Opuscula Math. 36, no. 1 (2016), 49-68
http://dx.doi.org/10.7494/OpMath.2016.36.1.49

Opuscula Mathematica

# On a linear-quadratic problem with Caputo derivative

Dariusz Idczak
Stanislaw Walczak

Abstract. In this paper, we study a linear-quadratic optimal control problem with a fractional control system containing a Caputo derivative of unknown function. First, we derive the formulas for the differential and gradient of the cost functional under given constraints. Next, we prove an existence result and derive a maximum principle. Finally, we describe the gradient and projection of the gradient methods for the problem under consideration.

Keywords: fractional Caputo derivative, linear-quadratic problem, existence and uniqueness of a solution, maximum principle, gradient method, projection of the gradient method.

Mathematics Subject Classification: 26A33, 49J15, 49K15, 49M37.

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• Dariusz Idczak
• University of Lodz, Faculty of Mathematics and Computer Science, Banacha 22, 90-238 Lodz, Poland
• Stanislaw Walczak
• University of Lodz, Faculty of Mathematics and Computer Science, Banacha 22, 90-238 Lodz, Poland
• Communicated by Marek Galewski.
• Revised: 2015-03-09.
• Accepted: 2015-04-10.
• Published online: 2015-09-19.