We study one class of nonlinear fluid dynamic models with controls in the initial condition and the source term. The model is described by a nonlinear inhomogeneous hyperbolic conservation law with state and control constraints. We consider the case when the greatest lower bound of the cost functional can be unattainable on the set Ξ of admissible pairs or the set Ξ is possibly empty. Using the methods of vector-valued optimization theory, we show that this optimal control problem admits the existence of the so-called weakened approximate solution which can be interpreted as generalized solution to some vector optimization problem of special form.
2000 Mathematics Subject Classification
46B40 49J45 90C29 49N90 76N15
Key words and phrases
Nonlinear conservation laws control and state constraints vector optimization problem weakened approximate solutions entropy solutions
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