Optimal control of linear systems with fractional derivatives

Abstract

Problem of time-optimal control of linear systems with fractional Caputo derivatives is examined using technique of attainability sets and their support functions. A method to construct a control function that brings trajectory of the system to a strictly convex terminal set in the shortest time is elaborated. The proposed method uses technique of set-valued maps and represents a fractional version of Pontryagin’s maximum principle. A special emphasis is placed upon the problem of computing of the matrix Mittag-Leffler function, which plays a key role in the proposed methods. A technique for computing matrix Mittag-Leffler function using Jordan canonical form is discussed, which is implemented in the form of a MATLAB routine. Theoretical results are supported by examples, in which the optimal control functions, in particular of the “bang-bang” type, are obtained.

Analytical Solution of Linear Fractional Systems with Variable Coefficients Involving Riemann–Liouville and Caputo Derivatives

“…Again, we will refer to the series on the right-hand side of Equation (24) as the generalized Peano-Baker series [14,16]. In view of Lemma 1 and of Equations (9) and (11), the following lemma holds true.…”

“…Explicit solutions to linear systems of differential equations provide a basis to solve control problems. Analytical solutions of the linear systems of fractional differential equations with constant coefficients were derived in the papers [8,9] and then applied to solving control problems and differential games in [10][11][12][13].…”

Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order

“…In 1976, V.K. Veber [4] introduced the Mittag-Leffler function of the matrix argument (see also [11]). Note that in the paper [4] the function E γ α,β (z) with natural values of the parameter γ ∈ N naturally comes into sight.…”

“…In the works [11,12] I. Matychyn and V. Onyshchenko investigated the solutions of the initial problems for systems of equations with fractional Riemann-Liouville and Caputo derivatives analytically and numerically using the Mittag-Leffler matrix function.…”

Linear quadratic optimization for fractional order differential algebraic system of Riemann-Liouville type

“…One can observe that the positive definite assumption of the matrix Q and R impliesQ 22 in equation 21is also positive definite. Therefore, one can use the theory in [Matychyn and Onyshchenko, 2018] and [Li and Chen, 2008] regarding the standard fractional linear quadratic optimization problem. Using the results in [Matychyn and Onyshchenko, 2018], the optimal state pairs (v * , ζ * 1 ) for optimization problem (20) exists and unique if…”