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

The necessary optimality conditions with Lagrange multipliers lambda(t) E Rn 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 tf and the final state x (tf) are fixed. According to convexity assumptions prove that necessary optimality conditions are sufficient optimality conditions.