MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY NGUYEN HAI SON NO-GAP OPTIMALITY CONDITIONS AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY SEMILINEAR ELLIPTIC EQUATIONS Major: Mathematics Code: 9460101 ABSTRACT OF DOCTORAL DISSERTATION OF MATHEMATICS Hanoi – 2019 The dissertation is completed at: Hanoi Univesity of Science and Technology Supervisors: Dr Nguyen Thi Toan Dr Bui Trong Kien Reviewer 1: Prof Dr Sc Vu Ngoc Phat Reviewer 2: Assoc Prof Dr Cung The Anh Reviewer 3: Dr Nguyen Huy Chieu The dissertation will be defended before approval committee at Hanoi Univesity of Science and Technology Time …… , date… month… year……… The dissertation can be found at: Ta Quang Buu Library - Hanoi Univesity of Science and Technology Vietnam National Library Introduction Optimal control theory has many applications in economics, mechanic and other fields of science It has been systematically studied and strongly developed since the late 1950s, when two basic principles were made: the Pontryagin Maximum Principle and the Bellman Dynamic Programming Principle Up to now, optimal control theory has developed in many various research directions such as non-smooth optimal control, discrete optimal control, optimal control governed by ordinary differential equations (ODEs), optimal control governed by partial differential equations (PDEs), In the last decades, qualitative studies for optimal control problems governed by ODEs and PDEs have obtained many important results One of them is to give optimality conditions for optimal control problems Better understanding of second-order optimality conditions for optimal control problems governed by semilinear elliptic equations is an ongoing topic of research for several researchers This topic is great value in theory and in applications Second-order sufficient optimality conditions play an important role in the numerical analysis of nonlinear optimal control problems, and in analyzing the sequential quadratic programming algorithms and in studying the stability of optimal control Second-order necessary optimality conditions not only provide criterion of finding out stationary points but also help us in constructing sufficient optimality conditions Let us briefly review some results on this topic For distributed control problems, i.e., the control only acts in the domain Ω in Rn , E Casas, T Bayen et al derived second-order necessary and sufficient optimality conditions for problem with pure control constraint, that is, a(x) ≤ u(x) ≤ b(x) a.e x ∈ Ω, (1) and the appearance of state constraints In particular, E Casas established secondorder sufficient optimality conditions for Dirichlet control problems and Neumann control problems with only constraint (1) when the objective function does not contain control variable u In addition, C Meyer and F Trăoltzsch derived secondorder sufficient optimality conditions for Robin problems with mixed constraint of the form a(x) ≤ λy (x) + u(x) ≤ b(x) a.e x ∈ Ω and finitely many equalities and inequalities constraints, where y is the state variable For boundary control problems, i.e., the control u only acts on the boundary Γ, E Casas et al and F Trăoltzsch derived second-order necessary and sufficient optimality conditions with pure pointwise constraints, i.e., a(x) ≤ u(x) ≤ b(x) a.e x A Răosch and F Trăoltzsch gave the second-order sufficient optimality conditions for the problem with the mixed pointwise constraints which has unilateral linear form c(x) ≤ u(x) + γ (x)y (x) for a.e x ∈ Γ We emphasize that in above results, a, b ∈ L∞ (Ω) or a, b ∈ L∞ (Γ) Therefore, the control u belongs to L∞ (Ω) or L∞ (Γ) This implies that corresponding Lagrange multipliers are measures rather than functions In order to avoid this disadvantage, B T Kien et al recently established second-order necessary optimality conditions for distributed control of Dirichlet problems with mixed state-control constraints of the form a(x) ≤ g (x, y (x)) + u(x) ≤ b(x) a.e x ∈ Ω with a, b ∈ Lp (Ω) and pure state constraints This motivates us to develop and study the following problems (OP 1) : Establish second-order necessary optimality conditions for Robin boundary control problems with mixed state-control constraints of the form a(x) ≤ g (x, y (x)) + u(x) ≤ b(x) a.e x ∈ Γ with a, b ∈ Lp (Γ) (OP 2) : Give second-order sufficient optimality conditions for optimal control problems with mixed state-control constraints when the objective function does not depend on control variables Solving problems (OP 1) and (OP 2) is the first goal of the dissertation After second-order necessary and sufficient optimality conditions are established, they should be compared to each other According to J F Bonnans, if the change between necessary and sufficient second-order optimality conditions is only between strict and non-strict inequalities, then we say that the no-gap optimality conditions are obtained Deriving second-order optimality conditions without a gap between second-order necessary optimality conditions and sufficient optimality conditions, is a difficult problem In some papers, J F Bonnans derived second-order necessary and sufficient optimality conditions with no-gap for an optimal control problem with pure