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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: Dr Nguyen Thi Toan 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, June 11th , 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, June 11th , 2019 Nguyen Hai Son ii CONTENTS i ii iii COMMITTAL IN THE DISSERTATION ACKNOWLEDGEMENTS CONTENTS TABLE OF NOTATIONS INTRODUCTION Chapter 1.1 1.2 1.3 PRELIMINARIES AND AUXILIARY RESULTS Variational analysis 1.1.1 Set-valued maps 1.1.2 Tangent and normal cones Sobolev spaces and elliptic equations 13 1.2.1 Sobolev spaces 13 1.2.2 Semilinear elliptic equations 20 Conclusions 24 Chapter NO-GAP OPTIMALITY CONDITIONS FOR DISTRIBUTED CONTROL PROBLEMS 25 2.1 Second-order necessary optimality conditions 26 2.1.1 An abstract optimization problem 26 2.1.2 Second-order necessary optimality conditions for optimal control problem 27 2.2 Second-order sufficient optimality conditions 40 2.3 Conclusions 57 Chapter NO-GAP OPTIMALITY CONDITIONS FOR BOUNDARY CONTROL PROBLEMS 58 3.1 Abstract optimal control problems 59 3.2 Second-order necessary optimality conditions 66 3.3 Second-order sufficient optimality conditions 75 3.4 Conclusions 89 Chapter UPPER SEMICONTINUITY AND CONTINUITY OF THE SOLUTION MAP TO A PARAMETRIC BOUNDARY CONTROL PROBLEM 91 4.1 Assumptions and main result 92 4.2 Some auxiliary results 94 iii 4.2.1 Some properties of the admissible set 94 4.2.2 First-order necessary optimality conditions 98 4.3 Proof of the main result 100 4.4 Examples 104 4.5 Conclusions 109 GENERAL CONCLUSIONS LIST OF PUBLICATIONS 110 111 REFERENCES 112 iv TABLE OF NOTATIONS N := {0, 1, 2, } R R+ set of x ∈ R with x ≥ |x| absolute value of x ∈ R RN N -dimensional Euclidean vector space x x set of natural numbers set of real numbers norm of a vector x norm of vector x in the space X X X∗ topological dual of a normed space X X ∗∗ topological bi-dual of a normed space X x∗ , x canonical pairing x, y 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 closed unit ball in a normed space X dist(x; Ω) distance from x to Ω {xk } sequence of vectors xk xk → x xk converges strongly to x (in norm topology) xk xk converges weakly to x 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 (x), ∇f (x) Fr´echet derivative of f at x f (x), ∇2 f (x) Fr´echet second-order derivative of f at x Lx , ∇x L Fr´echet derivative of L in x Lxy , ∇2xy L Fr´echet second-order derivative of L in x and y ∆ The Laplace operator ϕ : X → IR extended-real-valued function suppϕ the support of ϕ F :X⇒Y multifunction from X to Y domF domain of F rgeF range of F gphF graph of F kerF kernel of F T (K, x) Bouligand tangent cone of the set K at x T (K, x) adjoint tangent cone of the set K at x T (K, x, d) second-order Bouligand tangent set of the set K at x in direction d T2 (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 Ω ⊂⊂ Ω Ω ⊂ Ω and Ω is compact Lp (Ω) space of Lebesgue measurable functions f closure of the set Ω and Ω |f (x)|p dx < +∞ L∞ (Ω) ¯ C(Ω) space of bounded functions almost every Ω ¯ space of