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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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