Abstract
In this paper, we study the Galerkin spectral approximation to an unconstrained convex distributed optimal control problem governed by the time fractional diffusion equation. We construct a suitable weak formulation, study its well-posedness, and design a Galerkin spectral method for its numerical solution. The contribution of the paper is twofold: a priori error estimate for the spectral approximation is derived; a conjugate gradient optimization algorithm is designed to efficiently solve the discrete optimization problem. In addition, some numerical experiments are carried out to confirm the efficiency of the proposed method. The obtained numerical results show that the convergence is exponential for smooth exact solutions.
Highlights
Optimal control problems (OCPs) can be found in many scientific and engineering applications, and it has become a very active and successful research area in recent years
Considerable work has been done in the area of OCPs governed by integral order differential equations, the literature on this field is huge, and it is impossible to give even a very brief review here
We refer the interested reader in fractional optimal control problem (FOCP) to [ – ] for some recent work on the subject
Summary
Optimal control problems (OCPs) can be found in many scientific and engineering applications, and it has become a very active and successful research area in recent years. Fractional differential equations (FDEs) have gained considerable importance due to their application in various sciences, such as control theory [ , ], viscoelastic materials [ , ], anomalous diffusion [ – ], advection and dispersion of solutes in porous or fractured media [ ], etc. A general formulation and a solution scheme for the fractional optimal control problem (FOCP) were first proposed in [ ], where the fractional variational principle and the Lagrange multiplier technique were used. Following this idea, Frederico and Torres [ , ] formulated a Noether-type theorem in the general context and studied fractional conservation laws. We refer the interested reader in FOCP to [ – ] for some recent work on the subject
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