TAILIEUCHUNG - First and second order necessary optimality conditions for discrete optimal control problems
Discrete optimal control problems with varying endpoints are considered. First and second order necessary optimality conditions are obtained without normality assumptions. | Yugoslav Journal of Operations Research 16 (2006), Number 2, 153-160 FIRST AND SECOND ORDER NECESSARY OPTIMALITY CONDITIONS FOR DISCRETE OPTIMAL CONTROL PROBLEMS Boban MARINKOVIĆ Faculty of Mining and Geology, University of Belgrade Belgrade, Serbia mboban@ Received: October 2004 / Accepted: June 2005 Abstract: Discrete optimal control problems with varying endpoints are considered. First and second order necessary optimality conditions are obtained without normality assumptions. Keywords: Discrete optimal control, mathematical programming, abnormal extremal. 1. INTRODUCTION Consider discrete optimal control problem with varying endpoints. N −1 minimize ∑ fi ( xi , ui ) ; (1) xi +1 = ϕi ( xi , ui ) , i = 0, N − 1 , (2) K ( x0 , xN ) = 0 , (3) i =1 where fi ( x, u ) : R n × R r → R is twice continuously differentiable function, ϕi ( x, u ) : R n × R r → R n and K ( x0 , xN ) : R n × R n → R k are twice continuously differentiable mappings. Here xi ∈ R n is state variable, ui ∈ R r is a control parameter, N is given number of steps. Vector ε = ( x0 , x1 ,., xn ) is called a trajectory, ω = (u0 , u1 ,., u N −1 ) is called a control, x0 is a starting point and xN is an end point of corresponding trajectory. 154 B. Marinković / First and Second Order Necessary Optimality Conditions Let x0 be a starting point and let be a control. Then the pair ( x0 , ω ) defines the corresponding directory ε = ( x0 , x1 ,., xn ) . If the condition (3) is satisfied then we say that the pair ( x0 , ω ) is feasible. The discrete optimization problem is to minimize the function N −1 J ( x0 , ω ) = ∑ fi ( xi , ui ) i =0 on the set of feasible pairs. The aim of this paper is to obtain first and second order necessary optimality conditions for the problem (1)-(3) without normality assumptions. 2. FIRST ORDER NECESSARY OPTIMALITY CONDITIONS Let ( xˆ0 , ωˆ ) be a feasible pair and let εˆ = ( x0 , x1 ,., xN ) be a corresponding trajectory. Suppose that the pair
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