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.

There exists a number of definitions of the fractional order derivative. The classical one is the definition by Riemann-Liouville [1]. The RiemannLiouville fractional derivatives have a singularity at zero. That is why differential equations involving these derivatives require initial conditions of special form lacking clear physical interpretation. These shortcomings do not occur with the regularized fractional derivative in the sense of Caputo. Both the Riemann-Liouville and Caputo derivatives possess neither semigroup nor commutative property. That is why so-called sequential derivatives were introduced by Miller and Ross [2].In this paper we treat sequential derivatives of special form [3]. Their relation to the Riemann-Liouville and Caputo fractional derivatives and to each other is established.Differential games for the systems with the fractional derivatives of Riemann-Liouville, Caputo, as well as with the sequential derivatives are studied. Representations of such systems' solutions involving the MittagLeffler generalized matrix functions [4] are given. The use of asymptotic representations of these functions in the framework of the Method of Resolving Functions [5,6,7] allows to derive sufficient conditions for solvability of corresponding game problems. These conditions are based on the modified Pontrjagin's condition [5]. The results are illustrated on a model example where a dynamic system of order π pursues another system of order e.

Problem of time-optimal control of linear systems with fractional dynamics is treated in the paper from the convex-analytic standpoint. A linear system of fractional differential equations involving Riemann-Liouville derivatives is considered. A method to construct a control function that brings trajectory of the system to the terminal state in the shortest time is proposed in terms of attainability sets and their support functions.

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