Shooting methods for numerical solutions of control problems constrained. by linear and nonlinear hyperbolic partial differential equations

Transcription

1 Shooting methods for numerical solutions of control problems constrained by linear and nonlinear hyperbolic partial differential equations by Sung-Dae Yang A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Applied Mathematics Program of Study Committee: L. Steven Hou, Major Professor Oleg Emanouvilov Hailiang Liu Michael Smiley Sung-Yell Song Iowa State University Ames, Iowa 24 Copyright c Sung-Dae Yang, 24. All rights reserved.

2 ii Graduate College Iowa State University This is to certify that the doctoral dissertation of Sung-Dae Yang has met the dissertation requirements of Iowa State University Committee Member Committee Member Committee Member Committee Member Major Professor For the Major Program

3 iii TABLE OF CONTENTS ABSTRACT v INTRODUCTION Shooting Methods for Numerical Solutions of Distributed Optimal Control Problems Constrained by Linear and Nonlinear Wave Equations Distributed optimal control problems for the wave equations The solution of the eact controllability problem as the limit of optimal control solutions Shooting methods for D control problems Algorithm for D control problems D Computational results Shooting methods in 2D control problems Algorithm for 2D control problems D computational results Shooting Methods for Numerical Solutions of Eact Controllability Problems Constrained by Linear and Nonlinear Wave Equations with Local Distributed Controls Eact controllability problems for the wave equations The solution of the eact local controllability problem as the limit of optimal control solutions

4 iv 3.3 Computational results Eact controllability problems with linear cases Eact controllability problems with nonlinear cases Shooting Methods for Numerical Solutions of Eact Boundary Controllability Problems for the -D Wave Equation Introduction The solution of the eact controllability problem as the limit of optimal control solutions An optimality system of equations and a continuous shooting method The discrete shooting method Computational eperiments for controllability of the linear wave equation An eample with known smooth eact solution Generic eamples with minimum L 2 (Σ)-norm boundary control Computational eperiments for controllability of semilinear wave equations7 5 Shooting Methods for Numerical Solutions of Distributed Optimal Control Problems Constrained by First Order Linear Hyperbolic Equation Distributed optimal control problems for first order linear hyperbolic equation An optimality system of equations Computational results CONCLUSION BIBLIOGRAPHY ACKNOWLEDGMENTS

5 v ABSTRACT We consider shooting methods for computing approimate solutions of control problems constrained by linear or nonlinear hyperbolic partial differential equations. Optimal control problems and eact controllability problems are both studied, with the latter being approimated by the former with appropriate choices of parameters in the cost functional. The types of equations include linear wave equations, semilinear wave equations, and first order linear hyperbolic equations. The controls considered are either distributed in part of the time-space domain or of the Dirichlet type on the boundary. Each optimal control problem is reformulated as a system of equations that consists of an initial value problem (IVP) for the state equations and a terminal value problem for the adjoint equations. The optimality systems are regarded as a system of an IVP for the state equation and an IVP for the adjoint equations with unknown initial conditions. Then the optimality system is solved by shooting methods, i.e. we attempt to find adjoint initial values such that the adjoint terminal conditions are met. The shooting methods are implemented iteratively and Newton s method is employed to update the adjoint initial values. The convergence of the algorithms are theoretically discussed and numerically verified. Computational eperiments are performed etensively for a variety of settings: different types of constraint equations in -D or 2-D, distributed or boundary controls, optimal control or eact controllability.

6 INTRODUCTION In this thesis we study numerical solutions of optimal control problems and eact controllability problems for linear and semilinear hyperbolic partial differential equations defined over the time interval [, T ] [, ) and on a bounded, C 2 (or conve) spatial domain Ω R d, d = or 2 or 3. The optimal control problems for the controls being either distributed in part of the time-space domain or of the Dirichlet type on the boundary are reformulated as a system of equations (an optimality system) that consists of an initial value problem for the underlying (linear or semilinear) hyperbolic partial differential equations and a terminal value problem for the adjoint hyperbolic partial differential equations by applying Lagrange multipliers. We develop shooting algorithms to solve the optimality system as follows : The optimality systems are regarded as a system of an IVP for the state equation and an IVP for the adjoint equations with unknown initial conditions. Then the optimality system is solved by shooting methods, i.e. we attempt to find adjoint initial values such that the adjoint terminal conditions are met. The shooting methods are implemented iteratively and Newton s method is employed to update the adjoint initial values. Let target functions W L 2 (Ω), Z L 2 (Ω) or H (Ω), U L 2 ((, T ) Ω) and an initial condition w L 2 (Ω), z L 2 (Ω) or H (Ω) be given. Let f L 2 ((, T ) Ω) denote the distributed control and g [L 2 (, T )] 2 denote the boundary control. We wish to find a control f or g that drives the states u and u t to W, Z at time T and u to U in (, T ) Ω. In Chapter 2 we consider an optimal control approach with distributed controls

