Điều kiện tối ưu không cách biệt và tính ổn định nghiệm của các bài toán điều khiển tối ưu được cho bởi các phương trình elliptic nửa tuyến tính

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 DOCTORAL DISSERTATION OF MATHEMATICS Hanoi - 2019 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 DOCTORAL DISSERTATION OF MATHEMATICS SUPERVISORS: 1 Dr Nguyen Thi Toan 2 Dr Bui Trong Kien Hanoi - 2019 COMMITTAL IN THE DISSERTATION I assure that my scientific results are new and righteous Before I published these results, there had been no such results in any scientific document I have responsibilities for my research results in the dissertation Hanoi, April 3rd , 2019 On behalf of Supervisors Author Dr Nguyen Thi Toan Nguyen Hai Son i ACKNOWLEDGEMENTS This dissertation has been carried out at the Department of Fundamental Mathematics, School of Applied Mathematics and Informatics, Hanoi University of Science and Technology It has been completed under the supervision of Dr Nguyen Thi Toan and Dr Bui Trong Kien First of all, I would like to express my deep gratitude to Dr Nguyen Thi Toan and Dr Bui Trong Kien for their careful, patient and effective supervision I am very lucky to have a chance to work with them, who are excellent researchers I would like to thank Prof Jen-Chih Yao for his support during the time I visited and studied at Department of Applied Mathematics, Sun Yat-Sen University, Kaohsiung, Taiwan (from April, 2015 to June, 2015 and from July, 2016 to September, 2016) I would like to express my gratitude to Prof Nguyen Dong Yen for his encouragement and many valuable comments I would also like to especially thank my friend, Dr Vu Huu Nhu for kind help and encouragement I would like to thank the Steering Committee of Hanoi University of Science and Technology (HUST), and School of Applied Mathematics and Informatics (SAMI) for their constant support and help I would like to thank all the members of SAMI for their encouragement and help I am so much indebted to my parents and my brother for their support I thank my wife for her love and encouragement This dissertation is a meaningful gift for them Hanoi, April 3rd , 2019 Nguyen Hai Son ii CONTENTS COMMITTAL IN THE DISSERTATION AC K CO NT TA BL IN TR i Chapter 0 8 PRELIMINARIES AND AUXILIARY RESULTS 0.1 Variational analysis 8 0.1.1 Set-valued maps 8 0.1.2 Tangent and normal cones 9 0.2 Sobolev spaces and elliptic equations 13 0.2.1 Sobolev spaces 13 0.2.2 Semilinear elliptic equations 20 0.3 Conclusions 24 Chapter 1 PR 2 OB 5 1.1 2 NO-GAP OPTIMALITY CONDITIONS FOR DISTRIBUTED CONTROL Se 1 62 1.1.2 1 6 2 Se7 1.2 4 Se 1.3 05 Co 7 Chapter 2 NO-GAP OPTIMALITY CONDITIONS FOR BOUNDARY CONTROL 58 PROBLEMS 2.1 A 2.2 bs Se co 2.3 Se 2.4 co C on Chapter 3 UPPER SEMICONTINUITY AND 5 69 67 85 9 CONTINUITY OF THE SOLUTION MAP TO A PARAMETRIC BOUNDARY CONTROL PROBLEM 91 3.1 Assumptions and main result 92 94 3.2 Some auxiliary results 3 3.Some properties of the admissible set 2 3.First-order necessary optimality conditions 3.32 Proof of the main result 100 3.4 Examp 3.5 Concl GENE REF ERE RAL LIST OF 4 TABLE OF NOTATIONS N := {1, 2, } R set of real numbers |x| absolute value of x ∈ R R n set of positive natural numbers n-dimensional Euclidean vector space ∅ empty set x∈A x is in A x ∈/ A x is not in A A ⊂ B(B ⊃ A) A is a