On the optimality of singular controls in goursat-darboux systems

Singular controls in control problems for distributed-parameter systems

“…Let U be a nonempty bounded set of the r-dimensional Euclidean space R r . Every piecewisecontinuous function u(t, x) (with finitely many discontinuity lines of the first kind) assuming its values in U is called an admissible control (see [17]).…”

“…The general case of studying a singular control in the Goursat-Darboux systems was carried out in [17] for the fist time.…”

“…Here, M ztx (z, u, ψ) is the variational derivative of the first kind of the functional M (z, u, ψ) in z tx at the point (t, x) [7]. Then the following second-order increment formula of the performance index S(u) [17] holds:…”

“…In [17], by using the increment formula (1.18), it was proved that for an optimal process (u(t, x), z(t, x)) singular in the sense of (1.6), the inequality…”

“…Second, the formulation of the optimality condition (1.19) does not involve the second derivatives of the objective function. Because of this, this condition was called the intermediate condition between the optimality conditions of the first and second orders by Ashchepkov and Vasil'ev [17]. The reason for which the optimality condition (1.19) is not effective for the control singular in the sense of (1.7) is the fact that on simple needle-shaped variations of the form (1.17), the quantities Δz(t, x) , Δz t (t, x) , and Δz x (t, x) have different magnitudes of smallness.…”