control constraint and the objective function is quadratic in both state variable y and control variable u This result was established by basing on polyhedric property of admissible sets and the theory of Legendre forms However, there is an open problem in this area Namely, the following problem that we need to study: (OP 3) : Find a theory of no-gap second-order optimality conditions for optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints Solving problem (OP 3) is the second goal of this dissertation Solution stability of optimal control problem is also an important topic in optimization and numerical method of finding solutions The study of solution stability is to investigate continuity properties of solution maps in parameters such as lower semicontinuity, upper semicontinuity, Hăolder continuity and Lipschitz continuity Let us consider the following parametric optimal problem:  F (y, u, µ) → inf, P (µ, λ) (2) (y, u) ∈ Φ(λ), where y ∈ Y, u ∈ U are state and control variables, respectively; µ ∈ Π, λ ∈ Λ are parameters, F : Y × U × Π → R is an objective function on Banach space Y × U × Π and Φ(λ) is an admissible set of the problem It is well-known that if the cost function F (·, ·, µ) is strongly convex, and the admissible set Φ(λ) is convex, then the solution map of problem (2) is single-valued Moreover, A Dontchev showed that under some certain conditions, the solution map is Lipschitz continuous w.r.t parameters By using techniques of implicit function theorem, K Malanowski proved that the solution map of problem (2) is also a Lipschitz continuous function in parameters if weak second-order optimality conditions and standard constraint qualifications are satisfied at the reference point When conditions mentioned above are invalid, the solution map may not be singleton In this situation, we have to use tools of set-valued analysis and variational analysis to deal with the problem In 2012, B T Kien et al obtained the lower semicontinuity of the solution map to a parametric optimal control problem for the case where the cost function is convex in both variables and the admissible sets are also convex Recently, the upper semicontinuity of the solution map has been given by B T Kien et al and V H Nhu for problems, where the cost functions may not be convex in the both variables and the admissible sets are not convex Notice that the authors only considered the problems governed by ordinary differential equations and semilinear elliptic equation with distributed control From the above, one may ask to study the following problem: (OP 4) : Establish sufficient conditions under which the solution map of parametric boundary control problem is upper semicontinuous and continuous Giving a solution for (OP 4) is the third goal of this dissertation The objective of this dissertation is to study no-gap second-order optimality conditions and stability of solution to optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints Namely, the main content of the dissertation is to concentrate on (i) establishing second-order necessary optimality conditions for boundary control problems with the control variables belong to Lp (Γ), < p < ∞; (ii) deriving second-order sufficient optimality conditions for distributed control problems and boundary control problems when objective functions are quadratic forms in the control variables, and showing that no-gap optimality condition holds in this case; (iii) deriving second-order sufficient optimality conditions for distributed control problems and boundary control problems when objective functions are independent of the control variables, and showing that in general theory of no-gap conditions does not hold; (iv) giving sufficient conditions for a parametric boundary control problem under which the solution map is upper semicontinuous and continuous in parameters The dissertation has four chapters and a list of references Chapter collects several basic concepts and facts on variational analysis, Sobolev spaces and partial differential equations Chapter presents results on the no-gap optimality conditions for distributed control problems Chapter provides results on the no-gap optimality conditions for boundary control problems The obtained results in Chapters and are answers for (OP 1), (OP 2) and (OP 3) Chapter presents results on the upper semicontinuity and continuity of the solution map to a parametric boundary control problem, which is a positive answer for problem (OP 4) The main results of the dissertation are the contents of three papers which were published in the journals Set-Valued and Variational Analysis, SIAM Journal on Optimization, and Optimization These results have been presented at: • The Conference on Applied Mathematics and Informatics at Hanoi University of Science and Technology in November 2016 • The 15th Conference on Optimization and Scientific Computation, Ba Vi in April 2017 • The 7th International Conference on High Performance Scientific Computing in March 2018 at Vietnam Institute for Advanced Study in Mathematics • The 9th Vietnam Mathematical Congress, Nha Trang in August 2018 • Seminar ”Optimization and Control” at the Institute of Mathematics, Vietnam Academy of Science and