Lagrange problem for fractional ordinary elliptic system via Dubovitskii–Milyutin method

Abstract

In the paper, we investigate a weak maximum principle for Lagrange problem described by a fractional ordinary elliptic system with Dirichlet boundary conditions. The Dubovitskii-Milyutin approach is used to find the necessary conditions. The fractional Laplacian is understood in the sense of Stone-von Neumann operator calculus.

Necessary optimality conditions for Lagrange problems involving ordinary control systems described by fractional Laplace operators

“…Two illustrative examples were presented. In particular, we observed that [22,Thm. 4] cannot be used to problem (16)- (17).…”

“…Similarly (using assumption (A) and analogous arguments as in [22,Prop. 3.2]), we obtain a differentiability property of the mapping F 0 k 1 , whereby the differential…”

“…We derived our result using the smooth-convex extremum principle due to Ioffe and Tikchomirov. The result of such a type for the problem with Dirichlet boundary conditions was proved in [22,Thm. 4] via Dubovitskii-Milyutin method.…”

“…The definitions of these operators come from the Stone-von Neumann operator calculus and are based on the spectral integral representation theorem for a self-adjoint operator in a Hilbert space (cf. [21,22] and Appendix A).…”

“…The aim of this paper is to derive the necessary optimality conditions for problems (OCP k ). Below, we formulate a result of such a type for the problem (OCP 1 ) obtained in [22]…”