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Fractional optimal control of a 2-dimensional distributed system using eigenfunctions. (English) Zbl 1170.70397
Summary: This paper presents an eigenfunctions expansion based scheme for Fractional Optimal Control (FOC) of a 2-dimensional distributed system. The fractional derivative is defined in the Riemann-Liouville sense. The performance index of a FOC problem is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) containing two space parameters and one time parameter. Eigenfunctions are used to eliminate the terms containing space parameters and to define the problem in terms of a set of generalized state and control variables. For numerical computation Grünwald-Letnikov approximation is used. A direct numerical technique is proposed to obtain the state and the control variables. For a linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for different number of eigenfunctions and time discretization. Numerical results show that only a few eigenfunctions are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced.
##### MSC:
 70Q05 Control of mechanical systems (general mechanics) 26A33 Fractional derivatives and integrals (real functions)
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