subset of B A* B A is not a subset of B A∩B A∪B A\B intersection of the sets A and B union of the sets A and B set difference of A and B A×B Descartes product of the sets A and B [x1 , x2 ] the closed line segment between x1 and x2 kxk norm of a vector x kxkX norm of vector x in the space X X ∗ X ∗∗ topological dual of a normed space X topological bi-dual of a normed space X hx∗ , xi hx, yi canonical pairing canonical inner product B(x, δ) open ball with centered at x and radius δ B(x, δ) closed ball with centered at x and radius δ BX open unit ball in a normed space X BX dist(x; Ω) closed unit ball in a normed space X distance from x to Ω {xk } sequence of vectors xk xk → x xk converges strongly to x (in norm topology) xk * x xk converges weakly to x ∀x for all x ∃x there exists x A := B A is defined by B f :X → Y function from X to Y f 0 (x), ∇f (x) 00 2 f (x), ∇ f (x) Fr´echet derivative of f at x Fr´echet second-order derivative of f at x 1 Lx , ∇x L Fr´echet derivative of L in x 2 2 Lxy , ∇xy L ϕ : X → IR function domϕ Fr´echet second-order derivative of L in xand y extended-real-valued effective domain of ϕ epiϕ epigraph of ϕ suppϕ the support of ϕ F :X ⇒ Y multifunction from X to Y domF rgeF domain of F range of F gphF kerF graph of F kernel of F T (K, x) Bouligand tangent cone of the set K at x [ adjoint tangent cone of the set K at x 2 second-order Bouligand tangent set of the set T (K, x) T (K, x, d) K at x in direction d 2[ T (K, x, d) second-order adjoint tangent set of the set K at x in direction d N (K, x) normal cone of the set K at x ∂Ω Ω¯ boundary of the domain Ω 0 Ω ⊂⊂ Ω Ω0 ⊂ Ω and Ω0 is compact Lp (Ω) the space of Lebesgue measurable functions f closure of the set Ω and L∞ (Ω) C (Ω¯ )  m,p Ω |f (x)|p dx < +∞ the space of bounded functions almost every Ω the space of continuous functions on Ω¯ ¯) M(Ω  W R the space of finite regular Borel measures m,p (Ω), W0 (Ω), W s,r (Γ), Sobolev spaces H m (Ω), H 0m (Ω) m,p W −m,p (Ω)(p−1 + p0−1 = 1) the dual space of W0 X ,→ Y X is continuous embedded in Y X ,→,→ Y i.e X is compact embedded in Y id est (that is) a.e almost every s.t subject to p 5 page 5 w.r.t ✷ with respect to The proof is complete 0 3 (Ω) INTRODUCTION 1 Motivation Optimal control theory has many applications in economics, mechanics and other fields of science It has been systematically studied and strongly developed since the late 1950s, when two basic principles were made One was the Pontryagin Maximum Principle which provides necessary conditions to find optimal control functions The other was the Bellman Dynamic Programming Principle, a procedure that reduces the search for optimal control functions to finding the solutions of partial differential equations (the Hamilton-Jacobi equations) 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), (see [1, 2, 3]) 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 For instance, J F Bonnans et al [4, 5, 6], studied optimality conditions for optimal control problems governed by ODEs, while J F Bonnans [7], E Casas et al [8, 9, 10, 11, 12, 13, 14, 15, 16, 17], C Meyer and F Troltzsch [18], B T Kien et al [19, 20, 21, 22], A R¨osch and F Tr¨oltzsch [23, 24] derived optimality conditions for optimal control problems governed by el- liptic equations It is known that if u¯ is a local minimum of F , where F : U → R is a differentiable functional and U is a Banach space, then F 0 (u¯) = 0 This a firstorder necessary optimality condition However, it is not a sufficient condition in case of F is not convex Therefore, we have to invoke other sufficient conditions and should study the second derivative (see [17]) 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 (see [13, 16, 17]) and in studying the stability of optimal control (see [25, 26]) 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 Example 3.4.1 Let Ω be the open unit ball in R2 with boundary Γ We consider the following problem Z F (y, u, µ) = 3 Z 4y(x) + t(x)y(x) + t(x)y (x) + µ1 (x) dx Ω + s.t 1 [ (u(x) + y(x) + µ2 (x))2 − 4y(x) − 2u(x)]dσ → inf Γ 2  −∆y + y + y3 = 0 ∂ν y = u + λ1 in Ω, on Γ, − 1 ≤ y(x) + u(x) + λ2 (x) ≤ 0 a.e x ∈ Γ, where t(x) := 1 − x2 − x2 , x = (x1 , x2 ) ∈ R2 Here we suppose that µ ≡ 0, λ¯ ≡ 0 We 1 2 have L(x, y, µ1 ) = 4y + t(x)y + t(x)y 3 + µ1 , 1 `(x, y, u, µ2 ) = (u + y + 2 − 4y − 2u, 2 µ2 ) Ly (x, y, µ1 ) = 4 + t(x) + 3t(x)y 2 , `y (x, y, u, µ2 ) = u + y + µ2 − 4, 2, f (x, y) = y 3 , g(x, y) = y, `u (x, y, u, µ2 ) = u + y + µ2 − fy (x, y) = 3y 2 , gy (x, y) = 1 Notice that F is convex in u and is not convex in y Firstly, we show that assumptions (A3.1)–(A3.4) are fulfilled In fact, it is easy to check that (A3.2) and (A3.4) are satisfied For assumption (A3.1), we have |L(x, y, µ1 )| ≤ 4|y| + |y| + |y 3 | + |µ1 | ≤ 5M + M 3 + 0 , 1 |`(x0 , y, u, µ2 )| ≤ 2(|u|2 + |y + µ |2) + 4|y| + 1 + |u|2 2 2 ≤ 2|u|2 + (M + 0 )2 + 4M + 1, |Ly (x, y, µ1 )| ≤ 5 + 3M 2 , |Ly (x, y1 , µ1 ) − Ly (x, y2 , µ1 )| = |3t(x)(y1 2 − y2 2 )| ≤ 6M |y1 − y2 |, |`y (x0 , y, u, µ2 )| + |`u (x0 , y, u, µ2 )| = |u + y + µ2 − 2| + |u + y + µ2 − 4| ≤ 2(|y| + |u|) + 2 0 + 6, |`y (x0 , y1 , u1 , µ2 ) − `y (x0 , y2 , u2 , µ2 )| = |`u (x0 , y1 , u1 , µ2 ) − `u (x0 , y2 , u2 , µ2 )| = |y1 − y2 | + |u1 − u2 | for a.e x ∈ Ω, x0 ∈ Γ, for all µ2 , y, u, ui , yi ∈ R satisfying |y|, |yi | ≤ M , i = 1, 2 and |µ1 | + |µ2 | ≤ 0 Hence assumption (A3.1) is satisfied For assumption (A3.3), we have |`u (x0 , y, u, µ2 ) − `u (x0 , y, u, µ2 )| = |µ2 − µ2 |, h`u (x, yˆ, u, µ2 ) − `u (x, yˆ, uˆ, µ2 ), u − uˆi = hu − uˆ, u − uˆi = | u − uˆ|2 , for a.e x0 ∈ Γ, for all µ2 , y, u ∈ R satisfying |µ2 | ≤ 0 and (yˆ, uˆ) ∈ S(µ, λ¯ ) Therefore, assumption (A3.3) is fulfilled Now, we suppose that (y¯, u¯) ∈ S(µ, λ¯ ) By Lemma 3.2.5, there exists a function φ ∈ H 1 (Ω) ∩ C (Ω¯ ) such that the following conditions are valid: (i) the adjoint equation:  −∆φ + φ + 3y¯2 φ = 4 + t(x) + 3t(x)y¯2 in Ω, ∂ν φ + φ = −2 on (3.33) Γ, (ii) the weakly minimum principle: (φ(x0 ) + u¯(x0 ) + y¯(x0 ) − 2)(v − y¯(x0 ) − u¯(x0 )) ≥ 0 a.e x0 ∈ Γ (3.34) for all v ∈ [−1, 0] We notice that ∆t(x) = −4 and t(x) = 0, ∂ν t(x) = −2(x2 + x2 ) = −2 ∀x ∈ Γ Hence 1 2 φ = t(x) is a unique solution of the adjoint equation (3.33) This implies that φ(x) = 0 a.e x ∈ Γ From (3.34), for a.e x ∈ Γ we have (u¯(x0 ) + y¯(x0 ) − 2)(v − y¯(x0 ) − u¯(x0 )) ≥ 0 a.e x0 ∈ Γ, ∀v ∈ [−1, 0] Since y¯(x0 ) + u¯(x0 ) ∈ [−1, 0] a.e x0 ∈ Γ, we get y¯(x0 ) + u¯(x0 ) − 2 < 0 a.e x0 ∈ Γ Hence v − y¯(x0 ) − u¯(x0 ) ≤ 0 ∀v ∈ [−1, 0], a.e x0 ∈ Γ This implies that y¯(x0 ) + u¯(x0 ) = 0 a.e x0 ∈ Γ and so y¯ + u¯ = 0 on Γ Consequently, y¯ is a solution of the following equation  −∆y + y + y 3 = 0 ∂ν y + y = 