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, 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 i ii iii COMMITTAL IN THE DISSERTATION ACKNOWLEDGEMENTS CONTENTS TABLE OF NOTATIONS INTRODUCTION Chapter 0.1 0.2 0.3 PRELIMINARIES AND AUXILIARY RESULTS Variational analysis 0.1.1 Set-valued maps 0.1.2 Tangent and normal cones Sobolev spaces and elliptic equations 13 0.2.1 Sobolev spaces 13 0.2.2 Semilinear elliptic equations 20 Conclusions 24 Chapter NO-GAP OPTIMALITY CONDITIONS FOR DISTRIBUTED CONTROL PROBLEMS 25 1.1 Second-order necessary optimality conditions 26 1.1.1 An abstract optimization problem 26 1.1.2 Second-order necessary optimality conditions for optimal control problem 27 1.2 Second-order sufficient optimality conditions 40 1.3 Conclusions 57 Chapter NO-GAP OPTIMALITY CONDITIONS FOR BOUNDARY CONTROL PROBLEMS 58 2.1 Abstract optimal control problems 59 2.2 Second-order necessary optimality conditions 66 2.3 Second-order sufficient optimality conditions 75 2.4 Conclusions 89 Chapter UPPER SEMICONTINUITY AND CONTINUITY OF THE SOLUTION MAP TO A PARAMETRIC BOUNDARY CONTROL PROBLEM 91 3.1 Assumptions and main result 92 3.2 Some auxiliary results 94 iii 3.2.1 Some properties of the admissible set 94 3.2.2 First-order necessary optimality conditions 98 3.3 Proof of the main result 100 3.4 Examples 104 3.5 Conclusions 109 GENERAL CONCLUSIONS LIST OF PUBLICATIONS 110 111 REFERENCES 112 iv TABLE OF NOTATIONS N := {1, 2, } R set of positive natural numbers |x| absolute value of x ∈ R Rn 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 A is not a subset of B B set of real numbers A∩B intersection of the sets A and B A∪B union of the sets A and B A\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 x x 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 xand y ϕ : X → IR extended-real-valued function domϕ effective domain of ϕ epiϕ epigraph of ϕ 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) the space of Lebesgue measurable functions f closure of the set Ω and Ω |f (x)|p dx < +∞ L∞ (Ω) ¯ C(Ω) the space of bounded functions almost every Ω ¯ the space of continuous functions on Ω ¯ M(Ω) the space of finite regular Borel measures  m,p m,p   W (Ω), W0 (Ω), W s,r (Γ), 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 continuous embedded in Y X →→ Y X is compact embedded in Y i.e id est (that is) a.e almost every s.t subject to p page w.r.t with respect to ✷ The proof is complete INTRODUCTION 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 Trăoltzsch [18], B T Kien et al [19, 20, 21, 22], A Răosch and F Trăoltzsch [23, 24] derived optimality conditions for optimal control problems governed by elliptic 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 (¯ u) = This a first-order 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 For distributed control problems, i.e., the control only acts in the domain Ω in Rn , E Casas, T Bayen et al [11, 13, 16, 27] derived second-order necessary and sufficient optimality conditions for problem with pure control constraint, i.e., a(x) ≤ u(x) ≤ b(x) a.e x ∈ Ω, (1) and the appearance of state constraints More precisely, in [11] the authors gave second-order necessary and sufficient conditions for Neumann problems with constraint (1) and finitely many equalities and inequalities constraints of state variable y while the second-order sufficient optimality conditions are established for Dirichlet problems with constraint (1) and a pure state constraint in [13] T Bayen et al [27] derived second-order necessary and sufficient optimality conditions for Dirichlet problems in the sense of strong solution In particular, E Casas [16] established second-order 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 [18], C Meyer and F Trăoltzsch derived second-order sufficient optimality conditions