VOLTERRA FUNCTIONAL-OPERATOR EQUATIONS AND DISTRIBUTED OPTIMIZATION PROBLEMS

2018 ◽  
pp. 745-756
Author(s):  
Vladimir Iosifovich Sumin
Keyword(s):  
Optimization Problems ◽  
Initial Boundary ◽  
Optimization Methods ◽  
Global Solvability ◽  
Optimal Controls ◽  
Initial Boundary Value Problems ◽  
Operator Equations ◽  
Functional Operator ◽  
Controlled Systems ◽  
Initial Boundary Value

A survey of the results obtained in the theory of optimization of distributed systems by the method of Volterra functional-operator equations is given. Topics are considered: the conditions for preserving the global solvability of controllable initial-boundary value problems, optimality conditions, singular controlled systems in the sense of J.L. Lions, singular optimal controls, numerical optimization methods substantiation and others.

A class of densely invertible parabolic operator equations

1969 ◽  
Vol 1 (3) ◽  
pp. 363-374 ◽  
Author(s):  
R.S. Anderssen
Keyword(s):  
Variational Methods ◽  
Variational Formulation ◽  
Lagrange Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Parabolic Differential Equations ◽  
Operator Equations ◽  
Initial Boundary Value

Before variational methods can be applied to the solution of an initial boundary value problem for a parabolic differential equation, it is first necessary to derive an appropriate variational formulation for the problem. The required solution is then the function which minimises this variational formulation, and can be constructed using variational methods. Formulations for K-p.d. operators have been given by Petryshyn. Here, we show that a wide class of initial boundary value problems for parabolic differential equations can be related to operators which are densely invertible, and hence, K-p.d.; and develop a method which can be used to prove dense invertibility for an even wider class. In this way, the result of Adler on the non-existence of a functional for which the Euler-Lagrange equation is the simple parabolic is circumvented.

scholarly journals Hyperbolic differential-operator equations on a whole axis

2004 ◽  
Vol 2004 (2) ◽  
pp. 99-113 ◽  
Author(s):  
Yakov Yakubov
Keyword(s):  
Differential Operator ◽  
Abstract Interpretation ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Operator Equations ◽  
Stable Solutions ◽  
Initial Boundary Value ◽  
Hyperbolic Differential Equations

We give an abstract interpretation of initial boundary value problems for hyperbolic equations such that a part of initial boundary value conditions contains also a differentiation on the timetof the same order as equations. The case of stable solutions of abstract hyperbolic equations is treated. Then we show applications of obtained abstract results to hyperbolic differential equations which, in particular, may represent the longitudinal displacements of an inhomogeneous rod under the action of forces at the two ends which are proportional to the acceleration.

scholarly journals An Introduction to Optimal Control

2014 ◽  
Vol 66 (2) ◽  
Author(s):  
Carlo Cossu
Keyword(s):  
Optimization Problems ◽  
Adjoint Equation ◽  
Gradient Methods ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Constrained Optimization Problems ◽  
Variational Techniques ◽  
Finite Dimensional ◽  
Full State ◽  
Initial Boundary Value

The goal of these lecture notes is to provide an informal introduction to the use of variational techniques for solving constrained optimization problems with equality constraints and full state information. The use of the Lagrangian augmented cost function and variational techniques by which the adjoint equation and the optimality condition are found are introduced by the use of examples starting from steady finite-dimensional problems to end with unsteady initial-boundary value problems. Gradient methods based on sensitivity and adjoint equation solutions are also mentioned.

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