Rayleigh–Ritz operator in inverse problems for higher order multilinear nonautonomous evolution equations

We study solvability questions for the problem on realization of operator functions for an invariant polylinear regulator of a higher-order differential system in an infinite-dimensional separable Hilbert space. This is a nonstationary coefficient-operator inverse problem for multilinear evolution equations whose dynamic order is higher than one (notice that nonautomonous hyperbolic systems belong to this class of problems). We analyze semiadditivity and continuity for a nonlinear Rayleigh–Ritz functional operator and obtain an analytic model of an invariant polylinear regulator. This model allows us to combine two bundles of trajectory curves induced by different invariant polylinear regulators in a differential system and obtain a family of admissible solutions of the initial differential system in terms of an invariant polylinear action. The obtained results can be applied in the general qualitative theory of nonlinear infinite-dimensional adaptive control systems described by higher-order multilinear nonautonomous differential systems (including neuromodelling).