On an inverse boundary control of a system of integro-differential equations of fractional order in a locally convex space

This paper investigates an inverse boundary optimal control problem for a nonlinear system of fractionalorder integro-differential equations in a complete locally convex space. The state equation is supplemented with unknown boundary data, which are identified simultaneously with the optimal control. By introducing an adjoint final-value problem, a Pontryagin-type function is constructed, and a necessary optimality condition is derived in the form of a nonlinear integral equation for the control function. The original boundary value problem is transformed into an equivalent coupled system of nonlinear integral equations with respect to the state function, the boundary value, and the control. The solvability of this system is established by the Banach contraction principle in spaces whose topology is generated by a family of seminorms. Sufficient conditions guaranteeing the existence and uniqueness of the optimal control, the corresponding state trajectory, and the identified boundary value are obtained. In addition, the continuous dependence of the solution on the boundary data is proved. The proposed approach provides a constructive framework for studying inverse optimal control problems for fractional systems in locally convex spaces and can be extended to more general classes of nonlocal and infinite-dimensional dynamical systems.

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