Abstract
This paper studies two optimal control problems for linear time-invariant systems of fractional order with lumped parameters whose dynamics is described by equations which contain Riemann-Liouville derivative. The first problem is to find control with minimal norm and the second one is to find control with minimal control time at given restriction for control norm. The problem setting with nonlocal initial conditions is considered which differs from other known settings for integer-order systems and fractional-order systems described in terms of equations with Caputo derivative. Admissible controls are allowed to belong to the class of functions which arep-integrable on half segment. The basic investigation approach is the moment method. The correctness and solvability of moment problem are validated for considered problem setting for the system of arbitrary dimension. It is shown that corresponding conditions are analogous to those derived for systems which are described in terms of equations with Caputo derivative. For several particular cases of one- and two-dimensional systems the posed problems are solved explicitly. The dependencies of basic values from derivative index and control time are analyzed. The comparison is performed of obtained results with known results for analogous integer-order systems and fractional-order systems which are described by equations with Caputo derivative.
Highlights
Affairs of dynamics and control for fractional-order systems attract sufficient attention of modern research community. This field develops impetuously and is characterized by both of significant theoretical and very actual applied results. This field is much wider than integer-order dynamics and contains some open problems concerning the foundations of fractional calculus
In this paper the optimal control problem is investigated for linear dynamical systems of fractional order described by equations with Riemann-Liouville derivative
The problem reduced to the moment problem
Summary
Affairs of dynamics and control for fractional-order systems attract sufficient attention of modern research community. The Riemann-Liouville definition is used frequently in theoretical investigations it has a physical sense but less similar to ordinary derivative This derivative is nonzero for constant function and initial and boundary problems for equations with derivative of this kind require posing nonlocal conditions. Another effective and quite universal approach to the search of optimal control is moment method [4] Based on these methods in [5,6,7,8] the approach was developed to investigation of optimal control problems for linear dynamic systems of fractional order with lumped parameters, described by equations with Caputo derivative. In this paper the moment method is applied to investigation of optimal control problems for dynamical systems of fractional order with lumped parameters, described by equations with Riemann-Liouville derivative. Explicit solutions of optimal control problem are obtained and analyzed, including comparison with analogous integer-order systems and systems described by equations with Caputo derivative
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