**Author details**

¼ *d*

*Functional Calculus*

ð 1

0

0≥ *x*<sup>0</sup>

ð Þ 0 .

*x*0

ð Þ¼ <sup>0</sup> *<sup>x</sup>*~<sup>0</sup>

can rewrite

<sup>þ</sup> *<sup>x</sup>*<sup>∗</sup> <sup>0</sup>

or, in other words

straints *x*ð Þ1 ∈ *M*0, *x*<sup>0</sup>

**6. Conclusion**

**122**

¼ *x*<sup>0</sup>

ð Þ� 1 *x*~<sup>0</sup>

0≥ *x*<sup>0</sup>

ð Þ� <sup>1</sup> *<sup>x</sup>*<sup>∗</sup> ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>0</sup>

integral on the right-hand side of Eq. (31):

*dt* ð Þ *x t*ðÞ� *x t* <sup>~</sup>ð Þ <sup>0</sup>

ð Þ *x t*ðÞ� *x t* ~ð Þ <sup>00</sup>

ð Þ� 1 *x*~<sup>0</sup>

Then substituting Eq. (32) into Eq. (31), we have

ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>∗</sup> h i ð Þ<sup>1</sup> � *<sup>x</sup>*<sup>0</sup>

¼ *x*<sup>1</sup> . Then it follows from Eq. (33) that

ð Þ� 1 *x*~<sup>0</sup>

*g x*ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>0</sup> ð Þ� ð Þ<sup>1</sup> *<sup>g</sup> <sup>x</sup>*~ð Þ<sup>1</sup> , *<sup>x</sup>*~<sup>0</sup> ð Þ ð Þ<sup>1</sup> <sup>≥</sup> *<sup>v</sup>* <sup>∗</sup> ð Þþ <sup>1</sup> *<sup>x</sup>*<sup>∗</sup> <sup>0</sup>

ð Þ� <sup>1</sup> *<sup>x</sup>*~<sup>0</sup> h i ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>∗</sup> ð Þ<sup>1</sup> <sup>∈</sup> *<sup>K</sup>*<sup>∗</sup>

*x*ð Þ� , satisfying the initial conditions *x*ð Þ¼ 0 *x*<sup>0</sup> and *x*<sup>0</sup>

, *<sup>x</sup>*<sup>∗</sup> ð Þ*<sup>t</sup>* � � � *<sup>d</sup>*

ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>∗</sup> h i ð Þ<sup>1</sup> � *<sup>x</sup>*<sup>0</sup>

Thus by elementary property of the definite integrals, we can compute the

, *<sup>x</sup>*<sup>∗</sup> ð Þ*<sup>t</sup>* � � � *<sup>x</sup>*<sup>∗</sup> <sup>00</sup> h i ð Þ*<sup>t</sup>* , *x t*ð Þ� *x t* <sup>~</sup>ð Þ � �*dt*

ð Þ� 0 *x*~<sup>0</sup> ð Þ <sup>0</sup> , *<sup>x</sup>*<sup>∗</sup> h i ð Þ <sup>0</sup> � *<sup>v</sup>* <sup>∗</sup> ð Þþ <sup>1</sup> *<sup>x</sup>*<sup>∗</sup> <sup>0</sup> <sup>h</sup> ð Þ<sup>1</sup> , *<sup>x</sup>*ð Þ� <sup>1</sup> *<sup>x</sup>*~ð Þ<sup>1</sup> i þ *<sup>v</sup>* <sup>∗</sup> ð Þþ <sup>0</sup> *<sup>x</sup>*<sup>∗</sup> <sup>0</sup> h i ð Þ <sup>0</sup> , *<sup>x</sup>*ð Þ� <sup>0</sup> *<sup>x</sup>*~ð Þ <sup>0</sup> *:* (33)

Now, remember that *x*ð Þ� , *x*~ð Þ� are feasible trajectories and *x*ð Þ¼ 0 *x*~ð Þ¼ 0 *x*<sup>0</sup> and

Now, thanking to the transversality conditions Eq. (*c*1) at the endpoint *t* ¼ 1, we

