ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

17. Control Theory and Optimization

1991 MS Classification: 49, 35 Zelikin, Mikhail I., Moscow State (Lomonosov) University, Moscow, Russia Riccati-type PDE and Multidimensional Calculus of Variations Consider the Dirichlet-type functional with the integrand

J = ∫_{D}(A_{ij}^{αβ}(t)∂x^{i}/∂t^{α}∂x^{j}/∂t^{β} + 2C_{ij}^{α}(t)x^{i}∂x^{j}/∂t^{α} + B_{ij}(t)x^{i}x^{j})dt. Here D is the r–dimensional manifold with the boundary ∂D, t ∈ ℝ^{r}, x ∈ ℝ^{n}; x(t) is a section of the vector bundle E with the base D. Consider also the following system of quadratic PDE

∂/∂t^{α}(A_{ik}^{αβ}W_{jβ}^{k} + C_{ji}^{β}) = B_{ij} − A_{ks}^{αβ}W_{iα}^{k}W_{jβ}^{s}, i, j = 1, …, n. Solutions to the equation define a connection W on the vector bundle E. A field of extremals of the functional J in E exists iff the curvature of the connection W equals zero:

∂W_{α}/∂t^{β}∂W_{β}/∂t^{α} = [W_{α}, W_{β}].