## 1 Introduction
In [38], the IO-FTS was studied for a class of impulsive dynamical linear systems, and both static output and state feedback controllers were designed to stabilize the impulsive systems. In [43], using coupled differential linear matrix inequality, a pair of necessary and sufficient conditions for the IO-FTS of impulsive linear systems were proposed. By applying the MDADT technique, the IO-FTS was considered for a class of discrete-time positive switched systems with delays [30] and a class of continuous-time positive switched systems with delays [29] respectively, and the sufficient conditions were presented to guarantee the systems had IO-FTS. However, the IO-FTS of positive hybrid systems based on FSM was not mentioned. Compared with some positive hybrid systems, time-varying impulsive positive hybrid systems are more general, and the existence of impulse makes it more practical. In addition, the concept of IO-FTS is defined with the finite-time interval, and the transient performance of the system can be obtained with this interval. In practical systems, the system performance is usually only concerned with the finite-time, for example, multiple guided missiles transmission, and so on. Therefore, the research on the IO-FTS of the time-varying impulsive positive hybrid systems has certain practical value and theoretical significance. Motivated by the above backgrounds, we consider a class of hybrid systems whose discrete event subsystem is modeled as an FSM, and the continuous variable subsystem consists of several continuous time-varying impulsive positive systems. Such systems are called hybrid systems based on FSM. In fact, they are event-driven systems. The main contributions of this paper are given as follows: firstly, the concept of IO-FTS is extended for such hybrid systems. Secondly, under two different classes of exogenous input signals, the sufficient conditions of IO-FTS of a single linear system are deduced by co-positive Lyapunov function. Furthermore, by combining the multiple co-positive Lyapunov functions and MDADT technique, the sufficient conditions of IO-FTS of hybrid systems are derived, and they have good flexibility and weak conservatism. Next, the output feedback controller for stabilization problem is also deduced, and the obtained results are presented under linear programming form. Finally, a numerical example is given to ensure the accuracy of the results. Our work is organized as follows: in Section 2, some definitions and FSM are introduced, and the problems which will be dealt with are stated. In Section 3, four theorems of IO-FTS for time-varying positive linear systems are derived and the corresponding proofs are given. In Section 4, four theorems of IO-FTS for time-varying positive hybrid systems are derived and the corresponding proofs are given. In Section 5, an example is given to show the effectiveness of the theorem. Some conclusions are given in Section 6. Notation: Throughout our paper, \(\mathbb{R}^{+}\) denotes the set of nonnegative real numbers, \(\mathbb{R}^{n}\) represents the vector of n-tuples of real numbers, \(\mathbb{R}^{n\times m}\) is the space of matrices with real entries, and \(I_{n}\) denotes an identity matrix. For a given matrix \(A\in\mathbb{R}^{n\times m}\), \(a_{ij}\) stands for the element in \(i\)th row and \(j\)th column of \(A\), and \(A\prec 0\) (\(\preceq 0\)) means all elements of \(A\) are positive (nonnegative), i.e., \(a_{ij}<0\) (\(\leq 0\)). \(A\succ 0\) (\(\succeq 0\)) means all elements of \(A\) are negative (non-positive), i.e., \(a_{ij}>0\) (\(\geq 0\)). A matrix \(A\) is called a Metzler matrix if its off-diagonal elements are all nonnegative real numbers. The symbol \(\mathcal{L}_{p}\) denotes the space of vector-valued signals whose \(p\)-th power is absolutely integrable over \([0,\infty]\). The restriction of \(\mathcal{L}_{p}\) to \(\Omega\subset[0,\infty]\) is denoted by \(\mathcal{L}_{p,\Omega}\).