A New Approach in Solving Fractional Nonlinear Control Problems

Motivated by the porous media equation as well as population dynamics models, we study a nonlinear fractional system of differential equations and investigate a control problem for such system. We look for the control that will govern the system from a given initial state to a desired final state. The strategy that will be employed in order to solve the problem is essentially novel and involve various techniques and tools from fractional calculus and functional analysis. The basic idea consists of linearizing the system, and then using the fixed point results to obtain a solution. Also, we are interested in concepts such as controllability and observability in the fractional framework, which further involve the Gramian matrix, the Kalman rank condition and the adjoint system.