Constrained fractional variational problems of variable order

Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint. In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and we present a new variational problem subject to a holonomic constraint. We establish necessary optimality conditions in order to determine the minimizers of the fractional problems. The terminal point in the cost integral, as well as the terminal state, are considered to be free, and we obtain corresponding natural boundary conditions.

[1]  Carlos F.M. Coimbra,et al.  The variable viscoelasticity oscillator , 2005 .

[2]  Jacky Cresson,et al.  Fractional embedding of differential operators and Lagrangian systems , 2006, math/0605752.

[3]  Agnieszka B. Malinowska,et al.  Fractional calculus of variations for a combined Caputo derivative , 2011, 1109.4664.

[4]  Agnieszka B. Malinowska,et al.  Advanced Methods in the Fractional Calculus of Variations , 2015 .

[5]  B. Ross,et al.  Integration and differentiation to a variable fractional order , 1993 .

[6]  A. Kılıçman,et al.  On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative , 2014 .

[7]  Delfim F. M. Torres,et al.  Optimality conditions for fractional variational problems with dependence on a combined Caputo derivative of variable order , 2015, Optimization.

[8]  S. Pooseh.  Computational Methods in the Fractional Calculus of Variations , 2013, 1312.4064.

[9]  Agnieszka B. Malinowska,et al.  Multiobjective fractional variational calculus in terms of a combined Caputo derivative , 2010, Appl. Math. Comput..

[10]  Agnieszka B. Malinowska,et al.  Noether’s theorem for fractional variational problems of variable order , 2013, 1303.4075.

[11]  Delfim F. M. Torres,et al.  Caputo derivatives of fractional variable order: Numerical approximations , 2015, Commun. Nonlinear Sci. Numer. Simul..

[12]  Agnieszka B. Malinowska,et al.  Fractional Variational Calculus with Classical and Combined Caputo Derivatives , 2011, 1101.2932.

[13]  Delfim F. M. Torres,et al.  Towards a combined fractional mechanics and quantization , 2012, 1206.0864.

[14]  YangQuan Chen,et al.  A Physical experimental study of variable-order fractional integrator and differentiator , 2011 .

[15]  Duarte Valério,et al.  Variable-order fractional derivatives and their numerical approximations , 2011, Signal Process..

[16]  Delfim F. M. Torres,et al.  Fractional Variational Calculus of Variable Order , 2011, 1110.4141.

[17]  Agnieszka B. Malinowska,et al.  Introduction to the Fractional Calculus of Variations , 2012 .