Optimality Conditions for Systems with Distributed Parameters Based on the Dubovitskii–Milyutin Theorem with Incomplete Information About the Initial Conditions
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...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 276; no. 2; pp. 199 - 215 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Springer International Publishing
06.10.2023
Springer Springer Nature B.V |
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| Abstract | 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. |
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| AbstractList | 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. 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. Keywords and phrases: optimal control problem, Neumann problem, second-order parabolic operator, Dubovitskii--Milyutin theorem, conical approximations, optimality conditions. AMS Subject Classification: 37N10, 76N15, 76U05 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. |
| Audience | Academic |
| Author | Bahaa, G. M. |
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| References | Bahaa (CR1) 2003; 20 Bahaa (CR5) 2012; 3 CR17 Walczak (CR28) 1984; 42 CR16 CR12 Bahaa (CR3) 2007; 24 CR10 Bahaa, Kotarski (CR11) 2008; 25 Lions, Magenes (CR25) 1972 Bahaa (CR2) 2005; 22 Girsanov (CR20) 1972 Bahaa, Tharwat (CR14) 2012; 33 Bahaa (CR4) 2008; 25 Bahaa (CR6) 2012; 2012 Bahaa (CR7) 2016; 33 CR9 Bahaa, Tharwat (CR13) 2011; 21 CR24 Wang (CR29) 1975; 13 Bahaa (CR8) 2016; 30 Tröltzsch (CR26) 1984 CR21 Walczak (CR27) 1984; 1 Ya, Dubovitskii and A. A. Milyutin (CR18) 1963; 149 Kowalewski, Kotarski (CR23) 1981; 7 Bahaa, Tharwat (CR15) 2012; A3 Ya, Dubovitskii and A. A. Milyutin (CR19) 1965; 5 Kotarski, El-Saify, Bahaa (CR22) 2002; 19 GM Bahaa (6735_CR3) 2007; 24 GM Bahaa (6735_CR11) 2008; 25 6735_CR17 6735_CR16 F Tröltzsch (6735_CR26) 1984 A Ya (6735_CR18) 1963; 149 GM Bahaa (6735_CR15) 2012; A3 A Kowalewski (6735_CR23) 1981; 7 IV Girsanov (6735_CR20) 1972 GM Bahaa (6735_CR6) 2012; 2012 6735_CR10 6735_CR12 6735_CR9 GM Bahaa (6735_CR13) 2011; 21 PKC Wang (6735_CR29) 1975; 13 GM Bahaa (6735_CR8) 2016; 30 A Ya (6735_CR19) 1965; 5 S Walczak (6735_CR28) 1984; 42 JL Lions (6735_CR25) 1972 S Walczak (6735_CR27) 1984; 1 GM Bahaa (6735_CR4) 2008; 25 6735_CR21 W Kotarski (6735_CR22) 2002; 19 6735_CR24 GM Bahaa (6735_CR14) 2012; 33 GM Bahaa (6735_CR1) 2003; 20 GM Bahaa (6735_CR5) 2012; 3 GM Bahaa (6735_CR7) 2016; 33 GM Bahaa (6735_CR2) 2005; 22 |
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| SubjectTerms | Boundary conditions Differential equations Heating Initial conditions Mathematics Mathematics and Statistics Neumann problem Optimal control Optimization Partial differential equations Theorems |
| Title | Optimality Conditions for Systems with Distributed Parameters Based on the Dubovitskii–Milyutin Theorem with Incomplete Information About the Initial Conditions |
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