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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: $$\min J(y)= \int^{t_2}_{t_1} f(t, y(t), y'(t), y(t-\tau), y'(t-\tau)) dt$$ such that \align \phi_\beta(t, y(t), y'(t), y(t-\tau), y'(t-\tau)) & = 0,\ \beta= 1,\dots, m< n,\\ \phi_\mu(t_1, y(t_1), t_2,y(t_2)) & = 0,\ \mu= 1,\dots, p\le 2n,\\ y(t) & = \alpha(t),\ t\in [t_1-\tau, t_1],\endalign is considered and new necessary conditions are obtained in the form of a multiplier rule: a linear combination of $f$ and $\phi_1,\dots,\phi_m$ 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 $$\min J(y)= {1\over 2} \int^2_0 y^2_2(t)dt$$ such that \align y_1'(t)+ y_1(t- 1)- y_2(t) & = 0,\\ y_1(t) & = 1,\ t\in [-1,0],\endalign is solved.

##### MSC:
 49J25 Optimal control problems with equations with ret. arguments (exist.) (MSC2000)
##### Keywords:
optimization problem; delay systems; multiplier rule
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