This article is devoted to exploring the optimal control problem for a system of integro-differential equations on the infinite interval. Sufficient conditions for the existence of optimal controls and trajectories have been obtained in terms of right-hand sides and the quality criterion function.

Integro-differential equation systems are the mathematical models for many natural science processes, such as those in fluid dynamics and kinetic chemistry, among others. Many of these equations have the control that minimizing specific functionals related to the dynamics of these processes.

This work specifically focuses on deriving sufficient optimality conditions for  integro-differential systems on the half-axis. The complexity of the research is in the following aspects: Firstly, the problem at hand involves optimal control with an infinite horizon, which makes the direct application of compactness criteria like the Arzela-Ascoli theorem impossible. Secondly, the problem is considered up to the moment $\tau$ when the solution reaches the boundary of the domain. This reach moment depends on the control $\tau = \tau(u)$. Hence, the solution to the problem is essentially represented by the triplet $(u^*, x^*, \tau^*)$ — the optimal control, the optimal trajectory, and the optimal exit time. We note that a particular case of this problem is the problem of optimal speed.

The main idea of proving the existence of an optimal solution relies on a compactness approach and involves the following steps: identifying a weakly convergent minimizing sequence of admissible controls, extracting a strongly convergent subsequence of corresponding trajectories, and justifying boundary transitions in equations and the quality criterion.

The work provides a problem statement, formulates, and proves the main result.

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