Abstract
We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann–Liouville integral operator is used in our new techniques. An accurate operational matrix of variable-order fractional integration for Bernoulli polynomials is introduced. Our methods proceed as follows. First, a specific approximation of the differentiation order of the state function is considered, in terms of Bernoulli polynomials. Such approximation, together with the initial conditions, help us to obtain some approximations for the other existing functions in the dynamical control-affine system. Using these approximations, and the Gauss—Legendre integration formula, the problem is reduced to a system of nonlinear algebraic equations. Some error bounds are then given for the approximate optimal state and control functions, which allow us to obtain an error bound for the approximate value of the performance index. We end by solving some test problems, which demonstrate the high accuracy of our results.
Highlights
The Bernoulli polynomials, named after Jacob Bernoulli (1654–1705), occur in the study of many special functions and, in particular, in relation with fractional calculus, which is a classical area of mathematical analysis whose foundations were laid by Liouville in a paper from 1832 and that is nowadays a very active research area [1]
When such dynamic constraints are described by fractional differential equations, one speaks of fractional optimal control problems (FOCPs) [9]
In this work we focus on the following variable-order fractional optimal control-affine problem (FOC-AP): min J =
Summary
The Bernoulli polynomials, named after Jacob Bernoulli (1654–1705), occur in the study of many special functions and, in particular, in relation with fractional calculus, which is a classical area of mathematical analysis whose foundations were laid by Liouville in a paper from 1832 and that is nowadays a very active research area [1]. The variable-order fractional calculus was introduced in 1993 by Samko and Ross and deals with operators of order α, where α is not necessarily a constant but a function α(t) of time [19] With this extension, numerous applications have been found in physics, mechanics, control, and signal processing [20,21,22,23,24]. To the best of our knowledge, numerical methods based on Bernoulli polynomials for such kind of FOCPs are not available in the literature For this reason, in this work we focus on the following variable-order fractional optimal control-affine problem (FOC-AP): min J =.
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