Controllability of semilinearstochastic delay evolution equations in Hilbert spaces
Semilinear stochastic control systems with time-variable delays in the state variables are considered. It is assumed that such systems are defined in an infinite-dimensional Hilbert space, that they contain both linear and nonlinear parts and that the admissible controls are unbounded with values in an infinite-dimensional Hilbert space. Using the well-known Banach fixed-point theorem and the semigroup theory of linear operators, a sufficient condition for exact global relative controllability in a given finite-time interval is formulated and proved. In the proof, methods taken from nonlinear functional analysis and the theory of stochastic differential equations are used. Finally, an illustrative example, namely a stochastic Burgers-type equation with one constant time delay, is investigated, and controllability conditions for this equation are stated. It should be pointed out that similar controllability problems have been recently considered in the publication [\textit{K. Balachandran}, \textit{P. Balasubramaniam} and \textit{J. P. Dauer}, Optim. Control Appl. Methods 16, 283-290 (1995; Zbl 0849.93008)].
scientific article; zbMATH DE number 1780412