Necessary conditions of optimality for optimal problems with delays and with a discontinuous initial condition (Q2730967)

The authors deal with the optimal control problem for the delay differential system NEWLINE\[NEWLINE\dot x(t)=f(t,x_1(\tau_1(t)),\dots ,x_s(\tau_s(t)),u_1(\theta_1(t)),\dots , u_\nu(\theta_\nu(t))),\;t\in [t_0,t_1]\subset [a,b]NEWLINE\]NEWLINE with the discontinuity condition NEWLINE\[NEWLINEx(t)=\phi(t),\;t\in [\tau,t_0),\;x(t_0)=x_0,\;\tau=\min(\tau_1(a),\dots ,\tau_s(a)).NEWLINE\]NEWLINE The element \(\sigma=(t_0,t_1,x_0,\phi,u)\) is said to be admissible if the corresponding solution \(x(t)=x(t;\sigma)\) satisfies the conditions \(q^i(t_0,t_1,x(t_0),x(t_1))=0,\;i=1,\dots ,l.\) The admissible element \(\widetilde \sigma = (\widetilde t_0,\widetilde t_1,\widetilde x_0,\widetilde \phi,\widetilde u)\) is said to be optimal if for an arbitrary admissible element \(\sigma \) the inequality \(q^0(\widetilde t_0,\widetilde t_1,\widetilde x(\widetilde t_0),\widetilde x(\widetilde t_1))\leq q^0(t_0, t_1, x(t_0),x(t_1))\); \(\widetilde x(t)=x(t,\widetilde \sigma)\;\) holds. The paper contains six theorems on necessary optimality conditions without proofs.