0 in Ω, on Γ By the uniqueness, we have y¯ = 0 and so u¯ = 0 Hence S(µ, λ¯ ) = {(0, 0)} Moreover, by Theorem 3.1.2, S(µ, λ) is continuous at (0, 0) The following example says 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 Example 3.4.2 Let Ω be the open unit ball in R2 with boundary Γ We consider problem P (µ, λ) of finding u ∈ L2 (Γ) and y ∈ H 1 (Ω) ∩ C (Ω¯ ) such that Z F (y, u, µ) = `(x, y(x), u(x), µ(x))dσ → inf, Γ subject to the state equation  −∆y + y + y 3 = 0 ∂ν y =u+λ in Ω, on Γ and pointwise constraint 0 ≤ u(x) + y(x) ≤ 1 a.e x ∈ Γ, where ` is given by 1 1 `(x, y, u, µ) = [1 sign(u + y + µ2 )](u + y + µ2 )2 [1 + sign(u + y − µ2 )](u + y − 2 2 +− 2µ ) 2 Here sign(v) is defined by sign(v) =  1 if v > 0, 0 if v = 0,  −1 if v < 0 It is obvious that F (y, u, µ) ≥ 0 Let us take µ = 0, λ¯ = 0 We shall show that the following assertions are valid: (i) P (µ, λ¯ ) has unique solution (y¯, u¯) = (0, 0) (ii) P (µ, λ) satisfies assumptions (A3.1)– (A3.4) (iii) If 0 ≤ |µ(x)| ≤ 1 then S(µ, λ) ⊃ {(y(µ, λ), sµ2 − y(µ, λ)), 0 < s < 1}, where y(µ, λ) is a solution of the equation  −∆y + y + y 3 = 0 ∂ν y + y = sµ2 + λ In fact, when µ = 0, in Ω, on Γ (3.35) λ¯ = 0, problem P (µ, λ¯ ) becomes Z (y(x) + u(x))2 dσ → inf F (y, u, µ) = Γ with constraints  −∆y + y + y 3 = 0 ∂ν y =u in Ω, on Γ and 0 ≤ y(x) + u(x) ≤ 1 a.e x ∈ Γ Obviously, S(0, 0) = {(0, 0)} We now prove that P (µ, λ) satisfies (A3.1)–(A3.4) It is easily seen that `(x, y, u, µ) =  2 2  (u + y + µ ) if u + y < −µ2 , 0 if − µ2 ≤ u + y ≤ µ2 , (u + y − µ2 )2 if u + y > µ2   Hence  2(u + y + µ2) if u + y < −µ2 ,  `y (x, y, u, µ) = `u (x, y, u, µ) = if − µ2 ≤ u + y ≤ µ2 , 0  2(u + y − µ2 ) if u + y > µ2 It is clear that assumptions (A3.2) and (A3.4) hold For assumption (A3.1), we have |`(x, y, u, µ)| ≤ (u + y + µ2 )2 + (u + y − µ2 )2 ≤ 2[u2 + (y + µ2 )2 + u2 + (y − µ2 )2 ] ≤ 4u2 + 4y 2 + 4µ4 ≤ 4u2 + 4M 2 + 40 4 , |`u (x, y, u, µ) + `y (x, y, u, µ)| ≤ 2|u + y + µ2 | + 2|u + y − µ2 | ≤ 4(|u| + |y|) + 4µ2 ≤ 4(|u| + |y|) + 40 2 for a.e x ∈ Γ, for all y, µ ∈ R satisfying |y| ≤ M, |µ| ≤ 0 For any y1 , y2 , u1 , u2 ∈ R, let us put T1 := `u (x, y1 , u1 , µ) − `u (x, y2 , u2 , µ) = `y (x, y1 , u1 , µ) − `y (x, y2 , u2 , µ) In case of y1 + u1 ≤ −µ2 , we get  |2(y1 + u1 + µ2) − 2(y2 + u2 + µ2)|  |T1 | = = if − µ2 ≤ y2 + u2 ≤ µ2 , |2(y1 + u1 + µ2 ) − 2(y2 + u2 − µ2 )| if y2 + u2 > µ2  ≤ ≤ ≤ 2 | y1 + u1 + µ2 |  y2 + u2 < −µ2 , |2(y1 + u1 + µ2 )|   2|(y 1 − y2 ) + (u1 − u2 )| 2 if if y2 + u < −µ2 , if − µ2 ≤ y2 + u2 ≤ µ2 , 2|(y1 − y2 ) + (u1 − u2 ) + 2µ2 | if y2 + u2 > µ2   2(|y 1 − y2 | + |u1 − u2 |) if y2 + u < −µ2 , 2|y1 + u1 − y2 − u2 | if − µ2 ≤ y2 + u2 ≤ µ2 , if y2 + u2 > µ2 2  2(|y1 − y2 | + |u1 − u2 |) + 4µ2   2(|y 1 − y2 | + |u1 − u2 |) 2 if y2 + u < −µ2 , if − µ2 ≤ y2 + u2 ≤ µ2 , 2(|y1 − y2 | + |u1 − u2 |)  2(|y1 − y2 | + |u1 − u2 |) + 2|y2 + u2 − y1 − u1 | if y2 + u2 > µ2  if y2 + u < −µ2 ,  2(|y 1 − y2 | + |u1 − u2 |) 2 2(|y1 − y2 | + |u1 − u2 |) if − µ2 ≤ y2 + u2 ≤ µ2 , 4(|y1 − y2 | + |u1 − u2 |) if y2 + u2 > µ2  ≤ 4(|y1 − y2 | + |u1 − u2 |) Similarly, for the other cases, we also have |T1 | ≤ 4(|y1 − y2 | + |u1 − u2 |) Hence assumption (A3.1) is satisfied Notice that `y (x, y, u, 0) = 2(y + u) for all x, y, u, µ Putting T2 := `u (x, y, u, µ) − `u (x, y, u, 0), we get  2  |2(y + u + µ ) − 2(y + u)| |T2 | = = | − 2(y + u)| if