Decidability of Hybrid Systems with Rectangular Differential Inclusion

Proof: 2 L(H D) (2 L(H C)) provided there is a sequence =< D(l 0); 0 ; 0 >< D(l 1); 1 ; 1 > : : : which is discrete-time (continuous-time) consistent. As discussed at beginning of Sect. 3, every discrete-time consistent sequence is continuous-time consistent. From Lemma 3-Lemma 7, every continuous-time consistent sequence is also discrete-time consistent. Thus 2 L(H D) i 2 L(H C). Theorem 9. L(H D) is a regular language. Proof: We will construct a nite state automaton which generates L(H D). Let M j be the largest integer with which x j is compared or initialized, and m j , the smallest such integer. Dene 0 j = fg, 0 = 0 1 2 1 1 12 0 n , and the nite set of states Q = L 2 0 .