J Optim Theory Appl (2013) 156:33–44 DOI 10.1007/s10957-012-0212-5 Controllability Results for Nonlinear Fractional-Order Dynamical Systems K. Balachandran · V. Govindaraj · L. Rodríguez-Germa · J.J. Trujillo Received: 7 July 2012 / Accepted: 11 October 2012 / Published online: 19 October 2012 © Springer Science+Business Media New York 2012 Abstract This paper establishes a set of sufficient conditions for the controllability of nonlinear fractional dynamical system of order 1 <α< 2 in finite dimensional spaces. The main tools are the Mittag–Leffler matrix function and the Schaefer’s fixed-point theorem. An example is provided to illustrate the theory. Keywords Controllability · Fractional Differential Equations · Mittag–Leffler Matrix Function · Schaefer’s Fixed-Point Theorem 1 Introduction Fractional differential equations have gained considerable popularity and importance during the past there decades or so. The class of fractional differential equations of various types play important roles and tools are used from not only mathematics but also physics, control systems, dynamical systems and engineering to create the math- ematical modeling of many physical phenomena. The fractional-order integration and differentiation represent a rapidly growing field both in theory and in applications to K. Balachandran · V. Govindaraj Department of Mathematics, Bharathiar University, Coimbatore 641 046, India K. Balachandran e-mail: kb.maths.bu@gmail.com V. Govindaraj e-mail: govindaraj.maths@gmail.com L. Rodríguez-Germa · J.J. Trujillo ( ) Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain e-mail: jtrujill@ullmat.es L. Rodríguez-Germa e-mail: lrgerma@ull.es