A nonlinear controlled system in a separable Banach space is considered. The control constraint is given by a set-valued map with time- and state-dependent convex compact values. Alongside the original system, we study an associated system in which the control constraint takes values equal to the extreme points of the values of the original set-valued mapping. A solution of the systems is a pair consisting of a trajectory and a control. Solutions of the second system are called extreme. We prove the existence of solutions to both systems and the compactness of the set of solutions to the original one. Moreover, we establish the density (relaxation) of the set of extreme solutions in the set of solutions to the original system. This result is analogous to the bang–bang property for linear controlled systems in finite-dimensional spaces.
Keywords: extreme points, radial proection, relaxation
Received February 25, 2026
Revised April 13, 2026
Accepted April 20, 2026
Funding Agency: The research was carried out at the expense of the state assignment within the framework of the topic ”Evolutionary and Dynamic Controlled Systems: Theory, Numerical Methods, and Applications” (scientific code FWEW-2026-0011, No. registration 126021217177-7).
Aleksandr Aleksandrovich Tolstonogov, Dr. Phys.-Math. Sci., Prof., Corresponding member of RAS, Matrosov Institute for System Dynamics and Control Theory of the Siberian Branch of the Russian Academy of Sciences, Irkutsk, 664033 Russia, e-mail: alexander.tolstonogov@gmail.com
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Cite this article as: A.A. Tolstonogov. Extreme solutions of a nonlinear control system: existence and relaxation. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 3, pp. 221–235.
