## 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
Full Text:

### References:

 [1] I. Ekeland and R. Temam, Analyse Convexe et problèmes variationnels. Dunod, Gauthier Villars, Paris-Bruxeles-Montréal, 1974. [2] R. Glowinski and J.L. Lions, to appear inActa Numerica, 1993. [3] J.L. Lions, Equations Differentielles Opérationnelles et Problèmes aux limites. Springer-Verlag, Berlin-New York, 1961. [4] J.L. Lions, Exact controllability for distributed systems. Some trends and some problems.Applied and Industrial Mathematics, R. Spigler (ed.), 59–84,Kluwer, 1991. · Zbl 0735.93006 [5] –, Controle Optimal de systèmes gouvernés par les équations aux dérivées partielles. Dunod, Gauthier-Villars, Paris, 1968. [6] T.R. Rockafellar, Duality and stability in extremum problems involving convex functions.Pac. J. Math. 21 (1967), 167–187. · Zbl 0154.44902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.