30,561 research outputs found
A numerical technique for solving multi-dimensional fractional optimal control problems using fractional wavelet method
This paper presents an efficient numerical method for solving fractional
optimal control problems using an operational matrix for a fractional wavelet.
Using well-known formulae such as Caputo and Riemann-Liouville operators to
determine fractional derivatives and integral fractional wavelets, operational
matrices were devised and utilised to solve fractional optimal control
problems. The proposed method reduced the fractional optimal control problems
into a system of algebraic equations. To validate the effectiveness of the
presented numerical approach, some illustrative problems were solved using
fractional Taylor and Taylor wavelets, and the approximate cost function value
derived by approximating state and control functions was compared. In addition,
convergence rate and error bound of the proposed method have been derived
Fractional Order Version of the HJB Equation
We consider an extension of the well-known Hamilton-Jacobi-Bellman (HJB)
equation for fractional order dynamical systems in which a generalized
performance index is considered for the related optimal control problem. Owing
to the nonlocality of the fractional order operators, the classical HJB
equation, in the usual form, does not hold true for fractional problems.
Effectiveness of the proposed technique is illustrated through a numerical
example.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Nonlinear Dynamics', ISSN 1555-1415, eISSN
1555-1423, CODEN: JCNDDM. Submitted 28-June-2018; Revised 15-Sept-2018;
Accepted 28-Oct-201
A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
We survey some representative results on fuzzy fractional differential
equations, controllability, approximate controllability, optimal control, and
optimal feedback control for several different kinds of fractional evolution
equations. Optimality and relaxation of multiple control problems, described by
nonlinear fractional differential equations with nonlocal control conditions in
Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Applied Mathematics', ISSN: 0377-0427.
Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication
20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515
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
A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions
In this paper, the fractional order of rational Bessel functions collocation
method (FRBC) to solve Thomas-Fermi equation which is defined in the
semi-infinite domain and has singularity at and its boundary condition
occurs at infinity, have been introduced. We solve the problem on semi-infinite
domain without any domain truncation or transformation of the domain of the
problem to a finite domain. This approach at first, obtains a sequence of
linear differential equations by using the quasilinearization method (QLM),
then at each iteration solves it by FRBC method. To illustrate the reliability
of this work, we compare the numerical results of the present method with some
well-known results in other to show that the new method is accurate, efficient
and applicable
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
Optimum Weight Selection Based LQR Formulation for the Design of Fractional Order PI{\lambda}D{\mu} Controllers to Handle a Class of Fractional Order Systems
A weighted summation of Integral of Time Multiplied Absolute Error (ITAE) and
Integral of Squared Controller Output (ISCO) minimization based time domain
optimal tuning of fractional-order (FO) PID or PI{\lambda}D{\mu} controller is
proposed in this paper with a Linear Quadratic Regulator (LQR) based technique
that minimizes the change in trajectories of the state variables and the
control signal. A class of fractional order systems having single non-integer
order element which show highly sluggish and oscillatory open loop responses
have been tuned with an LQR based FOPID controller. The proposed controller
design methodology is compared with the existing time domain optimal tuning
techniques with respect to change in the trajectory of state variables,
tracking performance for change in set-point, magnitude of control signal and
also the capability of load disturbance suppression. A real coded genetic
algorithm (GA) has been used for the optimal choice of weighting matrices while
designing the quadratic regulator by minimizing the time domain integral
performance index. Credible simulation studies have been presented to justify
the proposition.Comment: 6 pages, 5 figure
Numerical investigation of Differential Biological-Models via GA-Kansa Method Inclusive Genetic Strategy
In this paper, we use Kansa method for solving the system of differential
equations in the area of biology. One of the challenges in Kansa method is
picking out an optimum value for Shape parameter in Radial Basis Function to
achieve the best result of the method because there are not any available
analytical approaches for obtaining optimum Shape parameter. For this reason,
we design a genetic algorithm to detect a close optimum Shape parameter. The
experimental results show that this strategy is efficient in the systems of
differential models in biology such as HIV and Influenza. Furthermore, we prove
that using Pseudo-Combination formula for crossover in genetic strategy leads
to convergence in the nearly best selection of Shape parameter.Comment: 42 figures, 23 page
- …