scholarly journals On Modified Interval-Valued Variational Control Problems with First-Order PDE Constraints

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 472
Author(s):  
Savin Treanţă
Keyword(s):  
Partial Differential Equations ◽  
Control Problem ◽  
Optimality Conditions ◽  
Generalized Convexity ◽  
Inequality Constraints ◽  
Control Problems ◽  
First Order ◽  
First Order Pde ◽  
Convexity Assumptions ◽  
Interval Valued

In this paper, a modified interval-valued variational control problem involving first-order partial differential equations (PDEs) and inequality constraints is investigated. Specifically, under some generalized convexity assumptions, we formulate and prove LU-optimality conditions for the considered interval-valued variational control problem. In order to illustrate the main results and their effectiveness, an application is provided.

scholarly journals A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

1995 ◽  
Vol 36 (3) ◽  
pp. 261-273 ◽  
Author(s):  
Mohammad A. Kazemi
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Fréchet Derivative ◽  
Necessary Condition ◽  
Control Problems ◽  
Partial Differential ◽  
Frechet Derivative ◽  
First Order

AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

scholarly journals Necessary optimality conditions for minimax optimal control problems with mixed constraints

2021 ◽  
Author(s):  
Paola Patzi Aquino ◽  
Maria do Rosario de Pinho ◽  
Geraldo Nunes Silva
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Optimal Control Problems ◽  
Necessary Optimality Conditions ◽  
Inequality Constraints ◽  
Control Problems ◽  
Parameter Uncertainties ◽  
Minimax Optimal Control ◽  
Minimax Optimal Control Problems ◽  
Equality And Inequality Constraints

A weak maximal principle for minimax optimal control problems with mixed state-control equality and inequality constraints is provided. In the formulation of the minimax control problem we allow for parameter uncertainties in all functions involved: in the cost function, in the dynamical control system and in the equality and inequality constraints. Then a new constraint qualification of Mangassarian-Fromovitz type is introduced which allowed us to prove the necessary conditions of optimality. We also derived the optimality conditions under a full rank conditions type and showed that it is, as usual, a particular case of the Mangassarian-Fromovitz type condition case. Illustrative examples are presented.

OPTIMALITY CONDITIONS FOR COOPERATIVE PARABOLIC SYSTEMS GOVERNED BY SCHRÖDINGER OPERATORS WITH CONTROL CONSTRAINTS

2008 ◽  
Vol 01 (02) ◽  
pp. 131-146 ◽  
Author(s):  
G. M. Bahaa
Keyword(s):  
Control Problem ◽  
Optimality Conditions ◽  
Distributed Control ◽  
Performance Index ◽  
Integral Form ◽  
Parabolic Systems ◽  
Control Problems ◽  
Control Constraints ◽  
Distributed Control Problem ◽  
The Neumann Problem

A distributed control problem for cooperative parabolic systems governed by Schrödinger operator is considered. The performance index is more general than the quadratic one and has an integral form. Constraints on controls are imposed. Making use of the Dubovitskii-Milyutin Theorem given by Walczak (1984, On some control problems Acta Univ. Lod. Folia Math., 1, 187-196), the optimality conditions are derived for the Neumann problem. Finally, several mathematical examples for derived optimality conditions are presented.

scholarly journals Mathematical analysis of nonlocal PDEs for network generation

2019 ◽  
Vol 14 (5) ◽  
pp. 506
Author(s):  
Tobias Böhle ◽  
Christian Kuehn
Keyword(s):  
Partial Differential Equations ◽  
Numerical Simulations ◽  
Mathematical Analysis ◽  
Degree Distribution ◽  
Network Science ◽  
Steady States ◽  
Network Generation ◽  
First Order ◽  
First Order Pde ◽  
Implicit Ode

In this paper, we study a certain class of nonlocal partial differential equations (PDEs). The equations arise from a key problem in network science, i.e., network generation from local interaction rules, which result in a change of the degree distribution as time progresses. The evolution of the generating function of this degree distribution can be described by a nonlocal PDE. To address this equation we will rigorously convert it into a local first order PDE. Then, we use theory of characteristics to prove solvability and regularity of the solution. Next, we investigate the existence of steady states of the PDE. We show that this problem reduces to an implicit ODE, which we subsequently analyze. Finally, we perform numerical simulations, which show stability of the steady states.

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