Geometric and analytic views in existence theorems for optimal control. II. Distributed and boundary controls

The paper is devoted to the regularization of the classical optimality conditions (COC) — the Lagrange principle and the Pontryagin maximum principle in a convex optimal control problem for a parabolic equation with an operator (pointwise state) equality-constraint at the final time. The problem contains distributed, initial and boundary controls, and the set of its admissible controls is not assumed to be bounded. In the case of a specific form of the quadratic quality functional, it is natural to interpret the problem as the inverse problem of the final observation to find the perturbing effect that caused this observation. The main purpose of regularized COCs is stable generation of minimizing approximate solutions (MAS) in the sense of J. Warga. Regularized COCs are: 1) formulated as existence theorems of the MASs in the original problem with a simultaneous constructive representation of specific MASs; 2) expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions; 3) are sequential generalizations of the COCs and retain the general structure of the latter; 4) “overcome” the ill-posedness of the COCs, are regularizing algorithms for solving optimization problems, and form the theoretical basis for the stable solving modern meaningful ill-posed optimization and inverse problems.

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On an optimal control problem involving first order hyperbolic systems with boundary controls

Numerical Functional Analysis and Optimization ◽

10.1080/01630568208816111 ◽

1982 ◽

Vol 4 (2) ◽

pp. 171-190 ◽

Cited By ~ 9

Author(s):

K. G. Choo ◽

K. L. Teo ◽

Z. S. Wu

Keyword(s):

Optimal Control ◽

Optimal Control Problem ◽

Control Problem ◽

Hyperbolic Systems ◽

First Order ◽

Boundary Controls

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On an optimal control problem involving second order hyperbolic systems with boundary controls

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700011564 ◽

1983 ◽

Vol 27 (1) ◽

pp. 139-148 ◽

Cited By ~ 2

Author(s):

K.G. Choo ◽

K.L. Teo ◽

Z.S. Wu

Keyword(s):

Optimal Control ◽

Optimal Control Problem ◽

Control Problem ◽

Hyperbolic Systems ◽

Sufficient Conditions ◽

Second Order ◽

Optimal Controls ◽

Convergence Properties ◽

Boundary Controls ◽

Necessary And Sufficient

In this paper, we consider an optimal control problem involving second-order hyperbolic systems with boundary controls. Necessary and sufficient conditions are derived and a result on the existence of optimal controls is obtained. Also, a computational algorithm which generated minimizing sequences of controls is devised and the convergence properties of the algorithm are investigated.

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Generalized control with compact support for systems with distributed parameters

Archives of Control Sciences ◽

10.1515/acsc-2015-0001 ◽

2015 ◽

Vol 25 (1) ◽

pp. 5-20 ◽

Cited By ~ 17

Author(s):

Asatur ZH. Khurshudyan

Keyword(s):

Compact Support ◽

Exact Controllability ◽

Time Interval ◽

State Function ◽

Infinite Dimensional ◽

Distributed Parameters ◽

Boundary Controls ◽

Systems With Distributed Parameters ◽

Distributed And Boundary Controls ◽

Controllability Conditions

Abstract We propose a generalization of the Butkovskiy's method of control with compact support [1] allowing to derive exact controllability conditions and construct explicit solutions in control problems for systems with distributed parameters. The idea is the introduction of a new state function which is supported in considered bounded time interval and coincides with the original one therein. By means of techniques of the distributions theory the problem is reduced to an interpolation problem for Fourier image of unknown function or to corresponding system of integral equalities. Treating it as infinite dimensional problem of moments, its L1, L2 and L∞-optimal solutions are constructed explicitly. The technique is explained for semilinear wave equation with distributed and boundary controls. Particular cases are discussed.

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