Extremal Problems for Hyperbolic Systems with Boundary Conditions Involving Integral Time Lags

eng Article in English DOI: 10.14313/PAR_251/23

send Adam Kowalewski AGH University of Science and Technology, Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Institute of Automatic Control and Robotics, Al. Mickiewicza 30, 30-059 Cracow

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Abstract

Extremal problems for integral time lag hyperbolic systems are presented. The optimal boundary control problems for hyperbolic systems in which integral time lags appear in the Neumann boundary conditions are solved. Such systems constitute, in a linear approximation, a universal mathematical model for many processes in which transmission signals at a certain distance with electric, hydraulic and other long lines take place. The time horizon is fixed. Making use of Dubovicki-Milyutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.

Keywords

boundary control, hyperbolic systems, integral time lags, Neumann boundary conditions

Problemy ekstremalne dla systemów hiperbolicznych z warunkami brzegowymi, w których występują całkowe opóźnienia czasowe

Streszczenie

Zaprezentowano ekstremalne problemy dla systemów hiperbolicznych z całkowymi opóźnieniami czasowymi. Rozwiązano problem optymalnego sterowania brzegowego dla systemów hiperbolicznych drugiego rzędu, w których całkowe opóźnienia czasowe występują w warunkach brzegowych typu Neumanna. Tego rodzaju systemy stanowią w liniowym przybliżeniu uniwersalny model matematyczny procesów fizycznych, w których ma miejsce przesyłanie sygnałów na odległość w liniach długich typu elektrycznego, hydraulicznego i innych. Korzystając ze schematu Dubowickiego-Milutina wyprowadzono warunki konieczne i wystarczające optymalności dla problemu liniowo-kwadratowego.

Słowa kluczowe

całkowe opóźnienia czasowe, sterowanie brzegowe, systemy hiperboliczne drugiego rzędu, warunki brzegowe typu Neumanna

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