Main scientific interests are various problems of non-convex multivalued analysis. Theory of differential inclusions with non-convex right-hand side in a Banach space was created. For the first time the existence and relaxation theorems for continuous selectors passing through extreme points of multivalued mappings with closed convex decomposable values in spaces of Bochner integrable functions were proved. The last means the density of the set of extreme continuous selectors in the set of all continuous selectors. To this end the definition of seminorm with properties of relaxation and scalar compactnes in the spaces of Bocner integrable functions was introduced. Rich in content examples such seminorms having numerious applications were constracted. Similar theorems for continuous selectors passing through fixed points of multivalued mappings with non-convex closed decomposable values depending on a parametr were proved. Numerous applications of selectors theorems for study of non-convex problems in calculus of variations and control systems were examined. In particular "bang-bang" principle for nonlinear evolution control systems with non-convex mixed constraints for control and an analog of N. N. Bogolyubov theorem concerning an approximation of value of convexified integrand by values of non-convex original integrand along solutions of such control sdystems were proved.
Main publications:
A. Tolstonogov. Differential inclusions in a Banach space. Kluwer Academic Publishers, 2000.
A. A. Tolstonogov, D. A. Tolstonogov. $L_p$-continuous extreme selectors of multifunctions with decomposable values: existence theorems // Set-valued Analysis, v. 4, no. 2 (1996), 173–203.
A. A. Tolstonogov, D. A. Tolstonogov. $L_p$-continuous extreme selectors of multifunctions with decomposable values: relaxation theorems // Set-valued Analysis, v. 4, no. 3 (1996), 237–269.
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