7 2 defined on spatial domain Ω for solving the eact controllability problems for one and two-dimensional linear and semilinear wave equations defined on a time interval (, T ) and a bounded spatial domain Ω. Precisely we consider the following optimal control problem: minimize the cost functional J (u, f) = α 2 T + 2 T K(u) d dt + β Ω 2 f 2 d dt Ω Ω Φ (u(t, )) d + γ 2 Ω Φ 2 (u t (T, )) d (.) (where α, β, γ are positive constants) subject to the wave equation u tt u + Ψ(u) = f in (, T ) Ω, (.2) with the homogeneous boundary condition u Ω =, (t, ) (, T ) Ω. (.3) and the initial conditions u(, ) = w(), u t (, ) = z() Ω. (.4) We first develop the shooting algorithms for D and 2D distributed control problems, and then simulate with known smooth solution and generic eamples both linear and semilinear cases in D and 2D. In Chapter 3 we consider an optimal control approach with local distributed controls defined on spatial subdomain Ω ( Ω) for solving the eact controllability problems for one-dimensional linear and semilinear wave equations defined on a time interval (, T ) and a bounded spatial domain Ω. Precisely we consider the following optimal control

8 3 problem: minimize the cost functional J (u, f) = α 2 = α 2 T + 2 T T + 2 K(u) d dt + β Φ (u(t, )) d + γ Ω 2 Ω 2 f 2 d dt Ω T K(u) d dt + β Ω 2 χ Ω f 2 d dt Ω Ω Φ (u(t, )) d + γ 2 Ω Ω Φ 2 (u t (T, )) d Φ 2 (u t (T, )) d (.5) subject to the wave equation u tt u + Ψ(u) = χ Ω f in (, T ) Ω, (.6) with the homogeneous boundary condition u Ω =, (t, ) (, T ) Ω. (.7) and the initial conditions u(, ) = w(), u t (, ) = z() Ω. (.8) The shooting algorithms are applied to eact controllability problems with local distributed controls in the eamples of known smooth solution and generic eamples for both linear and semilinear cases. In Chapter 4 we consider an optimal boundary control approach for solving the eact boundary control problem for one-dimensional linear or semilinear wave equations defined on a time interval (, T ) and spatial interval (, X). The eact boundary control problem we consider is to seek a boundary control g = (g L, g R ) L 2 (, T) [L 2 (, T )] 2

9 4 and a corresponding state u such that the following system of equations hold: u tt u + f(u) = V in Q (, T ) (, X), u t= = u and u t t= = u in (, X), u t=t = W and u t t=t = Z in (, X), u = = g L and u = = g R in (, T ), (.9) where u and u are given initial conditions defined on (, X), W L 2 (, X) and Z H (, X) are prescribed terminal conditions, V is a given function defined on (, T ) (, X), f is a given function defined on R, and g = (g L, g R ) [L 2 (, T )] 2 is the boundary control. In this chapter we attempt to solve the eact controllability problems by an optimal control approach. Precisely, we consider the following optimal control problem: minimize the cost functional J (u, g) = σ u(t, ) W () 2 d + τ 2 ( g L 2 + g R 2 ) dt u t (T, ) Z() 2 d (.) subject to u tt u + f(u) = V in Q (, T ) (, ) u t= = u and u t t= = u in (, ) u = = g L and u = = g R in (, T ). (.) The shooting algorithms for solving the optimal control problem will be described for the slightly more general functional J (u, g) = α 2 T + τ 2 u U 2 d dt + σ 2 u t (T, ) Z() 2 d + 2 u(t, ) W () 2 d ( g L 2 + g R 2 ) dt (.2)