y + u < −µ2 , if − µ2 ≤ y + u ≤ µ2 ,  |2(y + u − µ2 ) − 2(y + u)| if  2  if y + u < −µ2 ,  2µ 2 | y + u| 2µ2  − µ2 ≤ y + u ≤ µ2 , if  if y + u > µ2  2µ2 if y + u < −µ2 , ≤ 2µ2 if − µ2 ≤ y + u ≤ µ2 , 2µ2 if y + u > µ2  y + u > µ2 ≤ 2µ2 for all x, u, y, µ On the other hand, we obtain h`u (x, 0, u, 0) − `u (x, 0, 0, 0), u − 0i = h2u, ui = 2|u|2 = 2|u − 0|2 Thus, assumption (A3.3) is fulfilled Finally, if y(µ, λ) is a solution of equation (3.35) and u(µ, λ) = sµ2 − y(µ, λ) then (y(µ, λ), u(µ, λ)) ∈ Φ(λ) Moreover, since y(µ, λ) + u(µ, λ) = sµ2 with 0 < s < 1, we get F (y, u, µ) = 0 This implies that (y(µ, λ), u(µ, λ)) ∈ S(µ, λ) 3.5 Conclusions In this chapter, we present a result on the solution stability of boundary optimal control problems Theorem 3.1.2 provides suitable conditions under which the solution map S(·, ·) is upper semicontinuous and continuous in parameters The obtained result is an answer for (OP 4) and is proved by the direct method, techniques of variational analysis and necessary optimality conditions GENERAL CONCLUSIONS The main results of this dissertation include: 1 Second-order necessary optimality conditions for boundary control problems with mixed pointwise constraints 2 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 3 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 4 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: 1 Second-order optimality conditions and stability of solutions for optimal control problems governed by partial differential equations It is an ongoing topic 2 No-gap second-order optimality conditions for optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints and pure state constraints 3 The role of second-order sufficient conditions in the study of solution stability for boundary optimal control problems LIST OF PUBLICATIONS [1] N H Son, B T Kien and A R¨osch (2016), Second-order optimality conditions for boundary control problems with mixed pointwise constraints, SIAM J Optim., 26, pp 1912–1943 [2] 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 [3] N H Son (2017), On the semicontinuity of the solution map to a parametric boundary control problem, Optimization, 66, pp 311-329 REFERENCES [1] A D Ioffe and V M Tihomirov (1979), Theory of Extremal Problems, NorthHolland Publishing Company, Amsterdam [2] X Li and J Yong (1995), Optimal Control Theory for Infinite Dimensional Systems, Birkh¨auser, Boston [3] F Troltzsch (2010), Optimal Control of Partial Differential Equations: Theory, Method and Applications, AMS, Providence, Rhode Island [4] J F Bonnans and A Hermant (2009), No-gap second-order optimality conditions for optimal control problems with a single state constraint and control, Math Program Ser B, 117, pp 21-50 [5] J F Bonnans and A Hermant (2009), Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints, Ann Inst H Poincar´e - Anal Non Lin´eaire, 26, pp 561–598 [6] J F Bonnans, X Dupuis and L Pfeiffer (2014), Second-order necessary conditions in Pontryagin form for optimal control problems, SIAM J Control Optim., pp 3887-3916 52, [7] J F Bonnans (1998), Second-order analysis for control constrained optimal control problems of semilinear elliptic systems, Appl Math Optim., 38, pp 303–325 [8] E Casas (1993), Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM J Control and Optim., 31, pp 