Technology Chapter No-gap optimality conditions for distributed control problems Let Ω be a bounded domain in RN with N ≥ and the boundary Γ of class C We consider the following distributed optimal control problem of finding a control function u ∈ Lp (Ω) and a state function y ∈ W 2,p (Ω) ∩ W01,p (Ω) which minimize F (y, u) = L(x, y (x), u(x))dx, (1.1) Ω (DP ) s.t − ∆y + h(x, y ) = u in Ω, y=0 on Γ, a(x) ≤ g (x, y (x)) + λu(x) ≤ b(x) a.e x ∈ Ω, (1.2) (1.3) where L : Ω × R × R → R and g : Ω × R → R are Carath´eodory functions, h : Ω × R → R is a continuous function of class C w.r.t the second variable such that h(x, 0) = and hy (x, y ) ≥ for all y ∈ R and a.e x ∈ Ω, a, b ∈ Lp (Ω) and λ = is a constant Hereafter, we assume that p > N2 1.1 1.1.1 Second-order necessary optimality conditions An abstract optimization problem Let U be Banach space and E be a separable Banach space with the duals U ∗ and E ∗ , respectively We consider the following problem (P ) f (u) subject to G(u) ∈ K, u∈U where K is a nonempty closed and convex set in E , G : U → E and f : U → R are second-order Frech´et differentiable on U Put Φad := G−1 (K ) Definition 1.1.1 A function u ¯ ∈ Φad is said to be a locally optimal solution of problem (P ) if there exists ε > such that f (u) ≥ f (¯ u) ∀u ∈ BU (¯ u, ) ∩ Φad Given a point u¯ ∈ Φad , problem (P ) is said to satisfy Robinson’s constraint qualification at u¯ if there exists ρ > such that BE (0, ρ) ⊂ ∇G(¯ u)(BU ) − (K − G(¯ u)) ∩ BE (1.4) In this case, we also say that u¯ is regular Problem (P ) is associated with the following Lagrangian: L(u, e∗ ) = f (u) + e∗ , G(u) with e∗ ∈ E ∗ We shall denoted by Λ(¯ u) the set of multipliers e∗ ∈ E ∗ such that ∇u L(¯ u, e∗ ) = ∇f (¯ u) + ∇G(¯ u)∗ e∗ = 0, e∗ ∈ N (K, G(¯ u)) The set Λ(¯ u) is a non-empty, convex and weakly star compact set in E ∗ To analyze second-order conditions, we need the following critical cone at u¯: C (¯ u) := {d ∈ U | ∇f (¯ u), d ≤ 0, ∇G(¯ u)d ∈ T (K, G(¯ u))} The set K is said to be polyhedric at z¯ ∈ K if for any v ∗ ∈ N (K, z¯), one has T (K, z¯) ∩ (v ∗ )⊥ = cl[cone(K − z¯) ∩ (v ∗ )⊥ ], where (v ∗ )⊥ = {v ∈ E | v ∗ , v = 0} Moreover, problem (P ) is said to satisfy the strongly extended polyhedricity condition at u¯ ∈ Φ if the set C0 (¯ u) is dense in C (¯ u), where C0 (¯ u) := {d ∈ C (¯ u) | ∇G(¯ u)d ∈ cone(K − G(¯ u))} Lemma 1.1.3.1 Suppose that u ¯ is regular, at which the strongly extended polyhedricity condition is fulfilled If u¯ is a locally optimal solution, then for each d ∈ C (¯ u), there exists a multiplier e∗ ∈ Λ(¯ u) such that ∇2uu L(¯ u, e∗ )(d, d) = ∇2 f (¯ u)(d, d) + e∗ , ∇2 G(¯ u)(d, d) ≥ 1.1.2 Second-order necessary optimality conditions for optimal control problem Recall that a couple (¯ y, u ¯) satisfying constraints (1.2)–(1.3), is said to be admissible for (DP ) Given an admissible couple (¯ y, u ¯), symbols g [x], h[x], L[x], Ly [x], L[·], etc., stand respectively for g (x, y¯(x), u¯(x)), h(x, y¯(x)), L(x, y¯(x), u¯(x)), Ly (x, y¯(x), u¯(x)), L(·, y¯(·), u ¯(·)), etc Definition 1.1.4 An admissible couple (¯ y, u ¯) is said to be a locally optimal solution of (DP ) if there exists > such that for all admissible couples (y, u) satisfying ¯ Lp (Ω) ≤ , one has y − y¯ W 2,p (Ω) + u − u F (y, u) ≥ F (¯ y, u ¯) J F Bonnans and A Shapiro (2000), Perturbation Analysis of Optimization Problems, Springer, New York We now impose the following assumptions for problem (DP ) which involve (¯ y, u ¯) (A1.1) L : Ω × R × R → R is a Carath´eodory function of class C with respect to variable (y, u), L(x, 0, 0) ∈ L1 (Ω) and for each M > 0, there exist a positive number kLM and a function rM ∈ L∞ (Ω) such that |Ly (x, y, u)| + |Lu (x, y, u)| ≤ kLM |y| + |u|p−1 + rM (x), |Ly (x, y1 , u1 ) − Ly (x, y2 , u2 )| ≤ kLM (|y1 − y2 | + |u1 − u2 |), |u1 − u2 |p−1−j |u2 |j |Lu (x, y, u1 ) − Lu (x, y, u2 )| ≤ kLM j=0,p−1−j>0 for all y, y1 , y2 ∈ R satisfying |y|, |yi | ≤ M and any u1 , u2 ∈ R Also for each M > 0, there is a number kLM > such that Lyy (x, y1 , u1 ) − Lyy (x, y2 , u2 ) ≤ kLM (|y1 − y2 | + |u1 − u2 |), Lyu (x, y1 , u1 ) − Lyu (x, y2 , u2 ) ≤ kLM (|y1 − y2 | + ε|u1 − u2 |p−1 ) ( with ε = if < p ≤ and ε = if p > ) and  = |Luu (x, y, u1 ) − Luu (x, y, u2 )| p−2−j ≤ kLM |u2 |j j=0,j for a.e x ∈ Ω, for all y, ui , yi ∈ R with |y|, |yi | ≤ M and i = 1, (A1.2) The function g is a continuous function and of class C w.r.t the second variable, and satisfies the following property: g (·, 0) ∈ Lp (Ω) and for each M > 0, there exists a constant Cg,M > such that gy (x, y ) + gyy (x, y ) ≤ Cg,M , gy (x, y1 ) − gy (x, y2 ) + gyy (x, y1 ) − gyy (x, y2 ) ≤ Cg,M |y2 − y1 | for a.e x ∈ Ω and |y|, |y1 |, |y2 | ≤ M (A1.3) λ = and λhy [x] + gy [x] ≥ a.e x ∈ Ω λ For each u ∈ Lp (Ω), equation (1.2) has a unique solution yu ∈ W 2,p (Ω) ∩ W01,p (Ω) and there exists a constant C > such that yu W 2,p (Ω) ≤C u Lp (Ω) Define a mapping H : W 2,p (Ω) ∩ W01,p (Ω) × Lp (Ω) → Lp (Ω) by setting H (y, u) = −∆y + h(·, y ) − u Then, H is of class C around (¯ y, u ¯) and its derivatives at (¯ y, u ¯) are given by hyy [·] 0 ∇H (¯ y, u ¯) = (−∆ + hy [·], −I ), ∇2 H (¯ y, u ¯) = ¯ Since p > N/2, y¯ = y (¯ u) ∈ C (Ω) Hence hy [·] ∈ L∞ (Ω) Therefore, for each u ∈ Lp (Ω), the following equation has a