10 5 where the term involving (u U) reflects our desire to match the candidate state u with a given U in the entire domain Q. Our computational eperiments of the proposed numerical methods will be performed eclusively for the case of α =. In Chapter 5 the linear optimal control problems we study are to minimize the cost functional J (u, f) = α 2 T Ω u U 2 ddt + β 2 Ω u(, T ) W () 2 d + 2 T Ω f 2 ddt. subject to first order linear hyperbolic equation u t + au = f(, t), in (, T ) Ω, (.3) with the boundary condition u(t, ) = z(t), t (, T ), (.4) and the initial condition u(, ) = w(), Ω. (.5) We particularly test some eamples of distributed optimal control problems with known smooth solution and generic initial and boundary data and with a >. Since this thesis covers several topics, the literature and new contributions of the thesis will be discussed in the contet of each chapter.

11 6 2 Shooting Methods for Numerical Solutions of Distributed Optimal Control Problems Constrained by Linear and Nonlinear Wave Equations Numerical solutions of distributed optimal Dirichlet control problems for linear and semilinear wave equations are studied. The optimal control problem is reformulated as a system of equations (an optimality system) that consists of an initial value problem for the underlying (linear or semilinear) wave equation and a terminal value problem for the adjoint wave equation. The discretized optimality system is solved by a shooting method. The convergence properties of the numerical shooting method in the contet of eact controllability are illustrated through computational eperiments. 2. Distributed optimal control problems for the wave equations We will study numerical methods for optimal control and controllability problems associated with the linear and nonlinear wave equations. We are particularly interested in investigating the relevancy and applicability of high performance computing (HPC) for these problems. As an prototype eample of optimal control problems for the wave equations we consider the following distributed optimal control problem: choose a control f and a corresponding u such that the pair (f, u) minimizes the cost

12 7 functional J (u, f) = α 2 T + 2 T K(u) d dt + β Ω 2 f 2 d dt Ω Ω Φ (u(t, )) d + γ 2 Ω Φ 2 (u t (T, )) d (2..) subject to the wave equation u tt u + Ψ(u) = f in (, T ) Ω, u Ω =, u(, ) = w(), u t (, ) = z(). (2..2) Here Ω is a bounded spatial domain in R d (d = or 2 or 3) with a boundary Ω; u is dubbed the state, and g is the distributed control. Also, K, Φ and Ψ are C mappings (for instance, we may choose K(u) = (u U) 2, Ψ(u) =, Ψ(u) = u 3 u and Ψ(u) = e u, Φ (u) = (u(t, ) W ) 2, Φ 2 (u) = (u t (T, ) Z) 2, where U, W, Z is a target function.) Using Lagrange multiplier rules one finds the following optimality system of equations that the optimal solution (f, u) must satisfy: u tt u + Ψ(u) = f in (, T ) Ω u Ω =, u(, ) = w(), u t (, ) = z() ; ξ tt ξ + [Ψ (u)] ξ = α 2 K (u) in Q ξ Ω =, ξ(t, ) = γ 2 Φ 2(u t (T, )), ξ t (T, ) = β 2 Φ (u(t, )) ; f + ξ = in Q. This system may be simplified as u tt u + Ψ(u) = ξ in (, T ) Ω u Ω =, u(, ) = w(), u t (, ) = z() ; ξ tt ξ + [Ψ (u)] ξ = α 2 K (u) in (, T ) Ω (2..3) ξ Ω =, ξ(t, ) = γ 2 Φ 2(u t (T, )), ξ t (T, ) = β 2 Φ (u(t, )).

