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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 $u(t,x)\in U, (t,x)\in Q_{T}=(0,T)\times \Omega$ the following optimal control problem is investigated: $$ f(t,x,y(t,x),u(t,x))\in \frac{\partial y(t,x)}{\partial t}-\triangle y(t,x)+G(\triangle y)(t,x)+\beta(y(t,x)),\text{ a.e. }(t,x)\in Q_{T}, $$ $$ \cases y(t,x)=\varphi(t,x), &(t,x)\in (-r,0)\times \Omega;\ y(0,x)=z(x),\ x\in \Omega;\\ y(t,x)=0, &(t,x)\in (0,T)\times \partial \Omega, \endcases$$ $$ \int_{Q_{T}}f^{0}(t,x,y(t,x),u(t,x))\,dt\, dx\rightarrow\min. $$ Here $\Omega\subset R^{n}$ is a given bounded region with $C^{2}$ boundary $\partial\Omega;$ further $$\align G(\triangle y)(t,x)&=\int_{-r}^{0}\triangle y(t+\theta,x)\mu(d\theta),\quad \triangle=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}},\\ \beta(y(t,x))&= \cases (-\infty,0],&y(t,x)=0,\\ \{0\},&y(t,x)>0. \endcases \endalign$$ The existence of optimal controls under a Cesar-type condition is proved, and the necessary conditions of Pontryagin type for optimal controls is derived.

49J40Variational methods including variational inequalities
49K25Optimal control problems with equations with ret.arguments (nec.) (MSC2000)
Full Text: DOI
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