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Optimization of distributed hyperbolic systems with multiple time delays given in the integral form

Kowalewski A.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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2010

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T. 14, z. 1

31-49

EN

In this paper optimal control problems for hyperbolic systems in which different multiple time delays appear in the integral form both in the state equations and in the boundary conditions are considered. Necessary and sufficient conditions of optimality for the Neumann problem are derived. The optimal boundary control is obtained in the feedback form. Making use of the results of Schwartz, the representation of the optimal feedback boundary control is given.

PL

W pracy rozważono problemy sterowania optymalnego liniowymi systemami hiperbolicznymi z różnymi wielokrotnymi całkowymi opóźnieniami czasowymi występującymi zarówno w równaniach stanu oraz w warunkach brzegowych. Udowodniono warunki wystarczające na istnienie jednoznacznych rozwiązań dla tego rodzaju równań hiperbolicznych z warunkami brzegowymi typu Neumanna z wielokrotnymi całkowymi opóźnieniami czasowymi. Założono ustalony czas sterowania. Zdefiniowano kwadratowe wskaźniki jakości. Nałożono ograniczenia na sterowanie rozłożone i brzegowe. Wyprowadzono warunki konieczne i wystarczające optymalności przy kwadratowych wskaźnikach jakości i ograniczeniach na sterowanie dla problemu brzegowego Neumanna. Uzyskano postać optymalnego sterowania brzegowego ze sprzężeniem zwrotnym.

2

Time-optimal control of infinite order hyperbolic systems with time delays

Kowalewski A.

International Journal of Applied Mathematics and Computer Science

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2009

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Vol. 19, no 4

597-608

EN

In this paper, the time-optimal control problem for infinite order hyperbolic systems in which time delays appear in the integral form both in state equations and in boundary conditions is considered. Optimal controls are characterized in terms of an adjoint system and shown to be unique and bang-bang. These results extend to certain cases of nonlinear control problems. The particular properties of optimal control are discussed.

3

A problem for the hyperbolic system of differential equation without initial conditions

Zaręba L.

Demonstratio Mathematica

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2005

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Vol. 38, nr 2

387--400

EN

In the present paper we consider a problem for the system of hyperbolic equations of the first order with periodic conditions and without initial conditions. Some conditions for uniqueness and existence of solution for this problem are given.

4

Non-Smoothness in the Asymptotics of Thin Shells and Propagation of Singularities. Hyperbolic Case

Karamian P., Sanchez-Hubert J, Sanchez Palencia E.

International Journal of Applied Mathematics and Computer Science

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2002

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Vol. 12, no 1

81-90

EN

We consider the limit behaviour of elastic shells when the relative thickness tends to zero. We address the case when the middle surface has principal curvatures of opposite signs and the boundary conditions ensure the geometrical rigidity. The limit problem is hyperbolic, but enjoys peculiarities which imply singularities of unusual intensity. We study these singularities and their propagation for several cases of loading, giving a somewhat complete description of the solution.

5

Thermomechanical coupled waves in a viscoplastic medium

Frischmuth K., Kosiński W.

Archives of Mechanics

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2002

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Vol. 54, nr 5-6

439-457

EN

Ideas of Perzyna's theory of material systems in the cases of viscoplastic materials and heat conductors are elaborated. Both theories are based on the notion of a method of preparation. They are combined into one coupled model for an elasto-viscoplastic heat conductor. We study its behavior by numerically calculating solutions for plane waves passing through a body.