13 8 Such control problems are classical ones in the control theory literature; see, e.g., [5] for the linear case and [6] for the nonlinear case regarding the eistence of optimal solutions as well as the eistence of a Lagrange multiplier ξ satisfying the optimality system of equations. However, numerical methods for finding discrete (e.g., finite element and/or finite difference) solutions of the optimality system are largely limited to gradient type methods which are sequential in nature and generally require many iterations for convergence. The optimality system involves boundary conditions at t = and t = T and thus cannot be solved by marching in time. Direct solutions of the discrete optimality system, of course, are bound to be epensive computationally in 2 or 3 spatial dimensions since the problem is (d + ) dimensional (where d is the spatial dimensions.) The computational algorithms we propose here are based on shooting methods for two-point boundary value problems for ordinary differential equations (ODEs); see, e.g., [, 2, 3, 4]. The algorithms we propose are well suited for implementations on a parallel computing platform such as a massive cluster of cheap processors. 2.2 The solution of the eact controllability problem as the limit of optimal control solutions The eact distributed control problem we consider is to seek a distributed control f L 2 ((, T ) Ω) and a corresponding state u such that the following system of equations hold: u tt u + Ψ(u) = f in Q (, T ) Ω, u t= = w and u t t= = z in Ω, u t=t = W and u t t=t = Z in Ω, u Ω = in (, T ). (2.2.4) Under suitable assumptions on f and through the use of Lagrange multiplier rules,

14 9 the corresponding optimal control problem: minimize (2..) with respect to the control f subject to (2..2). (2.2.5) In this section we establish the equivalence between the limit of optimal solutions and the minimum distributed L 2 norm eact controller. We will show that if α, β and γ, then the corresponding optimal solution (û αβγ, f αβγ ) converges weakly to the minimum distributed L 2 norm solution of the eact distributed controllability problem (2.2.4). The same is also true in the discrete case. Theorem Assume that the eact distributed controllability problem (2.2.4) admits a unique minimum distributed L 2 norm solution (u e, f e ). Assume that for every (α, β, γ) R + R + R + (where R + is the set of all positive real numbers,) there eists a solution (u αβγ, f αβγ ) to the optimal control problem (2.2.5). Then f αβγ L 2 (Q) f e L 2 (Q) (α, β, γ) R + R + R +. (2.2.6) Assume, in addition, that for a sequence {(α n, β n, γ n )} satisfying α n, β n and γ n, Then u αn β n γ n u in L 2 (Q) and Ψ(u αn β n γ n ) Ψ(u) in L 2 (, T ; [H 2 (Ω) H (Ω)] ). (2.2.7) f αn β n γ n f e in L 2 (Q) and u αn β n γ n u e in L 2 (Q) as n. (2.2.8) Furthermore, if (2.2.7) holds for every sequence {(α n, β n, γ n )} satisfying α n, β n

15 and γ n, then f αβγ f e in L 2 (Q) and u αβγ u e in L 2 (Q) as α, β, γ. (2.2.9) Proof. Since (u αβγ, f αβγ ) is an optimal solution, we have that α 2 u αβγ(t ) U L 2 (Q) + β 2 u αβγ(t ) W L 2 (Ω) + γ 2 tu αβγ (T ) Z H (Ω) + 2 f αβγ L 2 (Q) = J (u αβγ, f αβγ ) J (u e, f e ) = 2 f e L 2 (Q) so that (2.2.6) holds, u αβγ t=t W in L 2 (Ω) and ( t u αβγ ) t=t Z in H (Ω) as α, β, γ. (2.2.) Let {(α n, β n, γ n )} be the sequence in (2.2.7). Estimate (2.2.6) implies that a subsequence of {(α n, β n, γ n )}, denoted by the same, satisfies f αn β n γ n f in L 2 (Q) and f L 2 (Q) f e L 2 (Q). (2.2.) (u αβγ, f αβγ ) satisfies the initial value problem in the weak form: T T u αβγ (v tt v ) d dt + [Ψ(u αβγ ) f αβγ ]v d dt Ω Ω + (v t u αβγ ) t=t d v t= z d (u αβγ t v) t=t d Ω Ω Ω + (w t v) t= d = v C 2 ([, T ]; H 2 (Ω) H(Ω)) Ω (2.2.2) Passing to the limit in (2.2.2) as α, β, γ and using relations (2.2.) and (2.2.)

