The Necessary and Sufficient Optimality Conditions for a System of FOCPs with Caputo–Katugampola Derivatives

: The necessary optimality conditions with Lagrange multipliers λ(t) ∈ R n are studied and derived for a new class that includes the system of Caputo – Katugampola fractional derivatives to the optimal control problems with considering the end time free. The formula for the integral by parts has been proven for the left Caputo – Katugampola fractional derivative that contributes to the finding and deriving the necessary optimality conditions. Also, three special cases are obtained, including the study of the necessary optimality conditions when both the final time 𝑡 𝑓 and the final state 𝑥(𝑡 𝑓 ) are fixed. According to convexity assumptions prove that necessary optimality conditions are sufficient optimality conditions.


Introduction:
In recent years, the topic of the fractional calculus with optimal control problems (OCPs) has become taking on a wide field and growing interest of many researchers and readers, the main reason for this is that solving problems for many natural systems, scientific problems, engineering, and biological applications is more accurate than classical OCPs ones.
To see some of these applications, the feedback control into the logistic model 1 , nonanalytic dynamic systems 2 , application to identification problems 3 , economic growth model and so on 4, -7 .Fractional optimal control problems (FOCPs) are the generalization of the OCPs with fractional dynamical systems.The performance index of a FOCP is considered a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations (FDEs).
Agrawal, O. P. 8 , is using Riemann-Liouville FD in a general formulation and finds an approximate solution for a class of FOCPs.

Basic preliminaries
The basic definitions of FDs and integrals are presented with proof of some important theorems, which are used in later work: Definition 1: 18,21 Let  > 0,  > 0, and an interval[, ]of R, where 0 <  < .The left and right Riemann-Katugampola FIs of a function  ∈  1  Theorem 1: 22 A function  () is convex if for any two points  1 and  2 , then where and Theorem 2: Let  ∈ (0,1) and  > 0, then the left Caputo-Katugampola FD of a function

Integration by Parts Formula for Caputo-Katugampola FDs
Integral formula with a transformational relation between Caputo-Katugampola FD and Riemann-Katugampola FD has been proven in Theorem 3 and will rely on it to derive the necessary optimality conditions.13 Now, using integration by parts of Eq.13, to obtain: By using the definition of the right Riemann-Katugampola FD of ( 1− ()) of order (1 − ,  ) in Eq.14, to get: 


To study three basic special cases on the end time   or on (  ) in both cases when they are fixed and free in Corollary 1.
Corollary 1: Let (, ) be a minimizer of Eq.15 subject to the dynamic constraint in Eq.16 and the boundary condition in Eq.17, then 1) The transversality conditions aren't used in Theorem 4 if both   and (  ) are fixed.2) Only the transversality condition in Eq.29 is used in Theorem 4, if   is fixed and (  ) is free.
3) Only the transversality condition in Eq.28 is used in Theorem 4, if   is free and (  ) is fixed.Remark 1: If   is fixed and (  ) is free and

Studying the Sufficient Optimality Conditions for a Class of System Caputo-Katugampola FOCPs
Under some convexity assumptions sufficient conditions are studied for a class of system Caputo-Katugampola FOCPs in Theorem 5, as follows

Conclusions:
In this paper, a new system of FOCPs with Caputo-Katugampola FDs   ,     (),  = 1,2, … , , has been studied.we are assuming that the end time   free and a Lagrange multiplier vector () ∈   .The necessary optimality conditions for the system are obtained when  ∈ (0,1),  > 0 and  ∈  and consist of a Hamiltonian system, stationary condition, and transversality conditions which contributes to solving non-linear dynamical control systems with FDs to obtain approximate solutions for state and control variables with the help of the proposed numerical methods.A special case was deduced to study the system of FOCPs if both the final time and the final state are fixed, then in this case the optimality conditions obtained are applied without the transversality conditions.Also, the necessary optimality conditions have been proven to be sufficient for a system of FOCPs.