Bulletin of Dnipropetrovsk University. Series: Communications in Mathematical Modeling and Differential Equations Theory

3-19

Authors: O. G. Goman, D. V. Evdokimov, Yu. L. Menshikov, V. V. Popov

The article is devoted to 70-anniversary of the famous scientist, teacher, of the member of the Academy of Ukraine Sciences, of the rector of Dnipropetrovsk National University named O.Gonchara Professor Nikolai Viktorovich Polyakov which is the one of the founders of the Dierential equations Department of DNU.

26-55

Authors: P. I. Kogut, A. O. Putchenko

We discuss the existence of weak solutions to one class of Dirichlet boundary value problems (BVP) for non-linear elliptic equation. Because of the specic of nonlinearity, we cannot a priori expect to have a solution in the standard functional space. Instead of this we show that the original BVP admits the so-called approximate weak solution. To do so, we introduce a special family of perturbed optimal control problems (OCPs) where the class of ctitious controls are closely related with the properties of nonlinearity in right-hand side of the elliptic equation. The main question we discuss in this paper is about solvability of perturbed OCPs, uniqueness of their solutions, and asymptotic properties of optimal pairs as the perturbation parameter " > 0 tends to zero. As a result, we derive the sucient conditions of the existence of weak solutions to the given class of nonlinear Dirichlet BVP and give a way for the approximation of such solutions.

71-92

Authors: P. I. Kogut, P. I. Tkachenko

An optimal control problem for the Dirichlet boundary value problem for the nonlinear elliptic equation with p(x)-Laplacian is considered. It is shown that this problem has at least one solution with respect to certain set of admissible pairs. The corresponding theoretic framework regarding Sobolev-Orlicz spaces is given.

106-117

Authors: O. P. Kupenko

We study a Dirichlet optimal control problem for a nonlinear elliptic anisotropic p-Laplace equation with control and state constraints. The matrix-valued coecients we take as controls and in the linear part of dierential operator we consider coecients to be unbounded skew-symmetric matrix. We show that, in spite of unboundedness of the non-linear dierential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable.

192-203

Authors: S. O. Gorbonos

The necessary optimality conditions to one class optimal control problems for parabolic equations with unbounded coecients were developed in case when their solutions have non-variational character.

222-229

Authors: N. N. Osipchuk, V. I. Perehrest

Considered full planetary vortex solution in the presence of two regular member sand one singular, the in uence of the intensity of singular dipole on geometry and characteristics of vortex structures. The results of the research can be applied to clarify the law of planetary distances.