• Tatiana S. Zarodnyuk, Matrosov Institute for System Dynamics and Control Theory SB RAS (Irkutsk, Russia)

The developed algorithms for solving nonlinear optimal control problems and approximating non-convex reachable sets formed the basis for specialized software constructed for studying nonlinear controlled dynamical systems. Such systems arise in various scientific, technical, and industrial fields and are characterized by a high degree of complexity (nonlinear dynamical systems and non-convex objective functionals). Therefore, their effective solution requires the use of algorithms that take into account the specific characteristics of these problems. The paper proposes appropriate computational technologies based on both classical approaches based on sequential discretization of continuous problems and the application of Pontryagin's maximum principle, as well as on the specific properties of the dynamical systems, such as the linear connectivity of the reachable set and the hidden convexity of the admissible velocities set of controlled dynamical systems. Pre-optimization analysis methods (estimating the degree of the objective functionals convexity and constructing the boundary of the reachable set) are also implemented as programs that allow for the initial assessment of the computational complexity of applied non-convex optimization problems and the selection of effective numerical methods for their solution. Descriptions of the mathematical, software, and technological formulations of studied non-convex optimal control problems are provided. A framework for synthesizing multi-method non-local algorithms for optimizing controlled dynamical systems is presented.
The stages of constructing the computational scheme and the specifics of selecting algorithmic parameter values are described. A developed test collection of nonlinear optimal control problems is used to test software implementations of non-convex optimization algorithms to study their limiting properties and find effective modifications. The collection includes both published problems with known solutions and problems generated using the proposed test generation methodology. The developed algorithms and corresponding software were used to solve practical problems in various scientific and technical fields: flight dynamics and space navigation, quantum physics and computational chemistry, synthesis of composite structures, economics, medicine, technical ecology, and other areas.

algorithms for non-convex optimization, optimal control problems, nonlinear controlled dynamical systems

2026-06-05