On optimality of singular controls in an optimal control problem

In this paper, a Moskalenko type optimal control problem is considered. We consider the optimal control problem of minimizing the terminal type functional ^xi S(u,v ) = ф(у (x,)) + J G (x, z (i, , x ))dx, ^x0 under constraints u (,,x) e U с R^r, (t,x) e D =[t₀,t,]x [x₀,x,], v(x) e V с R^q, x e X - [x₀,x,], z_t (I,x) - f (I,x,z (I,x),u (I,x)), (t,x) e D, z(t₀,x) - у (x), x e X, У (^x0)^- У0. Here, f (t,x,z,u) (g (x,y,v)) is an n-dimensional vector function which is continuous on the set of variables, together with partial derivatives with respect to z (у) up to second order, t₀, t,, x₀, x₁(t₀ < t,,x₀ < x,) are given, ф(у) (G(x,z)) is a given twice continuously differentiable with respect to у (z) scalar function, U (V) is a given nonempty bounded set, and u (t, x) is an r-dimensional control vector function piecewise continuous with respect to t and continuous with respect to x , v(x) is a q-dimensional piecewise continuous vector of control actions. The necessary optimality conditions for singular controls in the sense of the Pontryagin maximum principle have been obtained.

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