Initialized hybrid automata with linear differential inclusions and rectangular constraints are hybrid automata where the invariants, guards, resets, and initial values are given by rectangular constraints, the flows are described by linear differential inclusions of the form $ax + b \lhd_1\dot{x} \lhd_2 cx+d$ (with $\lhd_1,\lhd_2 \in \{<,\leq\}$), and a variable $x$ is reset on mode change whenever the differential inclusion describing the dynamics for $x$ changes. Such automata strictly subsume initialized rectangular automata. Our main result is that while the control state reachability problem for such automata is undecidable, the time-bounded reachability problem is decidable.
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