Pulse Perturbation of Optimal Control

In the next three steps, we determine the final state y(tf) due to a pulse perturbation in ti at time t.  

Step 1 Integrating the state equation over the subinterval [ti — At, ti) for the controls u and ti, we get, respectively, 

Since there is no perturbation prior to t — Ati), y at this time is the same as y. Hence, the difference between the above two equations gives 

Since the left-hand side of the inequality cannot be negative, CjS cannot be zero or negative. 

We consider Ati to be sufficiently small so that the second order term vanishes. With this provision, the change of state at t is 

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