unique solution z ∈ W 2,p (Ω) ∩ W01,p (Ω) −∆z + hy (·, y¯)z = u in Ω, z=0 on Γ Hence the operator A := ∇y H (¯ y, u ¯) = −∆ + hy (·, y¯) is bijective By the classical implicit function theorem, there exist a neighborhood Y0 of y¯, a neighborhood U0 of u ¯ and a mapping ζ : U0 → Y0 such that H (ζ (u), u) = for all u ∈ U0 Moreover, ζ is of class C and its derivatives are given by the following formulae Lemma 1.1.9 Assume that ζ : U0 → Y0 is the control-state mapping defined by ζ (u) = yu Then ζ is of class C and for each u ∈ U0 , v ∈ Lp (Ω), zu,v := ζ (u)v is the unique solution of the linearized equation  −∆z + h (·, y )z = v in Ω, u,v y u u,v (1.11) zu,v = on Γ In other words, ζ (¯ u) = A−1 Moreover, for all v1 , v2 ∈ Lp (Ω), zu,v1 v2 := ζ (u)(v1 , v2 ) is the unique solution of the equation  −∆z u,v1 v2 + hy (·, yu )zu,v1 v2 + hyy (·, yu )zu,v1 zu,v2 = in Ω, (1.12) zu,v v = on Γ Let us put U = E = Lp (Ω) and K := {v ∈ Lp (Ω)| a(x) ≤ v (x) ≤ b(x) a.e x ∈ Ω} Then, from Lemma 1.1.8, we see that (¯ y, u ¯) is a locally optimal solution of problem (DP ) if and only if u¯ is a locally optimal solution of the following problem which has a form of (P ): f (u) := F (ζ (u), u) → inf s.t G(u) ∈ K, (1.13) (1.14) where G(u) := g (·, ζ (u)) + λu By Φp := G−1 (K ), we denote the admissible set of problem (1.13)–(1.14), i.e., Φp = {u ∈ Lp (Ω) | G(u) ∈ K} Definition 1.1.9 The function u ¯ ∈ Φp is said to be a locally optimal solution of problem (1.13)–(1.14) if there exists ε > such that f (u) ≥ f (¯ u) ∀u ∈ BU (¯ u, ) ∩ Φp Moreover, for every M > and > 0, there exists a positive number δ > such that ∂ 2h ∂ 2h ( x, y ) − (x, y2 ) < ∂y ∂y a.e x ∈ Ω ∀y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 −y2 | < δ (B 1.2) Function ϕ : Ω × R → R is a Carath´eodory function of class C with respect to the second variable, ϕ(x, 0) ∈ L1 (Ω) and for each M > there are a constant Cϕ,M > and a function ϕM ∈ L2 (Ω) such that ∂ 2ϕ ∂ϕ (x, y ) ≤ ϕM (x), (x, y ) ≤ Cϕ,M ∂y ∂y a.e x ∈ Ω, ∀y ∈ R with |y| ≤ M and for each > 0, there exists δ > such that ∂ 2ϕ ∂ 2ϕ (x, y1 ) − (x, y2 ) < ∂y ∂y a.e x ∈ Ω, ∀y1 , y2 ∈ R, |y1 |, |y2 | ≤ M, |y1 − y2 | < δ (B 1.3) Function g : Ω × R → R is a continuous function of class C with respect to the second variable, g (·, 0) ∈ L2 (Ω) and for each M > there are a constant Cg,M > and a function gM ∈ L2 (Ω) such that ∂ 2g ∂g (x, y ) ≤ gM (x), (x, y ) ≤ Cg,M ∂y ∂y a.e x ∈ Ω, ∀y ∈ R with |y| ≤ M Moreover, for every M > and > 0, there exists a positive number δ > such that ∂ 2g ∂ 2g (x, y1 ) − (x, y2 ) < ∂y ∂y a.e x ∈ Ω, ∀y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 −y2 | < δ (B 1.4) α, β ∈ L∞ (Ω) and β (x) ≥ for a.e x ∈ Ω Besides, the following is verified βa ∈ L∞ (Ωa ), βb ∈ L∞ (Ωb ) (1.26) Note that condition (1.26) holds whenever one of the following conditions is verified: (i) a, b ∈ L∞ (Ω); (ii) u¯ ∈ L∞ (Ω); (iii) β = Based on Casas, we enlarge C2 (¯ u) by defining the following critical cone  ≥ if x ∈ Ω , a τ C2 (¯ u) = v ∈ L (Ω) | ∇u f (¯ u), v ≤ τ zu¯,v , gy [x]zu¯,v (x)+λv (x) ≤ if x ∈ Ωb Obviously, C2 (¯ u) = C20 (¯ u) and C2 (¯ u) ⊂ C2τ (¯ u) for all τ > Theorem 1.2.4 Suppose that assumptions (A1.3) and (B 1.1) − (B 1.4) are fulfilled, there exist multipliers e∗ ∈ L2 (Ω) and φ¯ ∈ W 2,2 (Ω)∩W01,2 (Ω) satisfy conditions (1.17) and (1.18) If there exist positive constants γ, τ > such that ∇2uu L(¯ u, e∗ )(v, v ) ≥ γ zu¯,v 2 ∀v ∈ C2τ (¯ u), then there are constants ρ, r > such that f (u) ≥ f (¯ u) + r zu¯,u−¯u 2 11 ∀u ∈ BL2 (Ω) (¯ u, ρ) ∩ Φ2 Chapter No-gap optimality conditions for boundary control problems Let Ω be a bounded domain in RN with the boundary Γ of class C 1,1 and N ≥ We consider the problem of finding a control function u ∈ Lq (Γ) and a corresponding state function y ∈ W 1,r (Ω) which (BP ) minimize F (y, u) = L(x, y (x))dx + Ω  Ay + h(x, y ) = in Ω, s.t ∂ν y + b0 y = u on Γ, (x, y (x), u(x))dσ, (2.1) Γ a(x) ≤ g (x, y (x)) + u(x) ≤ b(x) a.e x ∈ Γ, (2.2) (2.3) where L, h : Ω × R → R and : Γ × R × R → R are Carath´eodory functions, g : Γ × R → R is continuous, a, b ∈ Lq (Γ), a(x) < b(x) for a.e x ∈ Γ, b0 ∈ L∞ (Γ), b0 ≥ 0, A denotes a second-order elliptic operator of the form N Ay (x) = Dj (aij (x)Di y (x)) + a0 (x)y (x); i,j=1 ¯ satisfy aij (x) = aji (x), a0 ∈ L∞ (Ω), a0 (x) ≥ for a.e coefficients aij ∈ C 0,1 (Ω) x ∈ Ω, a0 ≡ and there exists m > such that N m ξ aij ξi ξj ∀ξ ∈ RN ≤ for a.e x ∈ Ω i,j=1 and ∂ν denote the conormal-derivative associated with A Moreover, we assume that 1 1 > > 1− N r q N 2.1 (2.4) Abstract optimal control problems Let Y, U, V and E be either separable or reflexive Banach spaces with the dual spaces Y ∗ , U ∗ , V ∗ and E ∗ , respectively We consider the following problem: Min F (y, u), (2.5) s.t H (y, u) = 0, (2.6) G(y, u) ∈ K, (2.7) 12 where F : Y × U → R, H : Y × U → V , G : Y × U → E are given mappings and K is a nonempty closed convex subset in E We define the space Z := Y × U and the set Q := {z = (y, u) ∈ Z | H (y, u) = 0} Define Φad := Q ∩ G−1 (K ) An couple (¯ y, u ¯) ∈ Φad is said to be a locally optimal solution of problem (2.5)–(2.7) if there exists > such that for all (y, u) ∈ Φad satisfying y − y¯ Y + u − u¯ U ≤ , one has F (y, u) ≥ F (¯ y, u ¯) For a given point z¯ = (¯ y, u ¯) ∈ Φad , we need the following assumptions: (H 2.1) The mappings F, H, G are of class C around z¯ (H 2.2) ∇y H (¯ z ) : Y → V is bijective (H 2.3) The regularity condition is verified at z¯, i.e., there is a number δ > satisfying ∈ int ∇G(¯ z )(T (Q, z ) ∩ BZ ) − (K − G(¯ z )) ∩ BE (2.8) z∈BZ (¯ z ,δ)∩Q (H 2.4) ∇G(¯ z )(T (Q, z¯)) = E Definition 2.1.3 A couple z = (y, u) is called a critical direction of problem (2.5)– (2.7) at z¯ = (¯ y, u ¯) if the following conditions are satisfied: (i) Fy (¯ z )y +Fu (¯ z )u ≤ 0; (ii) Hy (¯ z )y +Hu (¯ z )u = 0; (iii) ∇G(¯ z )z ∈ T (K, G(¯ z )) The set of such critical directions will be denoted by C (¯ z ) Problem (2.5)–(2.7) is associated with the Lagrangian L(z, e∗ , v ∗ ) := F (z ) + v ∗ , H (z ) + e∗ , G(z ) , (2.9) where z = (y, u) ∈ Z, e∗ ∈ E ∗ , v ∗ ∈ V ∗ Let z¯ be a locally optimal solution of problem (2.5)–(2.7) and denote by Λ(¯ z ) the ∗ ∗ ∗ ∗ set of Lagrange multipliers (e , v ) ∈ E × V which satisfy ∇z L(¯ z , e∗ , v ∗ ) = 0, e∗ ∈ N (K, G(¯ z )) Lemma 2.1.4 Suppose that the assumptions (H 2.1) − (H 2.3) are fulfilled and z¯ is a locally optimal solution of (2.5)–(2.7) Then Λ(¯ z ) is nonempty and bounded In addition, if (H 2.4) is fulfilled then Λ(¯ z ) is singleton When K is polyhedric at G(¯ z ), we have the following result Lemma 2.1.5 Suppose that the assumptions (H 2.1)–(H 2.4) are fulfilled and let z¯ be a locally optimal solution of problem (2.5)–(2.7) Then, the set of critical directions C (¯ z ) satisfies C (¯ z ) = {d ∈ Z | ∇F (¯ z )d = 0, ∇H (¯ z )d = 0, ∇G(¯ z )d ∈ T (K, G(¯ z ))} In addition, if K is polyhedric at G(¯ z ) then C (¯ z ) = C0 (¯ z ), where C0 (¯ z ) := (∇F (¯ z ))⊥ ∩ Ker∇H (¯ z ) ∩ ∇G(¯ z )−1 (cone(K − G(¯ z ))) 13 Theorem 2.1.7 Let z¯ be a locally optimal solution of problem (2.5)-(2.7) Suppose that assumptions (H 2.1)–(H 2.4) are fulfilled and K is polyhedric at G(¯ z ) Then there exists (e∗ , v ∗ ) ∈ Λ(¯ z ) such that ∇2zz L(¯ z , e∗ , v ∗ )(d, d) = ∇2 F (¯ z )d2 + e∗ , ∇2 G(¯ z )d2 + v ∗ , ∇2 H (¯ z )d2 ≥ for all d ∈ C (¯ z ) 2.2 Second-order necessary optimality conditions Definition 2.2.1 An admissible couple (¯ y, u ¯) is said to be a locally optimal solution of (BP ) if there exists > such that for all admissible couples (y, u) satisfying y − y¯ W 1,r (Ω) + u − u ¯ Lq (Γ) ≤ , one has F (y, u) ≥ F (¯ y, u ¯) Let us impose some assumptions for problem (BP ) which involve (¯ y, u ¯) (A2.1) L : Ω × R → R is a Carath´eodory function of class C with respect to second variable, L(x, 0) ∈ L1 (Ω) and for each M > 0, there exists a positive number kLM such that |Ly (x, y )| + |Lyy (x, y )| ≤ kLM , |Ly (x, y1 ) − Ly (x, y2 )| + Lyy (x, y1 ) − Lyy (x, y2 ) ≤ kLM |y1 − y2 | for a.e x ∈ Ω, for all y, yi ∈ R with |y|, |yi | ≤ M , i = 1, (A2.2) : Γ × R × R → R is a Carath´eodory function of class C with respect to variable (y, u), (x, 0, 0) ∈ L1 (Γ) and for each M > 0, there exist a positive number k M and a function rM ∈ L∞ (Γ) such that | y (x, y, u)| + | u (x, y, u)| ≤ k M |y| + |u|q−1 + rM (x), | y (x, y1 , u1 ) − y (x, y2 , u2 )| ≤ k M (|y1 − y2 | + |u1 − u2 |), | u (x, y1 , u1 ) − u (x, y2 , u2 )| ≤ k M |y1 − y2 | + |u1 − u2 |q−1−j |u2 |j , j≥0, q−1−j>0 yy (x, y1 , u1 ) − yy (x, y2 , u2 ) ≤ k M (|y1 − y2 | + |u1 − u2 |), yu (x, y1 , u1 ) − yu (x, y2 , u2 ) ≤ k M (|y1 − y2 | + εq |u1 − u2 |q−1 ), and | uu (x, y1 , u1 ) − uu (x, y2 , u2 )| ≤ k M |y1 −y2 | + εq j≥0, q−2−j>0 |u1 −u2 |q−2−j |u2 |j for a.e x ∈ Γ, for all y, ui , yi ∈ R satisfying |y|, |yi | ≤ M , i = 1, and εq = if < q ≤ and εq = if q > (A2.3) h : Ω × R → R is a Carath´eodory function and of class C w.r.t the second variable, and satisfies the following property: h(·, 0) ∈ LN r/(N +r) (Ω), hy (x, y ) ≥ 14 a.e x ∈ Ω and for each M > 0, there exists a constant Ch,M > such that hy (x, y ) + hyy (x, y ) ≤ Ch,M , hyy (x, y1 ) − hyy (x, y2 ) ≤ Ch,M |y2 − y1 | for a.e x ∈ Ω and |y|, |y1 |, |y2 | ≤ M (A2.4) g : Γ × R → R is a Carath´eodory function and of class C w.r.t the second variable, g (·, 0) ∈ Lq (Γ) a.e x ∈ Γ and for each M > 0, there exists a constant Cg,M > such that gy (x, y ) + gyy (x, y ) ≤ Cg,M , gyy (x, y1 ) − gyy (x, y2 ) ≤ Cg,M |y2 − y1 | for a.e x ∈ Γ and |y|, |y1 |, |y2 | ≤ M (A2.5) b0 + gy [x] ≥ a.e x ∈ Γ Let us define the mappings H : Z → V, H (z ) = H (y, u) := (Ay + h(x, y ); ∂ν y + b0 y − u), G : Z → E, G(z ) = G(y, u) := g (., y ) + u, and set K := {v ∈ Lq (Γ) : a(x) ≤ v (x) ≤ b(x) a.e x ∈ Γ} Then problem (BP ) reduces to the following problem: s.t Min F (z ) (2.14) H (z ) = 0, (2.15) G(z ) ∈ K (2.16) Denote by Φq := Q ∩ G−1 (K ) the admissible set of problem (2.14)-(2.16), where Q := {z = (y, u) ∈ Z | H (z ) = 0} We now use Theorem 2.1.6 to derive second-order necessary optimality conditions for problem (BP ) For this we have to show that under assumptions (A2.1)–(A2.4) all of hypotheses (H 2.1)-(H 2.4) are satisfied Lemma 2.2.2 Suppose that assumptions (A2.1)–(A2.4) are fulfilled Then F, H and G are of class C Lemma 2.2.3 Under assumption (A2.3), ∇y H (ˆ y, u ˆ) is bijective for all (ˆ y, u ˆ) ∈ Z Lemma 2.2.4 Suppose that assumptions (A2.3)–(A2.5) are fulfilled Then, the following assertions are valid: (i) (the regularity condition) for some constant δ > 0, one has 0∈ z )(T (Q, z ) ∩ BZ ) − (K − G(¯ z )) ∩ BE ] [∇G(¯ z∈BZ (¯ z ,δ)∩Q (ii) ∇G(¯ z )(T (Q, z¯)) = Lq (Γ) 15 (2.18) From Lemmas 2.2.3 and 2.2.4, we see that hypotheses (H 2.2)-(H 2.4) are valid Let us introduce the Lagrangian associated with problem (BP ) L(z, ψ, v ∗ ) =F (z ) + v ∗ H (z ) + ψG(z ) N L(·, y )dx + = Ω (·, y, u)dσ + Γ h(·, y )v1 dx − + Ω aij (·)Di yDj v1 + a0 (·)yv1 dx Ω (∂ν y + b0 y − u)v2 dσ + ∂ν yv1 dσ + Γ i,j=1 Γ (g (·, y ) + u)ψdσ, Γ where v ∗ = (v1 , v2 ) ∈ V ∗ = W 1,r (Ω) × W r ,r (Γ), ψ ∈ Lq (Γ)∗ = Lq (Γ) Here, we use the fact (X × Y )∗ = X ∗ × Y ∗ In case of v1 = φ, v2 = T φ, we denote L(z, ψ, φ) := L(z, ψ, v ∗ ) = (·, y, u)dσ + L(x, y )dx + Ω Γ h(·, y )φdx Ω N aij (·)Di yDj φ + a0 (·)yφ dx + + Ω (b0 y − u)T φdσ + Γ i,j=1 (g (·, y ) + u)ψdσ, Γ Let us consider the set-valued map K : Γ ⇒ R, defined by K(x) = [a(x), b(x)] a.e x in Γ Then K = {v ∈ Lq (Γ) | v (x) ∈ K(x) a.e x ∈ Γ} Let us set Γa = {x ∈ Γ | G(¯ z )(x) = g (x, y¯(x)) + u ¯(x) = a(x)}, Γb = {x ∈ Γ | G(¯ z )(x) = g (x, y¯(x)) + u ¯(x) = b(x)} Definition 2.2.6 A pair z = (y, u) ∈ W 1,r (Ω) × Lq (Γ) is said to be a critical direction for problem (BP ) at z¯ = (¯ y, u ¯) if the following conditions hold: (i) ∇F z )z = Ω (Ly [x]y (x)dx + Γ ( y [x]y (x) + u [x]u(x)) dσ ≤ 0;  (¯ − N D (a (·)D y ) + a (·)y + h [·]y = in Ω, i y i,j=1 j ij (ii) ∂ν y + b0 y = u on Γ;  ≥ a.e x ∈ Γ , a (iii) gy [x]y (x) + u(x) ≤ a.e x ∈ Γb We shall denote by Cq (¯ z ) the set of such critical directions Theorem 2.2.7 Suppose that assumptions (A2.1)-(A2.5) are fulfilled and z¯ is a local optimal solution of problem (BP ) There exists a unique couple (φ, ψ ) ∈ W 1,r (Ω) × Lq (Γ) with r ∈ (1, NN−1 ) such that the following hold: (i) The adjoint equation:  A∗ φ + h [·]φ = −L [·] in Ω, y y (2.21) ∂ν ∗ φ + b0 φ = − y [·] − gy [·]∗ ψ on Γ, A 16 where A∗ is the formal adjoint operator to A, and N ∂νA∗ φ = aij (x)Dj φ(x)νi (x); i,j=1 (ii) The stationary conditions in u: ∇u L(¯ z , ψ, φ) = u [·] −φ+ψ =0 on Γ; (iii) The complement condition with ψ :     ≤ a.e x ∈ Γa , ψ (x) ≥ a.e x ∈ Γb ,    = otherwise; (2.22) (2.23) (iv) The second-order nonnegativity condition: ∇2zz L(¯ z , ψ, φ)(y, u)2 ≥ ∀z = (y, u) ∈ Cq (¯ z ), where ∇2zz L(¯ z , ψ, φ)(y, u)2 = Lyy [x]y (x)2 + φhyy [x]y (x)2 dx Ω yy [x]y (x) + +2 yu [x]y (x)u(x) + uu [x]u(x) + ψ (x)gyy [x]y (x)2 dσ Γ 2.3 Second-order sufficient optimality conditions We consider (BP ) for the case p = and the objective function has the form F (y, u) := [ϕ(x, y (x)) + α(x)u(x) + β (x)u2 (x)]dσ, L(x, y (x))dx + Ω (2.30) Γ where ϕ : Γ × R → R is a Carath´eodory function and α, β ∈ L∞ (Γ) It is noted that by p > N − 1, we have N = In addition, we need the following assumption (A2.2) The function ϕ satisfies assumption (A2.1) with ϕ substituted for L and Γ substituted for Ω Moreover, there exists γ > such that β (x) ≥ γ for a.e x ∈ Γ Definition 2.3.1 The function F is said to satisfy the quadratic growth condition at z¯ ∈ Φ if there exist > 0, δ > such that F (z ) ≥ F (¯ z ) + δ z − z¯ for all z ∈ Φ2 satisfying z − z¯ Z ≤ 17 Z Theorem 2.3.2 Suppose that assumptions (A2.1), (A2.2) , (A2.3), (A2.4) are fulfilled, N = 2, z¯ ∈ Φ2 , there exist multipliers φ ∈ W 1,r (Ω), ψ ∈ L2 (Γ), r ∈ (1, 2) satisfying conditions (2.21)– (2.23) of Theorem 2.2.7, and ∇2zz L(¯ z , ψ, φ)(d, d) > 0, ∀d ∈ C2 (¯ z ) \ {0} (2.31) Then F satisfies the quadratic growth condition at z¯ In particular, z¯ is a locally optimal solution of problem (BP ) From Theorems 2.2.7 and 2.3.2, we obtain no-gap conditions in this case Remark 2.3.4 The above result holds if the objective function is of the form F (y, u) = [ϕ(x, y (x)) + α(x)u(x) + β (x)u2 (x) + α0 (x)y (x)u(x)]dσ, L(x, y (x))dx + Ω Γ where α0 ∈ L∞ (Γ) Example 2.3.5 illustrates how to use necessary and sufficient optimality conditions to find extremal points In the rest of this section, we shall derive second-order sufficient optimality conditions for problem (BP ) in the case, where F (y, u) is given by (2.30), where α(x) and β (x) may be zero (B 2.1) Function L : Ω × R → R is a Carath´eodory function of class C with respect to the second variable, L(x, 0) ∈ L1 (Ω) and for each M > there are a constant CL,M > and a function LM ∈ L2 (Ω) such that |Ly (x, y )| ≤ LM (x), |Lyy (x, y )| ≤ CL,M , for a.e x ∈ Ω, ∀y ∈ R, |y| ≤ M and for each > 0, there exists δ > such that |Lyy (x, y1 ) − Lyy (x, y2 )| < a.e x ∈ Ω, ∀y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 − y2 | < δ (B 2.2) The function ϕ satisfies assumption (B 2.1), where ϕ and Γ are replaced by L and Ω, respectively (B 2.3) Function h : R → R is of class C satisfying h(x, 0) = 0, hy (x, y ) ≥ for a.e x ∈ Ω, ∀y ∈ R and for every M > there is a constant Ch,M > such that |hy (x, y )| + |hyy (x, y )| ≤ Ch,M for a.e x ∈ Ω, ∀y ∈ R, |y| ≤ M Moreover, for every M > and > 0, there exists a positive number δ > such that |hyy (x, y1 ) − hyy (x, y2 )| < 18 a.e x ∈ Ω, for all y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 − y2 | < δ (B 2.4) Function g : Γ × R → R is a continuous function of class C with respect to the second variable, g (·, 0) ∈ L2 (Γ), and for each M > there is a constant Cg,M > such that |gy (x, y )| + |gyy (x, y )| ≤ Cg,M for a.e x ∈ Γ and ∀y ∈ R with |y| ≤ M Moreover, for every M > and > 0, there