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Optimal control of variational inequalities with delays in the highest order spatial derivatives. (English) Zbl 1117.49015

In a class of the measurable control functions $u\left(t,x\right)\in U,\left(t,x\right)\in {Q}_{T}=\left(0,T\right)×{\Omega }$ the following optimal control problem is investigated:

$f\left(t,x,y\left(t,x\right),u\left(t,x\right)\right)\in \frac{\partial y\left(t,x\right)}{\partial t}-▵y\left(t,x\right)+G\left(▵y\right)\left(t,x\right)+\beta \left(y\left(t,x\right)\right),\phantom{\rule{4.pt}{0ex}}\text{a.e.}\phantom{\rule{4.pt}{0ex}}\left(t,x\right)\in {Q}_{T},$
$\left\{\begin{array}{cc}y\left(t,x\right)=\varphi \left(t,x\right),\hfill & \left(t,x\right)\in \left(-r,0\right)×{\Omega };\phantom{\rule{4pt}{0ex}}y\left(0,x\right)=z\left(x\right),\phantom{\rule{4pt}{0ex}}x\in {\Omega };\hfill \\ y\left(t,x\right)=0,\hfill & \left(t,x\right)\in \left(0,T\right)×\partial {\Omega },\hfill \end{array}\right\$
${\int }_{{Q}_{T}}{f}^{0}\left(t,x,y\left(t,x\right),u\left(t,x\right)\right)\phantom{\rule{0.166667em}{0ex}}dt\phantom{\rule{0.166667em}{0ex}}dx\to min·$

Here ${\Omega }\subset {R}^{n}$ is a given bounded region with ${C}^{2}$ boundary $\partial {\Omega };$ further

$\begin{array}{cc}\hfill G\left(▵y\right)\left(t,x\right)& ={\int }_{-r}^{0}▵y\left(t+\theta ,x\right)\mu \left(d\theta \right),\phantom{\rule{1.em}{0ex}}▵=\sum _{i=1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},\hfill \\ \hfill \beta \left(y\left(t,x\right)\right)& =\left\{\begin{array}{cc}\left(-\infty ,0\right],\hfill & y\left(t,x\right)=0,\hfill \\ \left\{0\right\},\hfill & y\left(t,x\right)>0·\hfill \end{array}\right\\hfill \end{array}$

The existence of optimal controls under a Cesar-type condition is proved, and the necessary conditions of Pontryagin type for optimal controls is derived.

##### MSC:
 49J40 Variational methods including variational inequalities 49K25 Optimal control problems with equations with ret.arguments (nec.) (MSC2000)
##### References:
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