Observability inequalities for parabolic equations over measurable sets and some applications related to the bang-bang property for control problems. (English) Zbl 1381.49006
Summary: This article presents two observability inequalities for the heat equation over \(\Omega \times (0,T)\). In the first one, the observation is from a subset of positive measure in \(\Omega \times (0,T)\), while in the second, the observation is from a subset of positive surface measure on \(\partial\Omega \times (0,T)\). We will provide some applications for the above-mentioned observability inequalities, the bang-bang property for the minimal time, time optimal and minimal norm control problems, and also establish new open problems related to observability inequalities and the aforementioned applications.
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
| 58E25 | Applications of variational problems to control theory |
| 93B05 | Controllability |
| 93B07 | Observability |
| 35K05 | Heat equation |
Keywords:
parabolic equations; control theory; controllability; observability inequalities; bang-bang propertiesReferences:
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