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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)U,(t,x)Q T =(0,T)×Ω the following optimal control problem is investigated:

f(t,x,y(t,x),u(t,x))y(t,x) t-y(t,x)+G(y)(t,x)+β(y(t,x)),a.e.(t,x)Q T ,
y(t,x)=φ(t,x),(t,x)(-r,0)×Ω;y(0,x)=z(x),xΩ;y(t,x)=0,(t,x)(0,T)×Ω,
Q T f 0 (t,x,y(t,x),u(t,x))dtdxmin·

Here ΩR n is a given bounded region with C 2 boundary Ω; further

G(y)(t,x)= -r 0 y(t+θ,x)μ(dθ),= i=1 n 2 x i 2 ,β(y(t,x))=(-,0],y(t,x)=0,{0},y(t,x)>0·

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