Optimization of a fractional Mayer problem - existence of solutions, maximum principle, gradient methods

• Autorzy: Idczak D. , Walczak S.

Identyfikatory

DOI

http://dx.doi.org/10.7494/OpMath.2014.34.4.763

• Języki publikacji: EN

Abstrakty

EN

In the paper, we study a linear-quadratic optimal control problem of Mayer type given by a fractional control system. First, we prove a theorem on the existence of a solution to such a problem. Next, using the local implicit function theorem, we derive a formula for the gradient of a cost functional under constraints given by a control system and prove a maximum principle in the case of a control constraint set. The formula for the gradient is used to implement the gradient methods for the problem under consideration.

Słowa kluczowe

EN

fractional Riemann-Liouville derivative; Mayer problem; existence of an optimal solution; maximum principle; gradient method

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2014
• Tom: Vol. 34, no. 4
• Strony: 763--775
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Idczak D.

• Faculty of Mathematics and Computer Science University of Lodz, Banacha 22, 90-238 Lodz, Poland

autor

Walczak S.

• Faculty of Mathematics and Computer Science University of Lodz, Banacha 22, 90-238 Lodz, Poland

Bibliografia

• [1] L. Bourdin, D. Idczak, Fractional fundamental lemma and fractional integration by parts formula - Applications to critical points of Bolza functionals and to linear boundary value problems, submitted for publication.
• [2] D. Idczak, Optimal control of a coercive Dirichlet problem, SIAM J. Control Optim. 36 (1998) 4, 1250–1267.
• [3] D. Idczak, R. Kamocki, On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in Rn, Fract. Calc. Appl. Anal. 14 (2011) 4, 538–553.
• [4] D. Idczak, R. Kamocki, Fractional differential repetitive processes with Riemann-Liouville and Caputo derivatives, Multidim. Syst. and Sign. Process., published online 25 September 2013, DOI 10.1007/s11045-013-0249-0.
• [5] R. Kamocki, Some ordinary and distributed parameters fractional control systems and their optimization, Doctoral Thesis, University of Lodz, Lodz, 2012.
• [6] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives – Theory and Applications, Gordon and Breach, Amsterdam, 1993.
• [7] F.P. Vasiliev, Methods of Solving of Extreme Problems, Moscov, Science, 1981 [in Russian].
• [8] F.P. Vasiliev, Numerical Methods of Solving of Extreme Problems, Moscov, Science, 1988 [in Russian].