Iterative Methods for Mesh Approximations of Optimal Control Problems Controlled by Linear Equations with Fractional Derivatives

• Published in: Lobachevskii journal of mathematics Vol. 41; no. 12; pp. 2687 - 2701
• Main Authors: Lapin, A. V., Romanenko, A. D.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-12-2020; Springer Nature B.V
• Subjects: Algebra; Analysis; Approximation; Boundary conditions; Boundary value problems; Control stability; Convergence; Derivatives; Equations of state; Finite element method; Geometry; Iterative methods; Linear equations; Mathematical analysis; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Norms; Optimal control; Probability Theory and Stochastic Processes; Quadratures; fractional derivative; finite element approximation; iterative method; parabolic optimal control problem
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080220120227

We consider the constrained optimal control problems governed by a parabolic initial-boundary value problem with time-fractional derivative and mixed boundary conditions. Control is carried out on the right side of the equation and on the right side of Neumann boundary condition. Finite element method with the quadratures is used for the approximation of the problem with respect to the spatial variable, and
-approximation for the time-fractional derivative taken in different definitions. Stability estimates in
-norm via
-norms of the control functions are obtained for the discrete state equation. They are used to prove the convergence and convergence rate of the proposed iterative methods for discrete optimal control problems. The main results are generalized to a problem with a state equation with fractional derivatives in time and space. The results of a series of numerical tests and their analysis are presented.