Optimal control of a class of fractional heat diffusion systems

In this paper, a solution procedure for a class of optimal control problems involving distributed parameter systems described by a generalized, fractional-order heat equation is presented. The first step in the proposed procedure is to represent the original fractional distributed parameter model as an equivalent system of fractional-order ordinary differential equations. In the second step, the necessity for solving fractional Euler–Lagrange equations is avoided completely by suitable transformation of the obtained model to a classical, although infinite-dimensional, state-space form. It is shown, however, that relatively small number of state variables are sufficient for accurate computations. The main feature of the proposed approach is that results of the classical optimal control theory can be used directly. In particular, the well-known “linear-quadratic” (LQR) and “Bang-Bang” regulators can be designed. The proposed procedure is illustrated by a numerical example.

[1]  Dušan Petrovački.  The minimum time problem for a class of non-linear distributed parameter systems , 1980 .

[2]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[3]  R. Rees Fullmer,et al.  Boundary Stabilization and Disturbance Rejection for Time Fractional Order Diffusion–Wave Equations , 2004 .

[4]  O. Agrawal.  A General Formulation and Solution Scheme for Fractional Optimal Control Problems , 2004 .

[5]  H. O. Fattorini,et al.  Infinite dimensional linear control systems : the time optimal and norm optimal problems , 2005 .

[6]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[7]  Riewe,et al.  Nonconservative Lagrangian and Hamiltonian mechanics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  T. Atanacković,et al.  Variational problems with fractional derivatives: Euler–Lagrange equations , 2008, 1101.2961.

[9]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[10]  Spyros G. Tzafestas,et al.  Distributed Parameter Control Systems: Theory and Application , 1982 .

[11]  Ralf Metzler,et al.  FRACTIONAL DIFFUSION, WAITING-TIME DISTRIBUTIONS, AND CATTANEO-TYPE EQUATIONS , 1998 .

[12]  Zoran D. Jelicic,et al.  Optimality conditions and a solution scheme for fractional optimal control problems , 2009 .

[13]  W. Harmon Ray,et al.  Distributed parameter systems: Identification, estimation, and control , 1978 .

[14]  Teodor M. Atanackovic,et al.  On a numerical scheme for solving differential equations of fractional order , 2008 .

[15]  T. Atanacković.  A generalized model for the uniaxial isothermal deformation of a viscoelastic body , 2002 .

[16]  O. Agrawal,et al.  A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems , 2007 .

[17]  I. Podlubny.  Fractional differential equations , 1998 .

[18]  Teodor M. Atanackovic,et al.  On a Differential Equation with Left and Right Fractional Derivatives , 2007 .

[19]  D. Petrovacki,et al.  Optimal control of a heat conduction problem , 1975 .

[20]  Om P. Agrawal,et al.  Fractional variational calculus and the transversality conditions , 2006 .

[21]  Stevan Pilipović,et al.  A diffusion wave equation with two fractional derivatives of different order , 2007 .

[22]  O. Agrawal.  A numerical scheme for initial compliance and creep response of a system , 2009 .

[23]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[24]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[25]  Teodor M. Atanackovic,et al.  An introduction to modern variational techniques in mechanics and engineering , 2004 .

[26]  Teodor M. Atanackovic,et al.  On a class of differential equations with left and right fractional derivatives , 2007 .

[27]  Teodor M. Atanackovic,et al.  On a distributed derivative model of a viscoelastic body , 2003 .

[28]  Dumitru Baleanu,et al.  A Central Difference Numerical Scheme for Fractional Optimal Control Problems , 2008, 0811.4368.

[29]  O. Agrawal.  Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain , 2002 .

[30]  Ralf Metzler,et al.  Boundary value problems for fractional diffusion equations , 2000 .