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Nonlinear optimal control problems for parabolic equations. (English) Zbl 0551.49003

The author considers the following optimal control problem: the state y(x,t) is solution of a linear parabolic heat equation; the control variable is the diffusion coefficient k(x); we consider the set of admissible controls: \[ K=\{k(x)\quad measurable;\quad \alpha \leq k(x)\leq \beta;\quad \int^{1}_{0}k(x)dx=\gamma;\quad k\quad increa\sin g\}; \] (we assume \(\alpha <\gamma <\beta)\); the functional is given by: \(J(k)=\int^{1}_{0}| y(x,T)|^ 2dx.\)
Then the author proves that the unique solution \(k_ 0\) of the problem is given by: \(k_ 0(x)=\alpha\), if \(0\leq x\leq \theta\), and \(k_ 0(x)=\beta\), if \(\theta <x\leq 1\) with \(\theta\in [0,1]\) such that \(\theta \alpha +(1-\theta)\beta =\gamma\). The same characterization is proved when the functional is given by: \(J_ h(k)=\int^{1}_{0}h(x)y_ k(x,T)dx\) (where h is assumed to verify special hypothesis).
In a second part, the author applies a similar technique to different problems, in particular for the problem: \(y_ t=a(x)y_{xx}+b(x)y_ x- k(x)y\) plus boundary and initial conditions, where the control variable k(x) is belonging to \(K_*=\{K(x)\) monotone, increasing, \(0\leq k(x)\leq M\); \(\int^{1}_{0}k(x)dx=0\}\) (we assume \(M>0)\).
Reviewer: Chr.Saguez

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
35K05 Heat equation
93C20 Control/observation systems governed by partial differential equations
35B37 PDE in connection with control problems (MSC2000)
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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