In this paper, we address two minimal controllability problems, where the goal is to determine a minimal subset of state variables in a linear time-invariant system to be actuated to ensure controllability under additional constraints. First, we study the problem of characterizing the sparsest input matrices that assure controllability when the autonomous dynamics' matrix is simple. Secondly, we build upon these results to describe the solutions to the robust minimal controllability problem, where the goal is to determine the sparsest input matrix ensuring controllability when specified number of inputs fail. Both problems are NP-hard, but under the assumption that the dynamics' matrix is simple, we show that it is possible to reduce these two problems to set multi-covering problems. Consequently, these problems share the same computational complexity, i.e., they are NP-complete, but polynomial algorithms to approximate the solutions of a set multi-covering problem can be leveraged to obtain close-to-optimal solutions to either of the minimal controllability problems.

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This paper introduces the concept of structural hybrid systems to address, as a particular case, the model checking problem of switching (possible large scale) linear time invariant systems. Within the proposed setup, we provide necessary conditions to ensure properties such as controllability, at each time. We show that such model checking controllability properties can be implemented using efficient algorithms (with polynomial complexity). An example, based on the IEEE 5-bus power system, is presented which illustrates our model checking and design methodologies. This work was partially supported by grant SFRH/BD/33779/2009, from Fundac ¸ão para a Ciência e a Tecnologia (FCT) and the CMU-Portugal (ICTI) program, and by projects CONAV/FCT-PT (PTDC/EEACRO/113820/2009), FCT (PEst-OE/EEI/LA0008/2013) and MORPH (EU FP7 No. 288704).