continuous functions on Ω C k (Ω) space of k times continuously differentiable functions on Ω, k ≥ C ∞ (Ω) = ∩k≥0 C k (Ω) C 0,α (Ω) = f ∈ C(Ω) : supx,y∈Ω,x=y C k,α (Ω) ¯ M(Ω) = f ∈ C k (Ω) : Dj f ∈ C 0,α (Ω) ∀j, |j| ≤ k ¯ space of finite regular Borel measures on Ω  m,p m,p   W (Ω), W0 (Ω), W s,r (Γ), |f (x)−f (y)| |x−y|α Sobolev spaces    H m (Ω), H0m (Ω) W −m,p (Ω)(p−1 + p −1 = 1) the dual space of W0m,p (Ω) X →Y X is continuously embedded in Y X →→ Y X is compactly embedded in Y i.e id est (that is) a.e almost everywhere s.t subject to p page w.r.t with respect to ✷ The proof is complete This implies that lim In = From (4.24) and (4.26), we n→+∞ have |Jn | ≤ ClM |yn (x) − y¯(x)| |un (x) − u¯(x)|dσ Γ ≤ ClM yn − y¯ |un (x) − u¯(x)|dσ L∞ (Γ) Γ ≤ C yn − y¯ L∞ (Ω ≤ 2CM yn − y¯ un (x) − u¯(x) U Y for some positive constant C Hence, lim Jn = n→+∞ Combining the fact lim In = lim Jn = with (4.30) and (4.31) yields n→+∞ n→+∞ lim sup Kn ≤ (4.32) n→+∞ However, from (4.5) we have β(un (x) − u¯(x))2 dσ = β un − u¯ Kn ≥ L2 (Γ) Γ From this and (4.32), we have lim sup un − u¯ n→+∞ L2 (Γ) = and so un → u in L2 (Γ) The proof of lemma is complete From Lemmas 4.3.1 and 4.3.2, we obtain assertion (ii) of Theorem 1.1 (iii) The continuity of S(·, ·) ¯ In fact, let W1 be It remains to prove that S(·, ·) is lower semicontinuous at ( à, ) W1 ì W2 = ∅ Since an open set in Y and W2 be an open set in U such that S(¯ µ, λ) ¯ is singleton, S(¯ ¯ ⊂ W1 × W2 Hence there exist neighborhoods V1 of µ S(¯ µ, λ) µ, λ) ¯ and ¯ such that S(µ, λ) ⊂ W1 ì W2 for all (à, ) V1 ì V2 It follows from this and V2 of λ the fact S(µ, λ) = ∅ that S(µ, λ) ∩ W1 ì W2 = for all (à, ) V1 × V2 and so S(·, ·) ¯ Therefore, S(·, ·) is continuous at (¯ ¯ The proof is lower semicontinuous at (¯ µ, λ) µ, λ) of Theorem 4.1.2 is complete 4.4 Examples In this section, we give some examples illustrating Theorem 4.1.2 The first example ¯ is singleton and the solution map S(·, ·) is continuous at (¯ ¯ shows that S(¯ µ, λ) µ, λ) 104 Example 4.4.1 Let Ω be the open unit ball in R2 with boundary Γ We consider the following problem 4y(x) + t(x)y(x) + t(x)y (x) + µ1 (x) dx F (y, u, µ) = Ω + s.t [ (u(x) + y(x) + µ2 (x))2 − 4y(x) − 2u(x)]dσ → inf, Γ  −∆y + y + y = in Ω, ∂ν y = u + λ1 on Γ, − ≤ y(x) + u(x) + λ2 (x) ≤ a.e x ∈ Γ, ¯ ≡ We where t(x) := − x21 − x22 , x = (x1 , x2 ) ∈ R2 Here we suppose that µ ¯ ≡ 0, λ have L(x, y, µ1 ) = 4y + t(x)y + t(x)y + µ1 , (x, y, u, µ2 ) = (u + y + µ2 )2 − 4y − 2u, Ly (x, y, µ1 ) = + t(x) + 3t(x)y , y (x, y, u, µ2 ) = u + y + µ2 − 4, f (x, y) = y , g(x, y) = y, u (x, y, u, µ2 ) = u + y + µ2 − 2, fy (x, y) = 3y , gy (x, y) = Note that F is convex in u and is not convex in y Firstly, we show that assumptions (A4.1)–(A4.4) are fulfilled In fact, it is easy to check that (A4.2) and (A4.4) are satisfied For assumption (A4.1), we have |L(x, y, µ1 )| ≤ 4|y| + |y| + |y | + |µ1 | ≤ 5M + M + , | (x , y, u, µ2 )| ≤ 2(|u|2 + |y + µ2 |2 ) + 4|y| + + |u|2 ≤ 2|u|2 + (M + 0) + 4M + 1, |Ly (x, y, µ1 )| ≤ + 3M , |Ly (x, y1 , µ1 ) − Ly (x, y2 , µ1 )| = |3t(x)(y12 − y22 )| ≤ 6M |y1 − y2 |, | y (x , y, u, µ2 )| + | u (x , y, u, µ2 )| = |u + y + µ2 − 2| + |u + y + µ2 − 4| ≤ 2(|y| + |u|) + | y (x , y1 , u1 , µ2 ) − y (x + 6, , y2 , u2 , µ2 )| = | u (x , y1 , u1 , µ2 ) − u (x , y2 , u2 , µ2 )| = |y1 − y2 | + |u1 − u2 | for a.e x ∈ Ω, x ∈ Γ, for all µ2 , y, u, ui , yi ∈ R satisfying |y|, |yi | ≤ M , i = 1, and |µ1 | + |µ2 | ≤ Hence assumption (A4.1) is satisfied For assumption (A4.3), we have | u (x , y, u, µ2 ) − u (x , y, u, µ ¯2 )| = |µ2 − µ ¯2 |, ˆ, u, µ ¯2 ) − u (x, yˆ, uˆ, µ ¯2 ), u − uˆ u (x, y 105 = u − uˆ, u − uˆ = |u − uˆ|2 , for a.e x ∈ Γ, for all µ2 , y, u ∈ R satisfying |µ2 | ≤ ¯ Therefore, and (ˆ y , uˆ) ∈ S(¯ µ, λ) assumption (A4.3) is fulfilled ¯ By Lemma 4.2.5, there exists a function Now, we suppose that (¯ y , u¯) ∈ S(¯ µ, λ) ¯ such that the following conditions hold: φ ∈ H (Ω) ∩ C(Ω) (i) the adjoint equation:  −∆φ + φ + 3¯ y φ = + t(x) + 3t(x)¯ y2 in Ω, ∂ν φ + φ = −2 on Γ, (4.33) (ii) the weakly minimum principle: (φ(x ) + u¯(x ) + y¯(x ) − 2)(v − y¯(x ) − u¯(x )) ≥ a.e x ∈ Γ (4.34) for all v ∈ [−1, 0] We Note that ∆t(x) = −4 and t(x) = 0, ∂ν t(x) = −2(x21 + x22 ) = −2 ∀x ∈ Γ Hence φ = t(x) is a unique solution of the adjoint equation (4.33) This implies that φ(x) = a.e x ∈ Γ From (4.34), for a.e x ∈ Γ we have (¯ u(x ) + y¯(x ) − 2)(v − y¯(x ) − u¯(x )) ≥ a.e x ∈ Γ, ∀v ∈ [−1, 0] Since y¯(x ) + u¯(x ) ∈ [−1, 0] a.e x ∈ Γ, we get y¯(x ) + u¯(x ) − < a.e x ∈ Γ Hence v − y¯(x ) − u¯(x ) ≤ ∀v ∈ [−1, 0], a.e x ∈ Γ This implies that y¯(x ) + u¯(x ) = a.e x ∈ Γ and so y¯ + u¯ = on Γ Consequently, y¯ is a solution of the following equation  −∆y + y + y = in Ω, ∂ν y + y = on Γ ¯ = {(0, 0)} Moreover, By the uniqueness, we have y¯ = and so u¯ = Hence S(¯ µ, λ) by Theorem 4.1.2, 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 4.4.2 Let Ω be the open unit ball in R2 with boundary Γ We consider ¯ such that problem P (µ, λ) of finding u ∈ L2 (Γ) and y ∈ H (Ω) ∩ C(Ω) (x, y(x), u(x), µ(x))dσ → inf, F (y, u, µ) = Γ subject to the state equation  −∆y + y + y = in Ω, ∂ν y on Γ =u+λ 106 and pointwise constraint ≤ u(x) + y(x) ≤ a.e x ∈ Γ, where is given by 1 (x, y, u, µ) = [1 − sign(u + y + µ2 )](u + y + µ2 )2 + [1 + sign(u + y − µ2 )](u + y − µ2 )2 2 Here sign(v) is defined by sign(v) =    1 if v > 0, if v = 0,    −1 if v < It is obvious that F (y, u, µ) ≥ ¯ = We shall show that the following assertions hold: Let us take µ ¯ = 0, λ ¯ has unique solution (¯ (i) P (¯ µ, λ) y , u¯) = (0, 0) (ii) P (µ, λ) satisfies assumptions (A4.1)–(A4.4) (iii) If ≤ |µ(x)| ≤ then S(µ, λ) ⊃ {(y(µ, λ), sµ2 − y(µ, λ)), < s < 1}, where y(µ, λ) is a solution of the equation  −∆y + y + y = ∂ν y + y = sµ2 in Ω, +λ (4.35) on Γ ¯ = 0, problem P (¯ ¯ becomes In fact, when µ ¯ = 0, λ µ, λ) (y(x) + u(x))2 dσ → inf F (y, u, µ ¯) = Γ with constraints  −∆y + y + y = in Ω, ∂ν y on Γ, =u and ≤ y(x) + u(x) ≤ a.e x ∈ Γ Obviously, S(0, 0) = {(0, 0)} We now prove that P (µ, λ) satisfies (A4.1)–(A4.4) It is easily seen that (x, y, u, µ) =  2   (u + y + µ )    (u + y − µ2 )2 107 if u + y < −µ2 , if − µ2 ≤ u + y ≤ µ2 , if u + y > µ2 Hence y (x, y, u, µ) = u (x, y, u, µ) =    2(u + y + µ ) if u + y < −µ2 , − µ2 ≤ u + y ≤ µ2 , if    2(u + y − µ2 ) if u + y > µ2 It is clear that assumptions (A4.2) and (A4.4) hold For assumption (A4.1), 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 + 4µ4 ≤ 4u2 + 4M + 40 , | 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|) + 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 |T1 | = = ≤ ≤ ≤  2   |2(y1 + u1 + µ ) − 2(y2 + u2 + µ )| if y2 + u2 < −µ2 , |2(y1 + u1 + µ2 )| if − µ2 ≤ y2 + u2 ≤ µ2 ,    |2(y1 + u1 + µ2 ) − 2(y2 + u2 − µ2 )| if y2 + u2 > µ2    2|(y1 − y2 ) + (u1 − u2 )| 2|y + u + µ2 | 1    2|(y1 − y2 ) + (u1 − u2 ) + 2µ2 |    2(|y1 − y2 | + |u1 − u2 |) 2|y + u − y − u | 1 2    2(|y1 − y2 | + |u1 − u2 |) + 4µ2    2(|y1 − y2 | + |u1 − u2 |) if y2 + u2 < −µ2 , if − µ2 ≤ y2 + u2 ≤ µ2 , if y2 + u2 > µ2 if y2 + u2 < −µ2 , if − µ2 ≤ y2 + u2 ≤ µ2 , if y2 + u2 > µ2 if y2 + u2 < −µ2 , if − µ2 ≤ y + u2 ≤ µ2 , 2(|y − y | + |u − u |) 2    2(|y1 − y2 | + |u1 − u2 |) + 2|y2 + u2 − y1 − u1 | if y2 + u2 > µ2   if y2 + u2 < −µ2 ,  2(|y1 − y2 | + |u1 − u2 |)    2(|y1 − y2 | + |u1 − u2 |) if − µ2 ≤ y2 + u2 ≤ µ2 , 4(|y1 − y2 | + |u1 − u2 |) if y2 + u2 > µ2 ≤ 4(|y1 − y2 | + |u1 − u2 |) 108 Similarly, for the other cases, we also have |T1 | ≤ 4(|y1 − y2 | + |u1 − u2 |) Hence assumption (A4.1) is satisfied Note that y (x, y, u, 0) = 2(y + u) for all x, y, u, µ Putting u (x, y, u, µ) − u (x, y, u, 0), T2 := we get |T2 | = = ≤    |2(y + u + µ ) − 2(y + u)| | − 2(y + u)| if y + u < −µ2 , if − µ2 ≤ y + u ≤ µ2 ,    |2(y + u − µ2 ) − 2(y + u)| if   if y + u < −µ2 ,  2µ 2|y + u|    2µ    2µ    y + u > µ2 if − µ2 ≤ y + u ≤ µ2 , if y + u > µ2 if y + u < −µ2 , 2µ2 if − µ2 ≤ y + u ≤ µ2 , 2µ2 if y + u > µ2 ≤ 2µ2 for all x, u, y, µ On the other hand, we obtain u (x, 0, u, 0) − u (x, 0, 0, 0), u − = 2u, u = 2|u|2 = 2|u − 0|2 Thus, assumption (A4.3) is fulfilled Finally, if y(µ, λ) is a solution of equation (4.35) 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 < s < 1, we get F (y, u, µ) = This implies that (y(µ, λ), u(µ, λ)) ∈ S(µ, λ) 4.5 Conclusions In this chapter, we present a result on the solution stability of boundary optimal control problems Theorem 4.1.2 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 109 GENERAL CONCLUSIONS 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 control problem in two-dimensional 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 110 LIST OF PUBLICATIONS [1] N H Son, B T Kien and A Răosch (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 311-329 111 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, Birkhăauser, Boston [3] F Trăoltzsch (2010), Optimal Control of Partial Differential Equations: Theory, Method and Applications, AMS, 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x) = {(x1 , 0) | x1 ∈ R}, cone(D − x) =. .. (D, x) = T (D, x) = cone(D − x)) Putting D = {(x1 , x2 ) | x2 = 0} ∪ {(x1 , x2 ) | x1 = 0} ⊂ R2 and taking x = (0, 0), we have TC (D, x) = {(0, 0)}, T (D, x) = T (D, x) = cone(D − x) = D Let