectiveness of the numerical scheme are shown by comparison of analytical and numerical solutions for a numerical example. Finally, the variations of problem parameters are analyzed, with some figures obtained using MATLAB.

A Fractional Order Dynamical Trajectory Approach for Optimization Problem with HPM

In this paper, the homotopy perturbation method (HPM) is applied to solve nonlinear programming (NLP) problem on the basis of the fractional order differential equations system. The trajectory of the proposed fractional order dynamical system is approached to the optimal solution for optimization problem. The multistage strategy is used to show this behavior of the system trajectory in large timespan. The ability of the method to obtain approximate analytical solutions was shown by comparisons among the multistage HPM, the standard HPM and the fourth order Runge–Kutta method.

Optimal Boundary Control of Thermal Stresses in a Plate Based on Time-Fractional Heat Conduction Equation

This article presents an optimal control problem for a fractional heat conduction equation that describes a temperature field. The main purpose of the research was to find the boundary temperature that takes the thermal stress under control. The fractional derivative is defined in terms of the Caputo operator. The Laplace and finite Fourier sine transforms were applied to obtain the exact solution. Linear approximation is used to get the numerical results. The dependence of the solution on the order of fractional derivative and on the nondimensional time is analyzed.

Conformable Fractional Wave-Like Equation on a Radial Symmetric Plate

The generalization of physical processes by using local or nonlocalfractional operators has been an attractive research topic over the last decade. Fractionalization of integer order models gives quite reality to mathematical descriptions so that one should obtain the sub/super behaviors of real world problems. In this article, we are motivated to formulate a wave-like equation in terms of the left sequential conformable fractional derivative on a radial plate and also discuss on the differences among the statements of classical, existing fractional and conformable fractional wave equations.

Digital Variable Sampling Integral Control of Infinite Dimensional Systems Subject to Input Nonlinearity

In this technical note we present a novel sampled-data low-gain I-control algorithm for infinite-dimensional systems in the presence of input nonlinearity. The system is assumed to be exponentially stable with invertible steady-state gain. We use an integral controller with fixed integrator gain, chosen on the basis of state gain information and a time varying sampling period determined by the growth bound of the system. We compare this new algorithm with two other algorithms one with fixed gain and sampling period, the other with time-varying gain.