993-1006 [9] E Casas (1994), Pontryagins principle for optimal control problems governed by semilinear elliptic equations, International Series of Numerical Mathematics, Ed Birkhauser Verlad 118, pp 97-114 [10] E Casas and F Tr¨oltzsch (1999), Second-order necessary optimality conditions for some state-constrained control problems of semilinear equations Appl Math Optim., 39, pp 211-227 elliptic [11] E Casas and M Mateos (2002), Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM J Control and Optim., 40, pp 1431–1454 [12] E Casas and F Tr¨oltzsch (2002), Second order necessary and sufficient optimality condition for optimization problems and applications problem, SIAM J Optim., 13, pp 406–431 to control [13] E Casas, J C De Los Reyes and F Tr¨oltzsch (2008), Sufficient second-order opti- mality conditions for semilinear control problems with pointwise state constraints, SIAM J Optim., 19, pp 616–643 [14] E Casas and F Troltzsch (2009), First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations, SIAM J Control and Optim., 48, pp 688–718 [15] E Casas and F Troltzsch (2010), Recent advanced in the analysis of pointwise state-constrained elliptic optimal control problems, ESAIM: Control, Optim Caculus of Variations, 16, pp 581–600 [16] E Casas (2012), Second order analysis for bang-bang control problems of PDEs, SIAM J Control Optim., 50, pp 2355-2372 [17] E Casas and F Troltzsch (2012), Second order analysis for optimal control problems: Improving results expected from abstract theory, SIAM J Optim., 22, pp 266-279 [18] C Meyer and F Troltzsch (2006), On an elliptic optimal control problem with pointwise mixed control-state constraints, in Alberto Seeger, editor, Recent Advances in Optimization, volume 563 of Lecture Notes in Economics and Mathematical Systems, pp 187-204 Springer Berlin Heidelberg [19] B T Kien and V H Nhu (2014), Second-order necessary optimality conditions for a class of semilinear elliptic optimal control problems with mixed pointwise constraints, SIAM J Control and Optim., 52, pp 1166–1202 [20] B T Kien, V H Nhu and A Rosch (2015), Second-order necessary optimality conditions for a class of optimal control problems governed by partial differential equations with pure state constraints, J Optim Theory Appl., 165, pp 30–61 [21] B T Kien, V H Nhu and M M Wong (2015), Necessary optimality conditions for a class of semilinear elliptic optimal control problems with pure state constraints and mixed pointwise constraints, J Nonlinear Convex Anal., 16, pp 1363-1383 [22] V H Nhu, N H Son and J.-C Yao (2017), Second-order necessary optimality conditions for semilinear elliptic optimal control problems, Appl Anal., 96, pp 626-651 [23] A Rosch and F Troltzsch (2006), Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints, SIAM J Optim., 17, pp 776–794 ... 0.2 Sobolev spaces and elliptic equations 13 0.2.1 Sobolev spaces 13 0.2.2 Semilinear elliptic equations 20... spaces, and facts of partial differential equations relating to solutions of linear elliptic equations and semilinear elliptic equations For more details, we refer the reader to [1], [2], [3], [27],... 0.2.1 Sobolev spaces and elliptic equations Sobolev spaces First, we recall some relative concepts and properties which are introduced in many books on Sobolev spaces, elliptic equations and partial