Second-order optimality conditions and regularity of Lagrange multipliers for mixed optimal control problems
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Second-order optimality conditions and regularity of Lagrange multipliers for mixed optimal control problems N. B. Giang1 · N. Q. Tuan2 · N. H. Son3 Received: 13 March 2020 / Accepted: 16 October 2020 © Springer Nature Switzerland AG 2020
Abstract This paper deals with second-order optimality conditions and regularity of Lagrange multipliers for a class of optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints in which controls act both in the domain and on the boundary. We give some criteria under which the optimality conditions are of KKT type and the multipliers are of L p -spaces. Moreover, we show that the multipliers are Lipschitz continuous functions. Keywords Second-order optimality conditions · Lagrange multiplier · Regularity · Semilinear elliptic equation · Mixed pointwise constraint Mathematics Subject Classification 49K20 · 35J25
The research is funded by National University of Civil Engineering (NUCE) under grant number 205 2018/KHXD-TD.
B
N. H. Son [email protected] N. B. Giang [email protected] N. Q. Tuan [email protected]
1
Faculty of Information Technology, National University of Civil Engineering, 55 Giai Phong Str., Hanoi, Vietnam
2
Department of Mathematics, Hanoi Pedagogical University 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam
3
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam
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N. B. Giang et al.
1 Introduction Let Ω be a bounded domain in R N with the boundary Γ of class C 1,1 and N ∈ {2, 3}. We consider the following semilinear elliptic optimal control problem with mixed pointwise constraints. Find a couple of control functions (u, v) ∈ L ∞ (Ω) × L ∞ (Γ ) and a corresponding state function y ∈ W 1,r (Ω), r > N , which minimize F(y, u, v) = L(x, y(x), u(x))d x + (x, y(x), v(x))ds, Ω
Γ
(1) (P)
subject to Ay = f 1 (x, y, u) in Ω, ∂ν A y = f 2 (x, y, v) on Γ ,
(2)
a1 (x) ≤ g1 (x, y(x), u(x)) ≤ b1 (x) a.a. x ∈ Ω, a2 (x) ≤ g2 (x, y(x), v(x)) ≤ b2 (x) a.a. x ∈ Γ ,
(3) (4)
where L : Ω × R × R → R and : Γ × R × R → R are Carathéodory functions, f 1 , g1 : Ω × R × R → R, f 2 , g2 : Γ × R × R → R are continuous, a1 , b1 ∈ L ∞ (Ω), a1 (x) < b1 (x) a.a. x ∈ Ω and a2 , b2 ∈ L ∞ (Γ ), a2 (x ) < b2 (x ) a.a. x ∈ Γ ; the operator A is defined by Ay(x) = −
N
D j (ai j (x)Di y(x)) + a0 (x)y(x);
i, j=1
coefficients ai j ∈ C 0,1 (Ω) satisfy ai j = a ji , a0 ∈ L ∞ (Ω), a0 (x) ≥ 0 a.a. x ∈ Ω, and a0 (x) > 0 on a set of positive measure, and there exists m > 0 such that mξ 2 ≤
N
ai j ξi ξ j ∀ξ = (ξ1 , ξ2 , . . . , ξ N ) ∈ R N ,
i, j=1
and ∂ν A denote the conormal-derivative associated with A, that is, ∂ν A y(x) =
N
ai j (x)Di y(x)ν j (x),
i, j=1
where ν(x) = (ν1 (x), ..., ν N (x)) denotes the unit outward normal to Γ at the point x. Hereafter, the measure on the boundary Γ is the usual (N − 1)-dimensional measure induced by the parametrization (see [10], [22], [34]). The study of first- and second-order optimal
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