A generalized isoperimetric problem with a linear shift in the objective function (Q2704002)

In the present paper a necessary and sufficient condition for the solvability of the following isoperimetric problem is proved.NEWLINENEWLINENEWLINEThe problem is NEWLINE\[NEWLINEy[u]= T^2 \int^1_0 \langle A(u(\tau), x(\tau))\rangle d\tau\to \min,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\dot x(\tau)= u(\tau),\;R(x_0, u(\tau))= 0,\;G_A(x_0, u(\tau))\leq 0,\;x(0)= 0,\;x(\tau)\in X,\tag{2}NEWLINE\]NEWLINE in which the bilinear integrand of the objective function contains a linear shift.NEWLINENEWLINENEWLINELet \(U= \{u: R(x_0, u)= 0,\;G_p(x_0, u)\leq 0\}\) and \(\text{conv }U\) be the convex hull of the set \(U\). If \(U\) be a compact set, the problem (1), (2) is solvable if and only if \(\text{conv }U\cap X\neq\emptyset\). Moreover, different applications of this suggestion are given.