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29 Sensitivity Analysis and Real-Time Control of Optimal Control Problems Lemma 2. 36) Now we are in the position to state the main sensitivity result for control problems with mixed control-state constraints; cf. [39]: Theorem 3. (Solution differentiability for mixed control-state constraints) Let ✞✝ ✁ ✟ ✁ be admissible for OC ✄✁ such that assumptions (AC-1)–(AC-6) hold. 32) are satisfied for all ✟✁ . The functions ✒ ✤✣ ✫✔ ✞ ❑ ✠ ✂✁❆✄ ✒✓ ✫✔ ✄ ■✆ ✄ ✂ ❀✩✄ ✂❄ ✁ ✛ ✒ ✒✓ ✕✔✦✣✬ ✎✔ ✭ ✯ ✒ ❈✣✬ ✕✔❩❀ ✭ ✒ ❈✣ ✒✓ ✕✔✦✣✬ ✎✔❚✣ ✒ ❈✣✬ ✎✔❩❀ ✭ ✒ ❈✣ ✒● ✕✔✦✣✥ ✕✔ ✠ ✠ are -functions for all ✒ ❈✣✥ ✕✔❫✁ ✪ ✯✰✣ ✖✬ ❃ ✄ ✁ .

Maurer [44], Malanowski [35–38]. Extensions of SSC to control problems with free final time or to multiprocess control systems may be found in [5,48]. The SSC in the previous theorem are usually not suitable for a direct numerical verification in practical control problems. In order to obtain verifiable sufficient conditions we shall strengthen the SSC in Theorem 1 in the following way. 11) are satisfied. Firstly, this leads to the requirement that the strict Legendre-Clebsch condition ✛✜✒ ❩✣ ✔ ✂ ✞❑ ❃ ✞ ✏ ✪ ✬ ✤ ✡ ✌ ✯✰✣ ✡ ❯ ✁ ✪ ✰✯ ✣ ✖✬ ✣ ✯✰✣ ✡ ✝ ☞ ☎✂ is valid on the whole interval ✪ ✁ ✂ ✬ .

Recall the assumption (AG-3) where it was ✠ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ of junction points for the nominal sorequired that the set ✁ ✁ ✁ ✁ ✌✌✌ ✁ ✁ lution is finite with ✁ ✂ . Moreover, every ✠ ✂ ✝✕ ✚ ✁ junction is the junction for exactly one constraint ✪ ✬ with index ✠ . , the constraints are not active at the initial and final time. Hence, by construction we have ✁ ❀ ✭ ❍ ✣ ●✒ ✎✣ ✔ ✶✣  ✯❁❀ ✭ ✹ ❨ ❨ ✠ ❨ ✠ ✹ ✠ ●★ ❀ ✭ ✺✹❬✯ ✁ ✂ ❉✒ ✱❉✔ ❨ ✠ ✁✤✒ ❈✔✮✭ const. for ✁ ❨ ❨ ✁ ✣✼✱☞✭✟✯◆✣❉✳✵✣ ✁ ✁ ✁✶✣ ✑ ✁ ★ ★ ✁ ✁ In the interval ✒ ★ ✣ ★ ✔ the solution ✒❙✌✤✔ and the multiplier ✒✢✌✫✔ to the program ★ ✒❙✌✤✔ and the multiplier ★ ✒❙✌✤✔ of the following MP ✒❙✌✤✔ agree with the solution parametric program with equality constraints: ✠ ★ MP ✒ ✮✣ ❩✣✬ ✎✔ minimize ✑✆☎ ✝ ❍ ✏❱✒ ✮✣ ✣ ❩✣✥ ✕✔✺▲ ✒ ✮✣ ▼✣✥ ✕✔✮✭✟✯◆✣ ✡✁ ✁✰✒ ❈✔  ✁ The notation ✪ ✱✍✬ in brackets is used to distinguish these functions from the ✱ th component.