Robust Dissipativity for Uncertain Impulsive Dynamical Systems

In many engineering problems, stability issues are often linked to the theory of dissipative systems which postulates that the energy dissipated inside a dynamical system is less than the energy supplied from an external source. In the literature of nonlinear control, dissipativity concept was initially introduced by Willems in his seminal two-part papers [14, 15] in terms of an inequality involving the storage function and supply rate. The extension of the work of Willems to the case of affine nonlinear systems was carried out by Hill and Moylan (see [7, 8] and the references therein). Dissipativity theory along with its connections to Lyapunov stability theory have been mainly applied to dynamical systems possessing continuous motions. However, there are many real-world systems and natural processes which display special dynamic behavior that exhibits both continuous and discrete characteristics. For instance, many evolutionary processes, particularly some biological systems such as biological neural networks and bursting rhythm models in pathology, are characterized by abrupt changes of states at certain time instants. In addition, optimal control models in economics, frequencymodulated signal processing systems, and flying object motions may also exhibit the same feature. This feature is the familiar impulsive phenomenon, and the corresponding systems are called impulsive dynamical systems (see [1, 2, 9, 10, 11, 12, 17]). Recently, researchers have also introduced and studied the stability for other discontinuous dynamical systems such as hybrid systems [18], sampled-data systems [6], and discrete-event systems [13]. For all these systems, discontinuous system motions arise naturally. More recently, Haddad et al. have developed dissipativity and exponential dissipativity concepts