Abstract
This paper addresses the finite-time controllability and set controllability of impulsive probabilistic Boolean control networks (IPBCNs). Firstly, using the algebraic state space representation (ASSR) method, the transition probability matrix of IPBCNs is established. Secondly, a kind of finite step reachability matrix with probability one is constructed, based on which, several effective criteria are proposed for the finite-time controllability with probability one of IPBCNs. Thirdly, a necessary and sufficient condition is presented for the finite-time set controllability with probability one of IPBCNs by constructing the set controllability probability distribution vector. Finally, the obtained results are extended to switching topology case.
Highlights
As one of the most significant issues in modern control theory, the concept of controllability was initiated for linear systems in 1960s [16]
Controllability has been introduced into nonlinear systems [44], [53] and stochastic systems [1], [12]
We investigate the finite-time controllability and set controllability with probability one of impulsive probabilistic Boolean control networks (PBCNs) (IPBCNs)
Summary
As one of the most significant issues in modern control theory, the concept of controllability was initiated for linear systems in 1960s [16]. (iii) System (1) is said to be finite-time controllable with probability one at X0 ∈ Dn, if for any Xd ∈ Dn, there exist a positive integer s and a control sequence {U (t) : t ∈ {0, · · · , s − 1} \ } ⊆ Dm such that P{X (s; X0, U ) = Xd } = 1. It is evident that for any given open-loop control sequence u(t) : t ∈ {0, · · · , s − 1} \ , we have Qs ≥ Ms From the above construction, all the controllability information is contained in Qs. Since we are concerned about only the reachability with probability one, using the floor function, we obtain Qs ∈ B2n×2n , which is called the s-step reachability matrix with probability one. (iii) System (1) is finite-time controllable with probability one at x0, if and only if there exists a positive integer s such that s. Remark 1: system (1) has 2n different states, the upper bound of s satisfying (6) may be greater than 2n because of the influence of impulse and randomness
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