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Optimal control of quasilinear parabolic equations. (English) Zbl 0833.49002

The paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form. At first the existence and uniqueness of the solution for the state equation is proved, as well as the continuous dependence of the solution on the data. Then the optimal control problems with the Lagrangian type cost functional are considered, and the existence of optimal controls under some differentiability assumptions is proved. Sensitivity analysis of the state equation with respect to control is developed, and it is shown that under some assumptions the relation between the control and the state is Gâteaux- differentiable. Then the first-order optimality conditions are derived, and finally these conditions are generalized for some problems without the differentiability property.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
35K55 Nonlinear parabolic equations
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