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Zbl 1117.49015
Zhu, Shangwei
Optimal control of variational inequalities with delays in the highest order spatial derivatives.
(English)
[J] Acta Math. Sin., Engl. Ser. 22, No. 2, 607-624 (2006). ISSN 1439-8516; ISSN 1439-7617/e

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.
MSC 2000:
*49J40 Variational methods including variational inequalities
49K25 Optimal control problems with equations with ret.arguments (nec.)

Keywords: Variation inequalities; Delay; Optimal control; Maximum principle; Existence of optimal control

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