Optimal Control Problems for Linear Degenerate Fractional Equations
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OPTIMAL CONTROL PROBLEMS FOR LINEAR DEGENERATE FRACTIONAL EQUATIONS M. V. Plekhanova
UDC 517.97
Abstract. Optimal control problems for a linear degenerate evolution equation that is not solvable with respect to the fractional Gerasimov–Caputo derivative are examined. Solvability conditions for distributed control problems with various quality functionals are obtained. Applications of abstract results are illustrated by an example of a control problem for the system of equations of gravitationalgyroscopic waves. Keywords and phrases: optimal control, degenerate evolution equation, fractional differential equation, Caputo derivative. AMS Subject Classification: 49J20, 34G10, 35R11
CONTENTS 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong Solution of Inhomogeneous Cauchy Problem for a Fractional Linear Equation Order Distributed Controls for Linear Nondegenerate Equations . . . . . . . . . . . . . . . . . . Distributed Controls for Linear Degenerate Equations . . . . . . . . . . . . . . . . . . . . Control Problem for the System of Gravitational-Gyroscopic Waves . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
This paper is devoted to the study of the solvability of the optimal control problem for a linear equation in a Banach space of the form LDtα x(t) = M x(t) + f (t) + Bu(t),
t ∈ (t0 , T ),
x(k) (t0 ) = xk , k = 0, 1, . . . , m − 1, u ∈ U∂ ,
(1.1)
J(x, u) → inf, where L ∈ L(X ; Y) is a linear continuous operator acting from the Banach space X to the Banach space Y, M ∈ Cl(X ; Y) is a linear closed operator densely defined in X and acting in Y, B ∈ L(U ; Y), where U is also a Banach space, f : (t0 , T ) → Y is a given function, U∂ is the set of admissible controls, and J is the quality functional. Equation (1.1) belongs to the class of degenerate equations with respect to the fractional Gerasimov–Kaputo derivative Dtα (m − 1 < α ≤ m, m ∈ N), since we assume that the operator L has a nontrivial kernel: ker L = {0}. The rapid development of fractional calculus related to its advances in modeling of complex processes led to numerous studies of differential equations with fractional derivatives (see, e.g., [2, 9, 17, 20] and references therein). Works devoted to optimal control problems for fractional equations order are less common (see, e.g., [1, 4]); in this case, equations solvable with respect to the highest time Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018.
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derivative are usually considered. Many nonclassical equations and systems of mathematical physics are not equations of this type, for example, the quasistationary s
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