Optimal Control for a Fractional Differential-Integral Equation with Degeneration in a Locally Convex Space

This paper investigates a nonlinear optimal control problem for a fractional integro-differential equation of Gerasimov–Caputo type with degeneration in a locally convex space. The considered model contains a nonlinear Fredholm integral term together with the product of two nonlinear functions, which significantly complicates the analytical structure of the problem. Unlike the classical Banach-space framework, the analysis is carried out in a locally convex setting generated by a family of seminorms. This approach allows one to study classes of functional spaces that naturally arise in modern analysis, generalized function theory, and infinite-dimensional control problems. Using the Kilbas–Saigo (modifed Mittag-Leffler) function, the original fractional differential equation is transformed into an equivalent nonlinear integral equation. By constructing the Pontryagin functional, an additional integral equation for the control function is derived, leading to a coupled nonlinear system for the state and control variables. The existence and uniqueness of the solution pair are established by combining the method of successive approximations with a seminorm-based contraction principle and the Schauder–Tychonoff fixed point theorem in locally convex algebra. Furthermore, continuous dependence on the initial data and Lyapunov-type stability of the solution are proved using a specially constructed Lyapunov functional adapted to the fractional locally convex framework. The obtained results extend several known solvability and stability results for fractional optimal control problems from Banach spaces to the broader class of locally convex spaces.

Hali tarjima qilinmagan