Extremal Problems for Second Order Hyperbolic Systems with Time-Varying Delays

eng Article in English DOI: 10.14313/PAR_259/5

send Adam Kowalewski AGH University of Krakow, Institute of Automatic Control and Robotics, 30-059 Krakow, al. Mickiewicza 30,

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Abstract

Extremal problems for time-varying delay hyperbolic systems are presented. The optimal boundary control problems for hyperbolic systems in which time-varying delays appear both in the state equations and in the Neumann boundary conditions are solved. The time horizon is fixed. Making use of Dubovicki–Milutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.

Keywords

boundary control, electric long line, optimal control, second order hyperbolic system, time-varying delays

Problemy ekstremalne dla systemów hiperbolicznych drugiego rzędu ze zmiennymi opóźnieniami czasowymi

Streszczenie

Zaprezentowano ekstremalne problemy dla systemów hiperbolicznych ze zmiennymi opóźnieniami czasowymi. Rozwiązano problem optymalnego sterowania brzegowego dla systemów hiperbolicznych drugiego rzędu, w których zmienne opóźnienia czasowe występują zarówno w równaniach stanu oraz w warunkach brzegowych typu Neumanna. Tego rodzaju równania stanowią w liniowym przybliżeniu uniwersalny model matematyczny dla procesów fizycznych, w których ma miejsce przesyłanie sygnałów na odległość, takich jak linie długie typu elektrycznego, hydraulicznego i innych. Korzystając z metody Dubowickiego-Milutina wyprowadzono warunki konieczne i wystarczające optymalności dla problemu liniowo-kwadratowego.

Słowa kluczowe

sterowanie brzegowe, systemy hiperboliczne drugiego rzędu, zmienne opóźnienie

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