exists a positive number δ > such that |gyy (x, y1 ) − gyy (x, y2 )| < for a.e x ∈ Γ and for all y1 , y2 ∈ R with |y1 |, |y2 | ≤ M, |y1 − y2 | < δ (B 2.5) α, β ∈ L∞ (Γ) and β (x) ≥ for a.e x ∈ Γ In addition, the following is verified βa ∈ L∞ (Γa ), βb ∈ L∞ (Γb ) In what follows, we define · ∗ := · L2 (Ω) + · L2 (Γ) Associated with z¯, for each τ ≥ we define the following critical cone by C2τ (¯ z ) = z = (y, u) ∈ Z1 | z satisfies (C 2.1) − (C 2.3) (C 2.1) (C 2.2) (C 2.3) Ly [x]y (x)dx + (ϕy [x]y (x) + αu(x) + 2β u¯u(x)) dσ ≤ τ y ∗ , Γ Ω N − i,j=1 Dj (aij (·)Di y ) + a0 (·)y + hy [·]y = ∂ν y + b0 y = u  ≥ if x ∈ Γ , a gy [x]y (x) + u(x) ≤ if x ∈ Γb in Ω, on Γ, Obviously, C2 (¯ z ) ⊂ C2τ (¯ z ) for all τ ≥ Theorem 2.3.6 Suppose that N = 2, z¯ ∈ Φ∗ and assumption (A2.5) and assumptions (B 2.1)–(B 2.5) are fulfilled and that there exist multipliers ψ ∈ L2 (Γ) and φ ∈ W 1,r (Ω), s ∈ (1, 2) satisfying conditions (2.21)–(2.23), and positive constants γ, τ > such that ∇zz L(¯ z , ψ, φ)(z, z ) ≥ γ y ∗ ∀z = (y, u) ∈ C2τ (¯ z ) Then there are constants ρ, ε > such that F (z ) ≥ F (¯ z ) + ε y − y¯ ∗ ∀z = (y, u) ∈ BZ1 (¯ z , ρ) ∩ Φ∗ , where Φ∗ := {(y, u) ∈ Z1 | (y, u) satisfies (2.2) and (2.3)} 19 Chapter Upper semicontinuity and continuity of the solution map to a parametric boundary control problem Let Ω be a bounded domain in R2 with the boundary Γ of class C 1,1 We consider the following parametric elliptic optimal control problem (P ) Determine a control ¯ which function u ∈ L2 (Γ) and a corresponding state function y ∈ H (Ω) ∩ C (Ω), minimize the cost function L(x, y (x), µ(1) (x))dx + F (y, u, µ) = Ω (x, y (x), u(x), µ(2) (x))dσ, (3.1) Γ subject to  Ay + f (x, y ) = ∂ν y = u + λ(1) in Ω, (3.2) on Γ, a(x) ≤ g (x, y ) + u(x) + λ(2) ≤ b(x) a.e x ∈ Γ, (3.3) where L : Ω × R × R → R, l : Γ × R × R × R → R, f : Ω × R → R and g : Γ × R → R are functions, a, b ∈ L2 (Γ), a(x) < b(x) for a.e x ∈ Γ, (µ, λ) ∈ (L∞ (Ω) × L∞ (Γ)) × (L2 (Γ))2 is a vector of parameters with µ = (µ(1) , µ(2) ) and λ = (λ(1) , λ(2) ) The second-order elliptic operator A is defined as in Chapter with N = Let us put ¯ , U := L2 (Γ), Π := L∞ (Ω) × L∞ (Γ), Λ := (L2 (Γ))2 Y := H (Ω) ∩ C (Ω) For each λ ∈ Λ, the admissible set Φ(λ) := {(y, u) ∈ Y ×U |(3.2) and (3.3) are satisfied} Then problem (3.1)-(3.3) can be written in the form  F (y, u, µ) → inf, (3.4) P (µ, λ) (y, u) ∈ Φ(λ) Let us denote by S (µ, λ) the solution set of (3.1)-(3.3) or P (µ, λ) corresponding ¯ ) the reference point and by P (¯ ¯ ) the unperturbed problem to (µ, λ), and by (¯ µ, λ µ, λ 3.1 Assumptions and main result ¯ ) ì and Fix ( à, given function h > Notation hz stands for the derivative w.r.t z of a (A3.1) L : Ω ×R×R → R and l : Γ ×R×R×R → R are Carath´eodory functions such that y → L(x, y, µ(1) ) and (y, u) → (x , y, u, µ(2) ) are Fr´echet continuous differential 20 functions for a.e x ∈ Ω and x ∈ Γ, respectively and for all µ(1) , µ(2) ∈ R with |µ(1) − µ ¯(1) (x)| + |µ(2) − µ ¯(2) (x )| ≤ Furthermore, for each M > there exist CLM , ClM > and r1M ∈ L1 (Ω), r2M ∈ L1 (Γ), r3M ∈ L∞ (Ω), r4M ∈ L∞ (Γ) such that |L(x, y, µ(1) )| ≤ r1M (x), |Ly (x, y, µ(1) )| ≤ r3M (x), | (x , y, u, µ(2) )| ≤ r2M (x ) + ClM (1 + |u|2 ), |Ly (x, y1 , µ(1) ) − Ly (x, y2 , µ(1) )| ≤ CLM |y1 − y2 |, | y (x , y, u, µ(2) )|+| u (x , y, u, µ(2) )| ≤ ClM (|y| + |u|) + r4M (x ), | y (x , y1 , u1 , µ(2) )− y (x , y2 , u2 , µ(2) )| ≤ ClM (|y1 − y2 | + δ|u1 − u2 |), | u (x , y1 , u1 , µ(2) )− u (x , y2 , u2 , µ(2) )| ≤ ClM |y1 − y2 | + |u1 − u2 | for some δ ≥ 0, a.e x ∈ Ω, x ∈ Γ, for all µ(1) , µ(2) , y, ui , yi ∈ R satisfying |µ(1) − µ ¯(1) (x)| + |µ(2) − µ ¯(2) (x )| ≤ and |y|, |yi | ≤ M , i = 1, (A3.2) The function u → (x , y, u, µ(2) ) is convex for all (x , y, µ(2) ) ∈ Γ × R2 and |µ(2) − µ ¯(2) (x )| ≤ Moreover, for each M > there exist functions aM ∈ L2 (Γ) and bM ∈ L1 (Γ) satisfying (x , y, u, µ(2) ) ≥ aM (x )u + bM (x ) for a.e x ∈ Γ and for all y, u, µ(2) ∈ R with |y| ≤ M and |µ(2) − µ ¯(2) (x )| ≤ (A3.3) There exist a continuous function η : Γ × R3 → R and constants ≤ θ ≤ 1, α, β > such that | u (x , y, u, µ(2) ) − u (x u (x , y, u, µ ¯(2) (x )| ≤ η (x , |y|, |µ(2) |, |µ ¯(2) (x )|)|u|θ |µ(2) − µ ¯(2) (x )|α , , y¯(x ), u, µ ¯(2) (x )) − u (x , y¯(x ), u ¯, µ ¯(2) (x )), u − u¯(x ) ≥ β|u − u¯(x )|2 for a.e x ∈ Γ, for all y, u, µ(2) ∈ R with |µ(2) − µ ¯(2) (x )| ≤ (A3.4) f : Ω × R → R and g : Γ × R → R are Carath´eodory functions of class C w.r.t the second variable and satisfy the following properties: f (·, 0) = 0, fy (x, y ) ≥ a.e x ∈ Ω, g (·, 0) = 0, gy (x , y ) ≥ a.e x ∈ Γ and for each M > 0, there exist constants Cf M , CgM > such that fy (x, y ) ≤ Cf M , fy (x, y1 ) − fy (x, y2 ) ≤ Cf M |y1 − y2 |, gy (x , y ) ≤ CgM , gy (x , y1 ) − gy (x y2 ) ≤ CgM |y1 − y2 |, for a.e x ∈ Ω, x ∈ Γ and for all y, y1 , y2 ∈ R with |y|, |y1 |, |y2 | ≤ M We are now ready to state our main result of this chapter Theorem 3.1.1 Suppose that assumptions (A3.1)–(A3.4) are fulfilled Then the following assertions are valid: 21 (i) S (µ, ) = for all (à, ) ì Λ; ¯ ); (ii) S : Π × Λ → Y ì U is upper semicontinuous at ( à, ¯ ) is singleton then S (·, ·) is continuous at (¯ ¯ ) (iii) if, in addition, S (¯ µ, λ µ, λ 3.2 Some auxiliary results 3.2.1 Some properties of the