To the optimality of singular controls in the classical sense in one problem of optimal control of systems with variable structure

The paper deals with the problem of the minimization the functional 5(и,v) = _Ф1 (z(r, (t,x) е Д = '] x [x₀,X], v(t,x) е V с R^q, (t,x) е D₂ = [''] x [x₀,X], ^z« = ^f ('^, x ^z,u ^ ('^{, x} )^{е D}1 ^, z (t₀, x ) = a (x), x е[ x₀, X ], ^z ('^, x0 ) = ^bi (')^, ' ^е[^tо^,tl]^, y_a = g ('^{, x,} y^,v)^, ('^{, x})^{e D}2 ^, y(',x) = G(z(0,X], У (',x0 )= b₂ ('), ' g[G (^Z ('1, ^x0 ))= ^b2 ('1 ) . Here f (t,x,z,u) , (g(t,x,y, v)) is a given n (m)-dimensional vector-function continuous in D_l x R" x R^r (D₂ x R^m x R^q) together with their partial derivatives with respect to (z,и) ((y,v)) up to the second order inclusive, G (z) is a given twice continuously differentiable in Rⁿ m-dimensional vector-function, a(x), b ('), i = 1,2 , are given vector-functions of the corresponding dimensions satisfying a Lipschitz condition with some constants, t₀, ', t₂, x₀, X are given, so that t₀ < ^ < ^ , x₀ < X , control vector-function и(t,x) (v(',x)) everywhere is Lebesgue measurable on a rectangle D_t (D₂) , U (V) is a given non-empty, open, and bounded set. When the control domain is open, the first and second orders necessary optimality conditions are derived. Optimal singular controls in the classical sense are studied.

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