Optimality conditions of the maximum principle type in Goursat-Darboux systems (Q1116084)

For the two-dimensional control problem (P): \(\phi (x(s_ 1,t_ 1))\to \min\) for systems of Goursat-Darboux type \(x_{st}=f(x_ s,x_ t,x,u,s,t)\) on \([s_ 0,s_ 1]\times [t_ 0,t_ 1]\) with \(x(s_ 0,t)=a(t)\), \(x(s,t_ 0)=b(s)\), \(a(t_ 0)=b(s_ 0)\) and u(s,t)\(\in U\), the author proposes a reduction method which expresses necessary optimality conditions for (P) by two one-dimensional control problems along the characteristics. Basic to these investigations are needle-like variations of control of one variable each. The paper concludes with an example and considerations on the sufficiency of this ``maximum principle''.