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The cost of approximate controllability for heat equations: The linear case. (English) Zbl 1007.93034

Let us consider the system

y t -y+ay=v1 𝒪 inQ=Ω×(0,T),y=0onΩ×(0,T),y(x,0)=y 0 (x)inΩ,

where Ω is a bounded domain of d with C 2 -boundary Ω, 𝒪 is a (probably small) non-empty open subset of Ω, 1 𝒪 denotes the characteristic function of 𝒪, y 0 L 2 (Ω) is fixed and a(x,t) is a function in L (Q). Hence, the control v(x,t) is acting only on q=𝒪×(0,T). It is well known that this system is approximately controllable at any fixed time T>0 by taking controls vL 2 (q) in the sense that, for each ε>0 and y 1 L 2 (Ω), there exists vL 2 (q) such that the corresponding solution y v of the system satisfies y v (T)-y 1 L 2 (Ω) <ε. Of course, there exist infinite controls v satisfying this property. The objective of the paper is to obtain explicit bounds on the cost of approximate controllability for the system, i.e. the infimum of v L 2 (q) over all v satisfying the previous property. It is proved that this cost is of order exp(C/ε). The simultaneous finite-approximate controllability is also investigated. The proofs combine global Carleman estimates, energy estimates for parabolic equations and the variational approach to approximate controllability. When the coefficient a is constant, a different approach is used to show that the cost is of order exp(C/ε) and that this estimate is sharp.

MSC:
93C20Control systems governed by PDE
93B05Controllability
35K05Heat equation