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A Bliss-type multiplier rule for constrained variational problems with time delay. (English) Zbl 0903.49004

The optimization problem for delay systems:

minJ(y)= t 1 t 2 f(t,y(t),y ' (t),y(t-τ),y ' (t-τ))dt

such that

ϕ β (t,y(t),y ' (t),y(t-τ),y ' (t-τ))=0,β=1,,m<n,ϕ μ (t 1 ,y(t 1 ),t 2 ,y(t 2 ))=0,μ=1,,p2n,y(t)=α(t),t[t 1 -τ,t 1 ],

is considered and new necessary conditions are obtained in the form of a multiplier rule: a linear combination of f and ϕ 1 ,,ϕ m satisfy generalized Euler equations and appropriate boundary conditions. It is shown that every minimizing arc of the above problem satisfies a multiplier rule. The example

minJ(y)=1 2 0 2 y 2 2 (t)dt

such that

y 1 ' (t)+y 1 (t-1)-y 2 (t)=0,y 1 (t)=1,t[-1,0],

is solved.

MSC:
49J25Optimal control problems with equations with ret. arguments (exist.) (MSC2000)