Polyhedral optimization of discrete and partial differential inclusions of parabolic type

ABSTRACT

The purpose of this paper is to look into the optimization of the first mixed boundary value problems for partial differential inclusions of the parabolic type. More specifically, we discuss a constructive approach to the study and solution of optimization problems for partial differential inclusions based on the discrete-approximate method. We formulate necessary and sufficient conditions of optimality by reducing the original first mixed boundary value problems to finite-dimensional problems in mathematical programming. All of our conditions are presented through the notion of Euler-Lagrange inclusions and transversality conditions. The effectiveness of our approach and its superiority over well-known methods are demonstrated by the example of the linear optimal control problem. Then, by passing formally to the limit as the discrete steps approach zero, we derive sufficient conditions of optimality for parabolic differential inclusions in the discrete-approximate problem.

KEYWORDS:

• Partial differential inclusions
• optimality conditions
• Euler-Lagrange inclusion
• transversality

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 49K20
• 49M25
• 49M05
• 35Gxx
• 35K20

Disclosure statement

No potential conflict of interest was reported by the author(s).