Abstract
In this paper, the time and norm optimal control problems of controlled heat equations with a weight function are considered. For the time optimal problems, we study the following two cases: one is for equations with multi-domain control under null controllability, and the other is for equations under approximate null controllability. We prove the solvability, and obtain the bang-bang principle of the time optimal controls for aforementioned both cases. For the norm optimal control problems, we focus on equations with multi-time and multi-domain control, and present the solvability of these problems.
Highlights
Let T be a positive number and be an open bounded domain with smooth boundary in RN, N ≥
In Section, we consider two kind time optimal control problems: one is for equations with multi-domain control under null controllability, and the other is for equations under approximate null controllability
For any given partition K, by standard minimizing sequence method, there exists a solution to the following norm optimal control problem: K
Summary
Let T be a positive number and be an open bounded domain with smooth boundary in RN , N ≥. Consider the following controlled heat equation with a weight function:. We shall consider the time and norm optimal control problems of heat equations with a weight function. K }, i.e., the time optimal controls sequence of Problem (TP ) has the bang-bang property. Yn ≡ y(·, ·; {χωi uni }, y ) is a solution to the following equation:. We shall show that the time optimal control of Problem (TP ). K} and a subset E ⊂ [α, T∗ – α] with positive measure for some α > and a positive number ε , such that u∗i ∈ Uai d and Mi – u∗i L ( ) ≥ ε , for each t in the set E , where u∗i is the time optimal control respect to T∗.
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