*g x*ð Þ1 , *x*<sup>0</sup> ð Þ� ð Þ1 *g x*~ð Þ1 , *x*~<sup>0</sup> ð Þ ð Þ1

Thus, summing the inequalities Eqs. (34) and (35) for all feasible trajectories

ð Þ1 ∈ *M*1, we have the needed inequality:

According to proposed method, the problem with the differential inclusions described by polynomial linear differential operators is investigated. Obviously, this problem is an important generalization of problems with first-order differential inclusions. Thus, sufficient conditions of optimality for such problems are deduced. Here the existence of nonfunctional initial point or endpoint constraints generates different kinds of transversality conditions. Besides, there can be no doubt that investigations of optimality conditions of problems with second- and fourth-order Sturm-Liouville type differential inclusions can play an important role in the development of modern optimization and there is every reason to believe that this role

<sup>≥</sup> *<sup>v</sup>* <sup>∗</sup> ð Þþ <sup>1</sup> *<sup>x</sup>*<sup>∗</sup> <sup>0</sup> <sup>h</sup> ð Þ<sup>1</sup> , *<sup>x</sup>*ð Þ� <sup>1</sup> *<sup>x</sup>*~ð Þ<sup>1</sup> i � *<sup>x</sup>*<sup>∗</sup> ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>0</sup> ð Þ� <sup>1</sup> *<sup>x</sup>*~<sup>0</sup> h i ð Þ<sup>1</sup> (35) .

*g x*ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>0</sup> ð Þ� ð Þ<sup>1</sup> *<sup>g</sup> <sup>x</sup>*~ð Þ<sup>1</sup> , *<sup>x</sup>*~<sup>0</sup> ð Þ ð Þ<sup>1</sup> <sup>≥</sup>0 or *g x*ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>0</sup> ð Þ ð Þ<sup>1</sup> <sup>≥</sup> *<sup>g</sup> <sup>x</sup>*~ð Þ<sup>1</sup> , *<sup>x</sup>*~<sup>0</sup> ð Þ ð Þ<sup>1</sup> . □

ð Þ� 0 *x*~<sup>0</sup> ð Þ <sup>0</sup> , *<sup>x</sup>*<sup>∗</sup> h i ð Þ <sup>0</sup>

� *<sup>x</sup>*<sup>∗</sup> <sup>0</sup> h i ð Þ<sup>1</sup> , *<sup>x</sup>*ð Þ� <sup>1</sup> *<sup>x</sup>*~ð Þ<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>∗</sup> <sup>0</sup> h i ð Þ <sup>0</sup> , *<sup>x</sup>*ð Þ� <sup>0</sup> *<sup>x</sup>*~ð Þ <sup>0</sup> *:* (32)

ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>∗</sup> h i ð Þ<sup>1</sup> � *<sup>v</sup>* <sup>∗</sup> ð Þþ <sup>1</sup> *<sup>x</sup>*<sup>∗</sup> <sup>0</sup> h i ð Þ<sup>1</sup> , *<sup>x</sup>*ð Þ� <sup>1</sup> *<sup>x</sup>*~ð Þ<sup>1</sup> *:* (34)

ð Þþ <sup>1</sup> *<sup>x</sup>*<sup>∗</sup> h i ð Þ<sup>1</sup> , *<sup>x</sup>*ð Þ� <sup>1</sup> *<sup>x</sup>*~ð Þ<sup>1</sup>

*<sup>M</sup>*<sup>0</sup> ð Þ *<sup>x</sup>*~ð Þ<sup>1</sup> , *<sup>x</sup>*<sup>∗</sup> <sup>0</sup>

ð Þ<sup>1</sup> <sup>∈</sup>*<sup>K</sup>* <sup>∗</sup>

ð Þ¼ 0 *x*<sup>1</sup> and endpoint con-

*<sup>M</sup>*<sup>1</sup> *x*~<sup>0</sup> ð Þ ð Þ1

*dt <sup>x</sup>*<sup>∗</sup> <sup>0</sup> h i ð Þ*<sup>t</sup>* , *x t*ðÞ� *x t* <sup>~</sup>ð Þ *:*

Elimhan N. Mahmudov1,2

1 Department of Mathematics, Istanbul Technical University, Istanbul, Turkey

2 Azerbaijan National Academy of Sciences, Institute of Control Systems, Baku, Azerbaijan

\*Address all correspondence to: elimhan22@yahoo.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
