59,067 research outputs found
AN ANALYTICAL APPROCH FOR SOLVING FRACTIONAL FUZZY OPTIMAL CONTROL PROBLEM WITH FUZZY INITIAL CONDITIONS
Abstract. A fractional – fuzzy optimal control problem is an optimal control problem in which it is governedby a fuzzy system of fractional differential equation. The aim of this paper is to introduce an analytically solution for such Bolza problems when the initial state is also fuzzy. For this purpose, first the problem is turned to two fractional optimal control problems by concept of 훽-cut and complex numbers. Then, we apply a new method to solve these ractional optimal control problems, analytically by applying a new Riccati differential equation determined from PMP. Indeed this Riccati equation transfer each mentioned fractional optimal control problem to a fractional differential system. We show that if the new system has close solution, one is able to obtain the analytical solution of the fractional – fuzzy optimal control problems. A numerical simulation based on the new method is presented for different values of 훽 and fractional order and the results are compered. In the last section, a numerical example of fractional-fuzzy optima control problem is solved by the new method for different 훽 and 훾; and compared with the exact state; also, they are shown in figures for each cases.Key words: Fractional differential equation, optimal control, fuzzy
A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains
We propose and analyze a new discretization technique for a linear-quadratic
optimal control problem involving the fractional powers of a symmetric and
uniformly elliptic second oder operator; control constraints are considered.
Since these fractional operators can be realized as the Dirichlet-to-Neumann
map for a nonuniformly elliptic equation, we recast our problem as a
nonuniformly elliptic optimal control problem. The rapid decay of the solution
to this problem suggests a truncation that is suitable for numerical
approximation. We propose a fully discrete scheme that is based on piecewise
linear functions on quasi-uniform meshes to approximate the optimal control and
first-degree tensor product functions on anisotropic meshes for the optimal
state variable. We provide an a priori error analysis that relies on derived
Holder and Sobolev regularity estimates for the optimal variables and error
estimates for an scheme that approximates fractional diffusion on curved
domains; the latter being an extension of previous available results. The
analysis is valid in any dimension. We conclude by presenting some numerical
experiments that validate the derived error estimates
A FEM for an optimal control problem of fractional powers of elliptic operators
We study solution techniques for a linear-quadratic optimal control problem
involving fractional powers of elliptic operators. These fractional operators
can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic
problem posed on a semi-infinite cylinder in one more spatial dimension. Thus,
we consider an equivalent formulation with a nonuniformly elliptic operator as
state equation. The rapid decay of the solution to this problem suggests a
truncation that is suitable for numerical approximation. We discretize the
proposed truncated state equation using first degree tensor product finite
elements on anisotropic meshes. For the control problem we analyze two
approaches: one that is semi-discrete based on the so-called variational
approach, where the control is not discretized, and the other one is fully
discrete via the discretization of the control by piecewise constant functions.
For both approaches, we derive a priori error estimates with respect to the
degrees of freedom. Numerical experiments validate the derived error estimates
and reveal a competitive performance of anisotropic over quasi-uniform
refinement
A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives
We propose a direct numerical method for the solution of an optimal control
problem governed by a two-side space-fractional diffusion equation. The
presented method contains two main steps. In the first step, the space variable
is discretized by using the Jacobi-Gauss pseudospectral discretization and, in
this way, the original problem is transformed into a classical integer-order
optimal control problem. The main challenge, which we faced in this step, is to
derive the left and right fractional differentiation matrices. In this respect,
novel techniques for derivation of these matrices are presented. In the second
step, the Legendre-Gauss-Radau pseudospectral method is employed. With these
two steps, the original problem is converted into a convex quadratic
optimization problem, which can be solved efficiently by available methods. Our
approach can be easily implemented and extended to cover fractional optimal
control problems with state constraints. Five test examples are provided to
demonstrate the efficiency and validity of the presented method. The results
show that our method reaches the solutions with good accuracy and a low CPU
time.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Vibration and Control', available from
[http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised
03-Sept-2018; Accepted 12-Oct-201
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