Necessary optimality conditions for quasisingular controls in a step control problem
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NECESSARY OPTIMALITY CONDITIONS FOR QUASISINGULAR CONTROLS IN A STEP CONTROL PROBLEM R. R. Ismailov and K. B. Mansimov †
UDC 517.9
Necessary second-order optimality conditions are obtained for one class of optimal control problems for step systems. Keywords: step control problem, quasisingular control, necessary optimality condition, Pontryagin’s maximum principle.
In modeling many real processes, the need arises to study step control problems. For example, necessary first-order optimality conditions, such as Pontryagin’s maximum principle, are obtained in [1–11] for various optimal control problems for step systems described by ordinary differential and difference equations. We obtain here necessary optimality conditions for quasisingular [12] controls in one class of optimal control problems for multistage processes. Note that the control parameters appear also in the initial conditions [5]. 1. PROBLEM STATEMENT Let a controlled process be described by the system of nonlinear differential equations x& i = f i ( t , x i , u i , ), t Î[ t i - 1 , t i ] = Ti , i = 1, 3 ,
(1)
x1 ( t 0 ) = g 1 ( u1 ), x i ( t i - 1 ) = g i ( x i - 1 ( t i - 1 ), u i ), i = 2, 3,
(2)
with the initial conditions where t i , i = 0, 3, are given; f i ( t , x i , u i ), i = 1, 3, are given n-dimensional vector functions, continuous in the set of variables together with partial derivatives with respect to ( x i , u i ), i = 1, 3, up to the second order inclusively; g 1 ( u1 ) is a given twice continuously differentiable vector function; g i ( x i - 1 , u i ), i = 2, 3, are given vector functions, continuous in the set of variables together with the partial derivatives with respect to ( x i - 1 , u i ), i = 2, 3, up to the second order inclusively; u i ( t ), i = 1, 3, are r-dimensional measurable and bounded vector functions of controls; u i , i = 1, 3, are q-dimensional control parameters that satisfy the constraints u i ( t ) ÎU i Ì R r , t ÎTi , u i Î Vi Î R q ,
(3)
where U i and Vi are given nonempty, bounded, and convex sets. We will call the set ( u1 ( t ), u 2 ( t ), u 3 ( t ), u1 , u 2 , u 3 ) º ( u( t ), u ) satisfying the above-mentioned conditions the admissible control, and the corresponding absolutely continuous solution ( x1 ( t ), x 2 ( t ), x 3 ( t )) º x( t ) of the system (1), (2) the admissible trajectory.
Institute of Cybernetics, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan, †[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 101–115, January–February 2008. Original article submitted January 24, 2006. 78
1060-0396/08/4401-0078
©
2008 Springer Science+Business Media, Inc.
The problem is to minimize the functional S ( u, u ) = j( x1 ( t 1 ), x 2 ( t 2 ), x 3 ( t 3 ))
(4)
under the constraints (1)–(3), where j( x1 , x 2 , x 3 ) is a given twice continuously differentiable scalar function. We will call the admissible control ( u( t ), u ), which is the solution of the problem of minimizing the functional (4) under the constraints (1)–(3) the optimal control, and th
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