## A general formulation and solution scheme for fractional optimal control problems.(English)Zbl 1121.70019

The concept of fractional derivative (derivative of arbitrary order) leads to fractional differential equations and fractional dynamic systems (FDS). There are several definitions of fractional derivatives. Here the author formulates a fractional optimal control problem (FOCP) for fractional differential equations, written with the use of Riemann-Liouville definitions. This problem consists in finding the optimal control $$u(t)$$ that minimizes the integral performance index subject to system dynamic constraints and initial conditions. With the use of the calculus of variations, Lagrange multipliers and the formula for fractional integration by parts, necessary conditions of optimality of the FOCP are obtained in the form of Euler-Lagrange equations, including left and right Riemann-Liouville fractional derivatives. As a special case, the FOCP is studied for linear FDS with the performance index in the form of integral of quadratic form in the state and in the control. For this case, the author proposes numerical scheme based on the approximation of FOCP solutions by a set of basic functions (shifted Legendre polynomials). The numerical scheme results in a system of linear algebraic equations for unknown coefficients.

### MSC:

 70Q05 Control of mechanical systems 26A33 Fractional derivatives and integrals
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### References:

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