Nonlinear control problems with and without fractional derivatives

Abstract

The main subject of research of this thesis are nonlinear control problems with the Caputo fractional derivative of order α between 0 and 1, which in the case α=1 reduces to the classical first order derivative. We consider the question of global controllability. More precisely, we examine which conditions need to be satisfied so that, for any given initial and final data, one can find a control function for which the solution of the system reaches the desired state at the end of the given interval. In order to do so, we derive several auxiliary results regarding fractional differential equations and linear time-varying fractional control problems. We define the Riemann-Liouville and the Caputo state-transition matrices, which are essential part of the solutions of linear fractional systems, and derive estimates of those matrices. Further, we consider linear time-varying fractional control problems, introduce the controllability Gramian matrix, prove the equivalence between controllability and regularity of the Gramian, introduce the associated adjoint problem and prove the equivalence between controllability of the control problem and observability of the associated adjoint problem. Moreover, we apply the Hilbert uniqueness method and techniques from the calculus of variations to obtain the optimal control function in the weighted L2-space. Then, using properties of the solution of the linearized control problem and the Leray-Schauder fixed point theorem, we derive controllability result for one class of nonlinear control problems with unbounded dynamics. We consider the cases with fractional and with the integer-order derivative, since in the non-integer case the construction of the solution requires to take into account the memory embedded in the fractional derivative.