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1

Second-order characterization of convex functions and its applications

Nadi Mohammad Taghi, Yao Jen Chih, Zafarani Jafar

Journal of Applied Analysis

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2019

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Vol. 25, nr 1

49--58

EN

Some developments of the second-order characterizations of convex functions are investigated by using the coderivative of the subdifferential mapping. Furthermore, some applications of the second-order subdifferentials in optimization problems are studied.

2

On the multiobjective control problem

Kumar P., Sharma B.

Journal of Applied Analysis

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2018

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Vol. 24, nr 2

223--231

EN

In this paper, sufficient optimality conditions are established for the multiobjective control problem using efficiency of higher order as a criterion for optimality. The ρ-type 1 invex functionals (taken in pair) of higher order are proposed for the continuous case. Existence of such functionals is confirmed by a numer of examples. It is shown with the help of an example that this class is more general than the existing class of functionals.Weak and strong duality theorems are also derived for a mixed dual in order to relate efficient solutions of higher order for primal and dual problems.

3

Nondifferentiable (Phi,rho)-type I and generalized (Phi,rho)-type I functions in nonsmooth vector optimization

Antczak T.

Journal of Applied Analysis

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2013

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Vol. 19, nr 2

247--270

EN

In this paper, new classes of nondifferentiable generalized invex functions are introduced. Further, nonsmooth vector optimization problems with functions belonging to the introduced classes of (generalized) (Phi,rho)-type I functions are considered. Sufficient optimality conditions and duality results for such classes of nonsmooth vector optimization problems are established. It turns out that the presented results are proved also for such nonconvex vector optimization problems in which not all functions constituting them possess the fundamental property of invexity.

4

Sterowanie optymalne systemami o parametrach rozłożonych z opóźnieniami

Kowalewski A.

Pomiary Automatyka Robotyka

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2011

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R. 15, nr 12

57-59

PL

W artykule dokonano przeglądu problematyki sterowania optymalnego systemami o parametrach rozłożonych z opóźnieniami w warunkach brzegowych. Zaprezentowano podstawowe zagadnienia dotyczące systemów o parametrach rozłożonych takie jak: istnienie i jednoznaczność rozwiązań, sterowanie optymalne rozłożone i brzegowe, sterowanie optymalne ze sprzężeniem zwrotnym, sterowanie czasowo-optymalne, sterowalność oraz modele matematyczne systemów o parametrach rozłożonych.

EN

In this article the review of optimal control problems for distributed parameter systems with boundary conditions involving time delays is performed. The principal problems, namely: existence and uniqueness of solutions, distributed and boundary optimal control, optimal feedback control, controllability and mathematical models of distributed parameter systems are presented.

5

Optimality properties of controls with bang-bang components in problems with semilinear state equation

Felgenhauer U.

Control and Cybernetics

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2005

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Vol. 34, no 3

763-785

EN

In this paper we study optimal control problems with bang-bang solution behavior for a special class of semilinear dynamics. Generalizing a former result for linear systems, optimlity conditions are derived by a duality based approach. The results apply for scalar as well as for vector control functions and, in particular, for the case of the so-called multiple switches, too. Further, an iterative procedure for determining switching points is proposed, and convergence results are provided.

6

Optimality and sensitivity for semilinear bang-bang type optimal control problems

Felgenhauer U.

International Journal of Applied Mathematics and Computer Science

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2004

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Vol. 14, no 4

447-454

EN

In optimal control problems with quadratic terminal cost functionals and systems dynamics linear with respect to control, the solution often has a bang-bang character. Our aim is to investigate structural solution stability when the problem data are subject to perturbations. Throughout the paper, we assume that the problem has a (possibly local) optimum such that the control is piecewise constant and almost everywhere takes extremal values. The points of discontinuity are the switching points. In particular, we will exclude the so-called singular control arcs, see Assumptions 1 and 2, Section 2. It is known from the results by Agrachev et al. (2002) stating that regularity assumptions, together with a certain strict second-order condition for the optimization problem formulated in switching points, are sufficient for strong local optimality of a state-control solution pair. This finite-dimensional problem is analyzed in Section 3 and optimality conditions are formulated (Lemma 2). Using well-known results concerning solution sensitivity for mathematical programs in Rn (Fiacco, 1983) one may further conclude that, under parameter changes in the problem data, the switching points will change Lipschitz continuously. The last section completes these qualitative statements by calculating sensitivity differentials (Theorem 2, Lemma 6). The method requires a simultaneous solution of certain linearized multipoint boundary value problems.

7

Boundary optimal control problem for the phase-field transition system using fractional steps method

Moroşanu C.

Control and Cybernetics

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2003

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Vol. 32, no 1

5-32

EN

In this paper we prove the convergence of an iterative scheme of fractional steps type for boundary optimal control problem which is governed by the phase-field transition system. The existence of an optimal control and necessary optimality conditions are given for approximating problem. A gradient type algorithm and numerical implementation of this algorithm are discussed.

PL

W artykule dowodzi się zbieżności procedury iteracyjnej typu kroków ułamkowych dla zagadnienia sterowania optymalnego brzegiem w układzie pola ze zmianą fazy. Wykazano istnienie sterowania optymalnego i podano warunki konieczne optymalności dla zagadnienia aproksymującego. Rozpatrzono odpowiedni algorytm gradientowy oraz realizację numeryczną tego algorytmu.