Optimal control problem with comparatively large delay (Q2912507)

Let \(t_1, \delta, \delta_1 \in \mathbb {R}_+,\; f:\mathbb {R} \times \mathbb {R}^n \times \mathbb {R}^n \times \mathbb {R}^m \times \mathbb {R}^m \rightarrow \mathbb {R}^n,\; \tau:[0,t_1] \rightarrow ]- \delta, \delta[,\; \tau_1:[0,t_1] \rightarrow ]- \delta_1, \delta_1[,\; \phi:\mathbb {R} \times \mathbb {R}^n \rightarrow \mathbb {R},\; F:\mathbb {R} \times \mathbb {R}^n \times \mathbb {R}^n \times \mathbb {R}^m \times \mathbb {R}^m \rightarrow \mathbb {R}\) be differentiable mappings. Then, if \(\tau\) and \(\tau_1\) are constant or \(\delta\) and \(\delta_1\) are small enough, a necessary condition that \((x,u): [0,t_1] \rightarrow \mathbb {R}^n \times \mathbb {R}^m\) such that \(x'=f(\cdot,x, x(\cdot-\tau),u, u(\cdot-\tau_1))\) maximizes \(J\), where \(J(x,u)=\phi(t_1, x(t_1))+ \int_0 ^{t_1} F(t, x(t), x(t-\tau),u(t), u(t-\tau_1)) \text{d}t\), is that there exists \(p: [0,t_1] \rightarrow \mathbb {R}^n\) such that \(^t p(t_1)= \partial _2 \phi(t_1, x(t_1)),\) \(^t p'+\partial_2 H(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p)+\partial_3 H(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p)=0\) andNEWLINENEWLINE\(u' \big(\partial_4 H(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p) + \partial_5 H(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p)\big) + \tau' \big(\tau \partial_1 f(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1)) -f(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1))\big) \partial_3 H(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p) + \tau_1' (\tau_1 u''-u') \partial_5 H (\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p) - \tau \partial_1 f(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1)) \partial_3 H(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p) - \tau_1 u'' \partial_5 H(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p) +{{\tau^2} \over{2}} \partial _{1,1} f(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1)) \partial_3 H(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p) +{{\tau_1 ^2} \over {2}} u''' \partial_5 H(\cdot, x, x(\cdot-\tau),u, u(\cdot-\tau_1),p) =0,\)NEWLINENEWLINE where \(H:\mathbb {R} \times \mathbb {R}^n \times \mathbb {R}^n \times \mathbb {R}^m \times \mathbb {R}^m \times \mathbb {R}^n \rightarrow \mathbb {R},\; H(t,x,y,u,v,p)=F(t,x,y,u,v)+ ^t p f(t,x,y,u,v)\).