Necessary optimality conditions in the one boundary control problem for Qoursat - Darboux systems

In this paper, a boundary optimal control problem described by the Goursat-Darboux system is considered under the assumption that the control domain is open. We consider the problem of minimizing of the functional I (u) = ф(а ('1 )) + G (z ('1, x)), under constraints u (t) е U с R^r, t е T = [t₀,t₁ ] , ^z'x = ^B ('> ^x)zt + ^f (x ^z> ^zx)> ('>^x) ^е D = ['о> 'А ^х [^x0 = ^x1 ] > z(t,x₀ ) = а(t), t е T = [t₀,t₁], z (t₀, x )= b (x), x е X =[x₀, x₁ ], ^а ('о ) = ^b (^x0 )= ^ао = а = g(t^u), t е T, ^а ('о ) = ^ао . Here, f (t, x, z, z_x) is a given и-dimensional vector-function which is continuous with respect to set of variables, together with partial derivatives with respect to z, z_x up to second order, B (t, x) is a given measurable and bounded matrix function, b (x) is a given и-dimensional absolute continuous vector-valued function, t₀, t₁, x₀, x₁ (t₀ < t₁; x₀ < x₁) are given, a₀ is a given constant vector, g (t, a,u) given и-dimensional vector-function which is continuous with respect to the set of variables together with partial derivatives with respect to (a, u) up to second order, 9(a) and G(z) are given twice continuously differentiable scalar functions, U is a given nonempty, bounded, and open set, and u (t) is a measurable and bounded r-dimensional control vector-function. The first and second order necessary conditions of optimality are established.

Keywords

Authors

References