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

Abstract

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.

Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems

“…(2013) solved multi-dimensional problems with an inequality constraint by Bernstein polynomials. The optimal control of a linear time-invariant space–time fractional diffusion process is presented by Ozdemir and Avci (2014). A class of fractional optimal control problems has been solved directly without using the Hamiltonian equation (Keshavarz et al., 2015).…”

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

“…In the context of partial differential equations (PDEs), problems of this type are often referred to as PDE-constrained optimization problems and have been studied extensively over the last decades (see [37,85] for introductions to the field). Optimal control problems for FDEs have previously been studied in literature such as [2,3,54,55,65,74]. However, they were mostly considering one-dimensional spatial domains, since the direct treatment of higher dimensions was too expensive.…”

Existence results for mixed Hadamard and Riemann-Liouville fractional integro-differential inclusions

“…Afterwards, this branch of mathematics has found extensive applications in different fields of engineering and applied sciences such as electrical circuit [31], rotor-bearing system [10], finance system [29], biological system [35], thermoelectric system [17], reaction-diffusion system [18], etc. The memory (hereditary) property of fractional-order calculus provides a novel and better approach to model real-world phenomena than integerorder one such as viscoelasticity and diffusion processes [34,43]. Fractional order systems and controllers have also drawn considerable attention [16] and the concept of designing controllers based on fractionalorder systems has been developed, for example, sliding mode control of fractional-order chaotic systems [1,2,23,39,42].…”