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Measures of controllability. (English) Zbl 0812.93046

The author considers the control system \[ {\partial y\over \partial t}+ Ay= v(x, t) \tag{1} \] on an open set of \(\mathbb{R}^ n\), where \(A\) is a second-order elliptic operator, with initial condition \(y_ 0\) and zero boundary condition. Approximate controllability means that for each \(T> 0\) and \(y_ 1\in L^ 2\), the solution \(y(T, v)\) can be driven in an arbitrarily small neighborhood of \(y_ 1\).
The author defines and characterizes the measure of controllability of (1), namely the supremum (for \(y_ 0\), \(y_ 1\) ranging on fixed balls of \(L^ 2\)) of the infimum of \({1\over 2} \iint v^ 2 dx dt\) taken over all the control functions \(v\) such tht \(y(T, v)\) is in a fixed neighborhood of \(y_ 1\). The measure of controllability is seen as a function of \(A\).

MSC:

93C20 Control/observation systems governed by partial differential equations
49J20 Existence theories for optimal control problems involving partial differential equations
93B05 Controllability
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References:

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