Optimal control for quasilinear retarded parabolic systems (Q2725330)

Fourier's law for heat flux \({\mathbf q}(t, x) = -k \nabla y(t, x)\) leads to the classical heat equation \(c y_t(t, x) = k\Delta y(t, x)\) which is nonphysical in the sense of having infinite propagation of speed for finite heat pulses. Fourier's law was amended by \textit{M. E. Gurtin} and \textit{A. C. Pipkin} [Arch. Ration. Mech. Anal. 31, 113-126 (1968; Zbl 0164.12901)] assuming memory effects; this leads to the finite speed heat propagation equation NEWLINE\[NEWLINE cy_t(t, x) = \int_0^\infty \Delta y(t - s, x) \mu(ds) NEWLINE\]NEWLINE where \(\mu\) is a Borel measure. Taking this example as motivation, the authors set up a theory of abstract control systems of the form NEWLINE\[NEWLINE \begin{aligned} y'(t)&= Ay(t) + F(t, y_t, u(t)), \qquad 0 \leq t \leq T,\\ y(0)&= y_0, \quad y(t) = \varphi(t), \qquad -r \leq t < 0,\end{aligned}NEWLINE\]NEWLINE where \(u(t)\) is a control function and \(y_t\) is the past of the state \(y(t)\) up to the present time \(t\). The cost functional also involves \(y_t.\) The main result is a form of Pontryagin's maximum principle that fits this optimal control problem. As for comparison with existing results, the authors point out that in most of those results the delay terms involve only lower order space derivatives, while theirs include higher order derivatives.