Optimal Control for a Nonlinear System of Integro-Differential Equations of Fractional Order
Ключевые слова:
Fractional equation, product of a nonlinear function with a nonlinear integral, Pontryagin function, method of contracted mappings, existence, uniqueness and stability of a solutionАннотация
This paper investigates an optimal control problem for a nonlinear system of
fractional-order integro-differential equations involving the Gerasimov–Caputo operator. The
considered model contains a nonlinear integral interaction represented by the product of two
nonlinear functions, which substantially complicates both the analytical structure of the system
and the derivation of optimality conditions.
Unlike the classical approach in optimal control theory, where the control function is assumed to
be known when determining the state trajectory, the present work constructs the state function
and the control function simultaneously as a coupled solution pair of a nonlinear functionalintegral system.
Using fractional integration techniques, the original fractional integro-differential equation is
reduced to an equivalent nonlinear integral equation for the state function. By introducing an
adjoint final-value problem and constructing the Pontryagin function, a second nonlinear integral equation for the control function is derived. As a result, the optimal control problem is
transformed into a strongly coupled nonlinear system of integral equations.
A constructive analytical method based on successive approximations and the contraction mapping principle is developed. Explicit sufficient conditions ensuring existence and uniqueness of
the vector-valued control and state functions are established. Furthermore, continuous dependence of the solution on the initial vector is proved.
The obtained results provide a constructive framework for the analysis of nonlinear fractional
optimal control systems with memory effects and nonlocal interactions and extend several known
solvability results for fractional dynamical systems.
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