Sufficient optimality conditions for continuous and multistep processes (Q2768811)

The authors obtain sufficient optimality conditions for the optimal control problem for continuous processes NEWLINENEWLINE\[NEWLINEA(x,u) =\int_{t_0}^{T}f_0(t,x(t),u(t))\, dt+ F(x(T)-\tilde F(x(t_0))\to\min,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\dot x_{i}(t)= f_{i}(t,x(t),u(t)),\quad i=1,\ldots,n,NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINE (x(t),u(t))\in V_{t}\subset\mathbb R^n\times\mathbb R^r,\;t\in[t_{0},T],\;x(t_0)\in V_{t_0}^{x}\subset\mathbb R^n,\;x(T)\in V_{T}^{x}\subset\mathbb R^n,NEWLINE\]NEWLINE and for the optimal control problem NEWLINENEWLINE\[NEWLINEA(x,u)=\sum_{t=0}^{T-1}f_0(t,x(t),u(t))+F(x(T)-\tilde F(x(t_0))\to\min,NEWLINE\]NEWLINE NEWLINE\[NEWLINEx_{i}(t+1)=f_{i}(t,x(t),u(t)), \quad t\in\{0,1,\ldots,T-1\},\;i\in\{1,\ldots,n\},NEWLINE\]NEWLINE NEWLINE\[NEWLINE(x(t),u(t))\in V_{t}\subset\mathbb R^n\times\mathbb R^r,\;t\in\{0,1,\ldots,T-1\},\;x(t_0)\in V_{t_0}^{x}\subset\mathbb R^n,\;x(T)\in V_{T}^{x}\subset\mathbb R^nNEWLINE\]NEWLINE for multistep processes.