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Abstract. The study of properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. This paper aims to briefly survey recent results on stability and controllability of switched linear systems. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability. After that, we review the controllability results.

Dynamics of discrete-time neural networks have not been well documented yet in fractional-order cases, which is the first time documented in this manuscript. This manuscript is mainly considered on the stability criterion of discrete-time fractional-order complex-valued neural networks with time delays. When the fractional-order holds 1 < < 2, sufficient criteria based on a discrete version of generalized Gronwall inequality and rising function property are established for ensuring the finite stability of addressing fractional-order discrete-time-delayed complex-valued neural networks (FODCVNNs). In the meanwhile, when the fractional-order holds 0 < < 1, a global Mittag-Leffler stability criterion of a class of FODCVNNs is demonstrated with two classes of neuron activation function by means of two different new inequalities, fractional-order discrete-time Lyapunov method, discrete version Laplace transforms as well as a discrete version of Mittag-Leffler function. Finally, computer simulations of two numerical examples are illustrated to the correctness and effectiveness of the presented stability results.