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An Inverse Problems for Nonlinear Evolution Equations: Criteria of Existence of an Invariant Polylinear Controller for a Second-Order Differential System in a Hilbert Space

The solvability of the problem of realizing operator functions of an invariant polylinear controller (IPL-controller) of a non-stationary differential system (D-system) of the second order, which allows for two bundles of dynamic processes of the “trajectory, control” type that are induced in this D-system by two different polylinear controllers, to unite these bundles, through the action of the IPL-controller, into a subfamily of admissible solutions of the given Dsystem. The problem under consideration belongs to the type of nonstationary coefficient-operator inverse problems for evolution equations, including hyperbolic ones, in a separable Hilbert space and is solved on the basis of a qualitative study of the properties of continuity and semiadditivity of the RayleighRitz functional operator. The results obtained have applications in the theory of nonlinear infinite-dimensional adaptive dynamical systems for a class of higher-order polylinear differential models.