Convexity of the attainability set of a bilinear controlled system (Q1904778)

A bilinear control system composed of two scalar time-invariant bilinear dynamical systems is considered. The definition of the attainable set at time \(T\) from a given initial state is presented. Next, using methods of optimal control theory, convexity of the attainable set is proved. The convexity property depends on the position of the initial state and on control duration. Estimates of the attainable set are also discussed. Moreover, several remarks and comments on attainable sets for nonlinear control dynamical systems are given. Similar controllability and attainability problems for bilinear control systems can be found e.g. in [\textit{J.-P. Gauthier} and \textit{G. Bornard}, SIAM J. Control Optimization 20, No. 2, 377-384 (1982; Zbl 0579.93005)].