Boundary Value Problems for Impulsive Multi-Order Hadamard Fractional Differential Equations

Impulsive differential equations describe observed evolution processes of several real world phenomena in a natural manner, and exhibit several new phenomena such as noncontinuability and merging of solutions, rhythmical beating, etc. Dynamic processes associated with sudden changes in their states are governed by impulsive differential equations. The dynamical behavior of impulsive differential systems is much more complex than the one concerning dynamical systems without impulse effects. Many applied problems arising in control theory, population dynamics and medicines have stimulated the recent development in this field. Dynamic systems involving some continuous variable dynamic characteristics and certain reset maps that generate impulsive switching among them are termed as impulsive hybrid systems. Such systems are classified as impulsive differential systems (Benchohra et al. Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York, 2006; Lakshmikantham et al., Theory of Impulsive Differential Equations. World Scientific, Singapore, 1989; Samoilenko and Perestyuk, Impulsive Differential Equations. World Scientific, Singapore, 1995), sampled data or digital control systems (Krogh and Lynch, Hybrid Systems: Computation and Control. Springer, New York, 2000; Vaandrager and Van Schuppen, Hybrid Systems: Computation and Control. Springer, New York, 1999) and impulsive switched systems (Egbunonu and Guay, Nonlinear Analysis: Hybrid Systems 1:577–592, 2007; Engell et al. Modelling, Analysis and Design of Hybrid Systems. Springer, Heidelberg, 2002). Hybrid systems find their applications in embedded control systems interacting with the physical situation. Time and event-based behaviors are more accurately described by hybrid models as such models help to face challenging design requirements in the design of control systems. Examples include automotive control (Altafini et al. Hybrid Systems: Computation and Control. Springer, New York, 2002; Balluchi et al. Proceedings of the IEEE 7:888–912, 2000), mobile robotics (Balluchi et al. Hybrid Systems: Computation and Control. Springer, New York, 2001), process industry (Engell et al. Proceedings of the IEEE 7:1050–1068, 2000), real-time software verification (Alur et al. Proceedings of the 5th Annual IEEE Symposium on Logic in Computer Science, Philadelphia, PA, 1990, pp. 414–425), transportation systems (Lygeros et al. IEEE Transactions on Automatic Control 43:522–539, 1998; Varaiya, IEEE Transactions on Automatic Control 38:195–207, 1993), manufacturing (Pepyne and Cassandras, Proceedings of the IEEE 7:1108–1123, 2000).