A Convexity of Final States

This function has pulse perturbations vi and V2 in the two disjunct subintervals 11 and I2, respectively. Elsewhere in [0, if], the function is the same as the optimal control. Derived earlier in Section 5.4.1 (p. 131), the state change at time is 

For sufficiently small Ati and At2, the second order term vanishes, and the state change at (t2 — At2) — the onset of I2 — simplifies to 

To determine the state change at the end of I2, we repeat the steps of Section 5.4.1, which is concerned with a single pulse perturbation corresponding to the first subinterval Ii. 

for sufficiently small At2, the above equation when substituted in the result of Step 1, yields [compare with Equation (5.8), p. 133] 

In the above equation, Sy t2) indicates the state change at 2 if there were no previous perturbation. Thus, Ay(t2) is Sy t2) if Ay(t2 - At2) is zero. Obviously, for the first perturbation, Ay(ti) is 5y(ti) so that Equation (5.37) can be written as 

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