Friedman, Avner Nonlinear optimal control problems for parabolic equations. (English) Zbl 0551.49003 SIAM J. Control Optimization 22, 805-816 (1984). 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 Cited in 8 Documents 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.) Keywords:nonlinear parabolic control problem; control in the diffusion coefficient PDFBibTeX XMLCite \textit{A. Friedman}, SIAM J. Control Optim. 22, 805--816 (1984; Zbl 0551.49003) Full Text: DOI