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An existence theorem for a class of optimal problems with delayed argument. (English) Zbl 0974.49004
The authors consider the problem $$\dot x(t)=f(t,x,x(\tau(t)),u(t),u(\theta(t)))$$, $$t\in [t_0,t_1]\subset J$$, $$u(.)\in \Omega_1,$$ $$x(t)=\varphi(t),\;t\in [\tau(t_0),t_0)$$, $$x(t_0)=x_0$$, $$\varphi(.)\in \Omega_0$$, $$x_0\in K_1,$$ $$q^i(t_0,t_1,x_0,x(t_1))=0$$, $$i=0,\dots ,l$$, $$q^0(t_0,t_1,x_0,x(t_1))\to \min$$ with $$J=[a,b]$$, $$K_1$$, $$O\subset \mathbb R^n$$ – an open set; $$U\subset \mathbb R^r$$, $$V\subset \mathbb R^p$$ – compact sets. The function $$f:J\times O^2 \times U^2 \to \mathbb R^n$$ is measurable with respect to $$t\in J.$$ The main result is the existence theorem about the optimal element $$\widetilde z=(\widetilde t_0,\widetilde t_1,\widetilde x_0,\widetilde \varphi(.),\widetilde u(.))\in \Delta$$ fulfilling $$\widetilde I=I(\widetilde z)=\inf _{z\in \Delta}I(z),$$ where $$I(z)=q^0(t_0,t_1,x_0,x(t_1))$$, $$x(t)=x(t,z)$$.

##### MSC:
 49J25 Optimal control problems with equations with ret. arguments (exist.) (MSC2000)
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