admissible set Lemma 3.2.2 Under assumption (A3.4), the admissible set Φ(λ) is a nonempty and closed set for any λ ∈ Λ Lemma 3.2.3 Suppose that assumption (A3.4) is satisfied and {λn } is a sequence ˆ strongly in Λ Then the following assertions are valid: converging to λ ˆ ), there is a sequence {(yn , un )}, (yn , un ) ∈ Φ(λn ) which con(i) For any (ˆ y, u ˆ) ∈ Φ(λ verges to (ˆ y, u ˆ) strongly in Z (ii) For any sequence {(yn , un )}, (yn , un ) ∈ Φ(λn ), there exist a subsequence {(ynk , unk )} ˆ ) such that and (ˆ y, u ˆ) ∈ Φ(λ ynk → yˆ and 3.2.2 unk u ˆ First-order necessary optimality conditions Lemma 3.2.5 Suppose that (¯ y, u ¯) is a local optimal solution of problem (3.1)–(3.3) Then there exists a unique element φ ∈ H (Ω) such that the following conditions are fulfilled: (i) The adjoint equation:  A∗ φ + f (·, y¯)φ = L (·, y¯, µ(1) ) in Ω, y y ∂n ∗ φ + gy (·, y¯)φ = y (·, y¯, u ¯, µ(2) ) − u (·, y¯, u¯, µ(2) )gy (·, y¯) on Γ, A where A∗ is the formal adjoint operator to A, that is, ∗ A y (x) = − Di (aij (x)Dj y (x)) + a0 (x)y (x); i,j=1 (ii) The weak minimum principle: (φ(x)+ u (x, y¯(x), u¯(x), µ(2) (x)))(g (x, y¯(x)) + u¯(x) + λ(2) (x)) = v∈[a(x),b(x)] (φ(x) + ¯(x), u¯(x), µ(2) (x)))v u (x, y 22 a.e x ∈ Γ (3.14) Let us define K := {v ∈ L2 (Γ)| a(x) ≤ v (x) ≤ b(x) a.e x ∈ Γ} (3.15) Then from (3.14), we get (φ(x) + ¯(x), u¯(x), µ(2) (x)))(v (x) u (x, y − g (x, y¯) − u ¯(x)) − λ(2) (x))dσ ≥ 0, Γ ¯ that for all v ∈ K Furthermore, it follows from assumption (A3.1) and y¯ ∈ C (Ω) Ly (·, y¯, µ(1) ) ∈ L∞ (Ω), y (·, y¯, u ¯, µ(2) ) − u (·, y¯, u¯, µ(2) )gy (·, y¯) ∈ L2 (Γ) Therefore, we ¯ get φ ∈ H (Ω) ∩ C (Ω) 3.3 Proof of the main result (i) The non-emptiness of S (µ, λ) (ii) Upper semicontinuity of S (·, ·) ¯ ) We argue by contradiction Assume that S (·, ·) is not upper semicontinuous at (¯ µ, λ Then there exist open sets W1 in Y , W2 in U and sequences {(àn , n )} ì , {(yn , un )} ⊂ Y × U such that   ) W1 ì W2 , à,   S (¯ ¯ ), (3.16) (µn , λn ) → (¯ µ, λ µn − µ Π ≤ ,     (yn , un ) S (àn , n ) \ (W1 ì W2 ), ∀n ≥ By Lemma 3.2.3, we can assume after choosing a subsequence that yn → y¯ in Y and un u ¯ in U ¯ ) If we can show that (¯ ¯ ) and un → u¯ strongly for some (¯ y, u ¯) ∈ Φ(λ y, u ¯) ∈ S (¯ µ, λ in U as n → +∞ then (yn , un ) ∈ W1 × W2 for n large enough This contradicts to (3.16) and the proof is complete (iii) The continuity of S (·, ·) 3.4 Examples In this section, we give some examples illustrating Theorem 3.1.1 One shows that ¯ ) is singleton and the solution map S (·, ·) is continuous at (¯ ¯ ) Other says S (¯ µ, λ µ, λ that although the unperturbed problem has a unique solution, the perturbed problems may have several solutions and solution map is continuous at a reference point 23 General Conclutions The main results of this dissertation include: Second-order necessary optimality conditions for boundary control problems with mixed pointwise constraints No-gap second-order optimality conditions for distributed control problems and boundary control problems governed by semilinear elliptic equations with mixed pointwise constraints, where objective functions are quadratic forms in the control variables Second-order sufficient optimality conditions for distributed control problems and boundary control problems governed by semilinear elliptic equations with mixed pointwise constraints when objective functions may not depend on the control variables A set of conditions for a parametric boundary optimal control problems in twodimensional space, under which the solution map is upper semicontinuous and continuous in parameters in the case where the objective function is not convex, and the admissible set is not convex Some open problems related to the dissertation are continued to study: Second-order optimality conditions and stability of solutions for optimal control problems governed by partial differential equations It is an ongoing topic No-gap second-order optimality conditions for optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints and pure state constraints The role of second-order sufficient conditions in the study of solution stability for boundary optimal control problems 24 LIST OF PUBLICATIONS N H Son, B T Kien and A Rosch (2016), Second-order optimality conditions for boundary control problems with mixed pointwise constraints, SIAM J.Optim., 26, pp 1912-1943 B T Kien, V H Nhu and N H Son (2017), Second-order optimality conditions for a semilinear elliptic optimal control problem with mixed pointwise constraints, Set-Valued Var Anal., 25, pp 177-210 N H Son (2017), On the semicontinuity of the solution map to a parametric boundary control problem, Optimization, 66, pp 311-329 ... Sc Vu Ngoc Phat Reviewer 2: Assoc Prof Dr Cung The Anh Reviewer 3: Dr Nguyen Huy Chieu The dissertation will be defended before approval committee at Hanoi Univesity of Science and Technology... conditions for optimal control problems Better understanding of second-order optimality conditions for optimal control problems governed by semilinear elliptic equations is an ongoing topic of... of no-gap second-order optimality conditions for optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints Solving problem (OP 3) is the second goal of