for Robin control 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 For boundary control problems, i.e., the control u only acts on the boundary , E Casas and F Trăoltzsch [10, 12] derived second-order necessary optimality conditions while the second-order sufficient optimality conditions were established by E Casas et al in [12, 13, 17] with pure pointwise constraints, i.e., a(x) u(x) b(x) a.e x A Răosch and F Trăoltzsch [23] 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 papers, 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 (see [19]) In order to avoid this disadvantage, B T Kien et al [19, 20, 21] recently established second-order necessary optimality conditions for distributed control of Dirichlet problems with mixed statecontrol constraints of the form a(x) ≤ g(x, y(x)) + u(x) ≤ b(x) a.e x ∈ Ω with a, b ∈ Lp (Ω), < p < ∞ 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, ¯ and Lemma 3.2.5 that Proof It follows from (yn , un ) ∈ S(µn , λn ), (¯ y , u¯) ∈ S(¯ µ, λ) ¯ satisfying the following conditions there exist function φn , φ¯ ∈ H (Ω) ∩ C(Ω)  A∗ φ¯ + f (·, y¯)φ¯ = L (·, y¯, µ¯(1) ) y y ∂n ∗ φ¯ + gy (·, y¯)φ¯ = y (·, y¯, u¯, µ¯(2) ) − A in Ω, ¯, u¯, µ ¯(2) )gy (·, y¯) u (·, y  A∗ φ + f (·, y )φ = L (·, y , µ(1) ) n y n n y n n ∂n ∗ φn + gy (·, yn )φn = y (·, yn , un , µ(2) n )− A (3.20) on Γ, in Ω, (2) u (·, yn , un , µn )gy (·, yn ) on Γ (3.21) and ¯ (φ(x) + ¯(x), u¯(x), µ ¯ u (x, y (2) ¯ (2) (x))dσ ≥ 0, (x)))(v(x) − g(x, y¯) − u¯(x) − λ (3.22) Γ (φn (x) + (2) (2) u (x, yn (x), un (x), µn (x)))(v(x) − g(x, yn ) − un (x) − λn (x))dσ ≥0 Γ (3.23) for all v ∈ K, where K is defined by (3.15) We claim that there is a subsequence of {φn } converges φ0 in L2 (Ω) for some φ0 ∈ H (Ω) In fact, since yn → y¯ in Y and un y¯ Y, yn Y, u¯ in U , there is M > such that u¯ U, un U ≤M (3.24) for all n ≥ By assumptions (A3.1) and (A3.4), there are functions r3M ∈ L∞ (Ω), r4M ∈ L∞ (Γ) and positive constants ClM , CgM such that (1) |Ly (x, yn , µn )| ≤ r3M (x), (2) (2) | y (x , yn , un , µn )| + | u (x , yn , un , µn )| ≤ ClM (|yn | + |un |) + r4M (x ), (3.25) | u (x , yn , un , µ ¯(2) ) − (3.26) u (x , y¯, un , µ ¯(2) )| ≤ ClM |yn − y¯|, |gy (x , yn )| ≤ CgM for all n ≥ 1, x ∈ Ω and x ∈ Γ It follows from this, (3.21) and Lemma 3.2.1 that φn (1) Y ≤ Ly (·, yn , µn ) L2 (Ω) + (2) (2) y (·, yn , un , µn ) − u (x, yn , un , µn )gy (·, yn ) L2 (Γ) ≤ r3M L∞ (Ω) |Ω| + (1 + CgM )[(ClM M + r4M L∞ (Γ) )|Γ| + ClM ≤ r3M L∞ (Ω) |Ω| + (1 + CgM )[(ClM M + r4M L∞ (Γ) )|Γ| + ClM M ], un L2 (Γ) ] where |Ω| and |Γ| are the volumes of Ω and Γ, respectively Hence {φn } is bounded in H (Ω) By passing subsequences, we can assume that φn 102 φ0 in H (Ω) Since the embedding H (Ω) → L2 (Ω) is compact, we get φn → φ0 in L2 (Ω) Our claim is proved Putting G(y, u, λ) := g(x, y) + u + λ(2) , we have from (3.22) and (3.23) that ¯ (φ(x) + ¯(x), u¯(x), µ ¯ u (x, y (2) ¯ (x)))(G(yn , un , λn ) − G(¯ y , u¯, λ))dσ ≥ 0, (3.27) Γ (φn (x) + (2) ¯ − G(yn , un , λn ))dσ y , u¯, λ) u (x, yn (x), un (x), µn (x)))(G(¯ ≥ (3.28) Γ By adding (3.27) and (3.28), yields ¯ ¯ (φn (x) − φ(x))(G(y y , u¯, λ))dσ n , un , λn ) − G(¯ Γ (2) ( u (x, yn , un , µn (x)) − + ¯, u¯, µ ¯ u (x, y (2) (2) ¯ (2) )dσ (x)))(g(x, yn ) − g(x, y¯) + λn − λ Γ (2) ( u (x, yn (x), un (x), µn (x)) − + ¯(x), u¯(x), µ ¯ u (x, y (2) (x)))(un − u¯)dσ ≤ (3.29) Γ (2) ¯ (2) ¯ and un Since yn → y¯ in Y , λn → λ u¯ in L2 (Γ), we get g(·, yn ) + λn → g(·, y¯) + λ ¯ in L2 (Γ) Besides, it follows from (3.25) and the fact and G(yn , un , λn ) G(¯ y , u¯, λ) φn → φ0 in L2 (Ω) that u (·, yn , un , µn ) is bounded and φn − φ¯ → φ0 − φ¯ in L2 (Γ) Therefore, the first term and second term in the left hand side of (3.29) tend to as n → +∞ This implies that (2) ( u (x, yn (x), un (x), µn (x)) − lim sup n→+∞ ¯(x), u¯(x), µ ¯ u (x, y (2) (x)))(un − u¯)dσ ≤ Γ (3.30) On the other hand, we have (2) ( u (x, yn (x), un (x), µn (x)) − ¯(x), u¯(x), µ ¯ u (x, y (2) (x)))(un − u¯)dσ = In + Jn + Kn , Γ (3.31) where (2) In = (2) ( u (x, yn (x), un (x), µn (x)) − ¯ u (x, yn (x), un (x), µ (x)))(un − u¯)dσ, ( u (x, yn (x), un (x), µ ¯(2) (x)) − ¯(x), un (x), µ ¯ u (x, y ( u (x, y¯(x), un (x), µ ¯(2) (x)) − ¯(x), u¯(x), µ ¯(2) (x)))(un u (x, y Γ Jn = (2) (x)))(un − u¯)dσ, Γ Kn = − u¯)dσ Γ It follows from (3.4) that (2) (2) η(x, |yn (x)|, |µn (x)|, |¯ µ(2) (x)|)|un (x)|θ |µn (x)−¯ µ(2) (x)|α |un (x)−¯ u(x)|dσ |In | ≤ Γ By the continuity of η and (3.24), we have (2) η(x, |yn (x)|, |µn (x)|, |¯ µ(2) (x)|) ≤ η¯ := max (x,t1 ,t2 ,t3 )∈Γ×[0,M ]×[0,γ+ ]×[0,γ] 103 η(x, t1 , t2 , t3 ) < +∞ for all n ≥ 1, where γ = µ ¯(2) L∞ (Γ) Hence (2) η¯ µn − µ ¯(2) |In | ≤ α θ ¯(x)|dσ L∞ (Γ) |un (x)| |un (x) − u Γ By the Holder inequality and (3.24), we have (2) ¯(2) |In | ≤ C η¯ µn − µ α L∞ (Γ) ≤ C η¯M θ (M + u¯ L2 (Γ) ) θ ¯ L2 (Γ) L2 (Γ) un − u (2) µn − µ ¯(2) αL∞ (Γ) un for some constant C > This implies that lim In = From (3.24) and (3.26), 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 (3.30) and (3.31) yields n→+∞ n→+∞ lim sup Kn ≤ (3.32) n→+∞ However, from (3.5) we have β(un (x) − u¯(x))2 dσ = β un − u¯ Kn ≥ L2 (Γ) Γ From this and (3.32), we have lim sup un − u¯ n→+∞ L2 (Γ) = and so un → u in L2 (Γ) The proof of lemma is complete From Lemmas 3.3.1 and 3.3.2, 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, ) W1 ì W2 for all (à, ) V1 × V2 It follows from this and V2 of λ the fact S(µ, λ) = ∅ that S(µ, ) W1 ì W2 = for all (à, λ) ∈ V1 × V2 and so S(·, ·) ¯ Therefore, S(·, ·) is continuous at (¯ ¯ The proof is lower semicontinuous at (¯ µ, λ) µ, λ) of Theorem 3.1.2 is complete 3.4 Examples In this section, we give some examples illustrating Theorem 3.1.2 The first example ¯ is singleton and the solution map S(·, ·) is continuous at (¯ ¯ shows that S(¯ µ, λ) µ, λ) 104 Example 3.4.1 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 [ (u(x) + y(x) + µ2 (x))2 − 4y(x) − 2u(x)]dσ → inf Γ + s.t  −∆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) = 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 | + |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 (A3.1) is satisfied For assumption (A3.3), 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 (A3.3) is fulfilled ¯ By Lemma 3.2.5, there exists a function Now, we suppose that (¯ y , u¯) ∈ S(¯ µ, λ) ¯ such that the following conditions are valid: 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, (3.33) (ii) the weakly minimum principle: (φ(x ) + u¯(x ) + y¯(x ) − 2)(v − y¯(x ) − u¯(x )) ≥ a.e x ∈ Γ (3.34) for all v ∈ [−1, 0] We notice 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 (3.33) This implies that φ(x) = a.e x ∈ Γ From (3.34), 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 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 ¯ 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 are valid: Let us take µ ¯ = 0, λ ¯ has unique solution (¯ (i) P (¯ µ, λ) y , u¯) = (0, 0) (ii) P (µ, λ) satisfies assumptions (A3.1)–(A3.4) (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 (3.35) 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 (A3.1)–(A3.4) 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 (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 + 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 (A3.1) is satisfied Notice 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 (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 < s < 1, we get F (y, u, µ) = 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 109 GENERAL CONCLUSIONS The main results of this dissertation include: Second-order necessary 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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