Optimality Conditions for Systems with Distributed Parameters Based on the Dubovitskii–Milyutin Theorem with Incomplete Information About the Initial Conditions

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 276; no. 2; pp. 199 - 215
• Main Author: Bahaa, G. M.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 06.10.2023; Springer; Springer Nature B.V
• Subjects: Boundary conditions; Differential equations; Heating; Initial conditions; Mathematics; Mathematics and Statistics; Neumann problem; Optimal control; Optimization; Partial differential equations; Theorems; optimality conditions; Neumann problem; 37N10; optimal control problem; second-order parabolic operator; 76N15; Dubovitskii–Milyutin theorem; 76U05; conical approximations
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-023-06735-0

In this paper, we consider an optimal control problem for a system described by a parabolic linear partial differential equation with the Neumann boundary condition. We impose some constraints on the control. The performance functional has the integral form. The control time T is fixed. The initial condition is not given by a known function but belongs to a certain set (incomplete information about the initial state). To obtain optimality conditions for the Neumann problem, a generalization of the Dubovitskii–Milyutin theorem was applied. The problem formulated in this paper describes the process of optimal heating for which we do not have exact information about the initial temperature of the heating object. We also present an example in which admissible controls and one of the initial conditions are given by means of the norm constraints.