The problem of control of a two-dimensional transformation of spatial arguments in a parabolic functional-differential equationстатья
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Аннотация:Quasilinear parabolic functional-differential equations with a control transformation of the space argument of the unknown function arise in the modeling of nonlinear optical systems with nonlocal feedback and have been comprehensively studied by analytical and numerical methods. Numerous papers deal with sufficiently smooth invertible transformations: the Andronov–Hopf bifurcation was considered for a rotation in an annulus and a disk and for a smooth invertible transformation in an arbitrary domain; the Turing bifurcation for reflection of the argument; the method of normal forms for functional-differential equations with a rotation of arguments was used as well.
In optimization problems for argument transformations, it is unnatural to remain in the framework of the very narrow class of invertible admissible transformations, since the wide abilities of modern optical systems to perform noninvertible transformations (for example, focusing to a point or a line) are not used in this case. On the other hand, the results of numerical analysis show that localized phase profiles important for applications can be obtained just in the case of transformations which are not invertible in neighborhoods of localization points. The extension, traditional in optimal control theory, of the set of admissible controls-transformations to the class of arbitrary (not necessarily invertible) Lebesgue measurable transformations bounded almost everywhere requires the passage to a generalized treatment of a transformation and the statement of the corresponding new optimal control problems. In the special case of a transformation one-dimensional with respect to spatial variables, such an investigation was performed in previous papers. The control of a completely two-dimensional transformation typical for optical
models is a much more complicated problem not studied yet; it differs from the one-dimensional case by a much smaller margin of smoothness for solutions of functional-differential equations.
In the present paper, we suggest a statement of the control problem for a two-dimensional
transformation of the arguments and analyze its basic properties: the uniform quasidifferentiability of the objective functional, the H¨older condition for its gradient, the convergence of the gradient projection methods, and the solvability of the control problem on a compact set.