16 we obtain: T T u(v tt v ) d dt + [Ψ(u) f]v d dt Ω Ω + v t=t Z() d v t= z d W ()( t v) t=t d Ω Ω Ω + (w t v) t= d = v C 2 ([, T ]; H 2 (Ω) H(Ω)). Ω The last relation and (2.2.) imply that (u, f) is a minimum boundary L 2 norm solution to the eact control problem (2.2.4). Hence, u = u e and f = f e so that (2.2.8) and (2.2.9) follows from (2.2.7) and (2.2.). Remark If the wave equation is linear, i.e., Ψ =, then assumption (2.2.7) is redundant and (2.2.9) is guaranteed to hold. Indeed, (2.2.2) implies the boundedness of { u αβγ L 2 (Q)} which in turn yields (2.2.7). The uniqueness of a solution for the linear wave equation implies (2.2.7) holds for an arbitrary sequence {(α n, β n, γ n )}. Theorem Assume that i) for every (α, β, γ) R + R + R + there eists a solution (u αβγ, f αβγ ) to the optimal control problem (2.2.5); ii) the limit terminal conditions hold: u αβγ t=t W in L 2 (Ω) and ( t u αβγ ) t=t Z in H (Ω) as α, β, γ ; (2.2.3) iii) the optimal solution (u αβγ, f αβγ ) satisfies the weak limit conditions as α, β, γ : f αβγ f in L 2 (Q), u αβγ u in L 2 (Q), (2.2.4) and Ψ(u αβγ ) Ψ(u) in L 2 (, T ; [H 2 (Ω) H (Ω)] ) (2.2.5)

17 2 for some f L 2 (Q) and u L 2 (Q). Then (u, f) is a solution to the eact boundary controllability problem (2.2.4) with f satisfying the minimum boundary L 2 norm property. Furthermore, if the solution to (2.2.4) admits a unique solution (u e, f e ), then f αβγ f e in L 2 (Q) and u αβγ u e in L 2 (Q) as α, β, γ. (2.2.6) Proof. (u αβγ, f αβγ ) satisfies (2.2.2). Passing to the limit in that equation as α, β, γ and using relations (2.2.3), (2.2.4) and (2.2.5) we obtain: T T u(v tt v ) d dt + [Ψ(u) f]v d dt Ω Ω + v t=t Z() d v t= z d W ()( t v) t=t d Ω Ω Ω + (w t v) t= d = v C 2 ([, T ]; H 2 (Ω) H(Ω)). Ω This implies that (u, f) is a solution to the eact boundary controllability problem (2.2.4). To prove that f satisfies the minimum boundary L 2 norm property, we proceeds as follows. Let (u e, f e ) denotes a eact minimum boundary L 2 norm solution to the controllability problem (2.2.4). Since (u αβγ, f αβγ ) is an optimal solution, we have that α 2 u αβγ U 2 L 2 (Q) + β 2 u αβγ W 2 L 2 (Ω) + γ 2 tu αβγ Z 2 H (Ω) + 2 f αβγ 2 L 2 (Q) = J (u αβγ, f αβγ ) J (u e, f e ) = 2 f e 2 L 2 (Q) so that f αβγ 2 L 2 (Q) f e 2 L 2 (Q).

18 3 Passing to the limit in the last estimate we obtain f 2 L 2 (Q) f e 2 L 2 (Q). (2.2.7) Hence we conclude that (u, f) is a minimum boundary L 2 norm solution to the eact boundary controllability problem (2.2.4). Furthermore, if the eact controllability problem (2.2.4) admits a unique minimum boundary L 2 norm solution (u e, f e ), then (u, f) = (u e, f e ) and (2.2.6) follows from assumption (2.2.4). Remark If the wave equation is linear, i.e., Ψ =, then assumptions i) and (2.2.5) are redundant. Remark Assumptions ii) and iii) hold if f αβγ and u αβγ converges pointwise as α, β, γ. Remark A practical implication of Theorem is that one can prove the eact controllability for semilinear wave equations by eamining the behavior of a sequence of optimal solutions (recall that eact controllability was proved only for some special classes of semilinear wave equations.) If we have found a sequence of optimal control solutions {(u αn β n γ n, f αn β n γ n )} where α n, β n, γ n and this sequence appears to satisfy the convergence assumptions ii) and iii), then we can confidently conclude that the underlying semilinear wave equation is eactly controllable and the optimal solution (u αnβnγ n, f αnβnγ n ) when n is large provides a good approimation to the minimum boundary L 2 norm eact controller (u e, f e ). 2.3 Shooting methods for D control problems The basic idea for a shooting method is to convert the solution of a two-point boundary value problem into that of an initial value problem (IVP). The IVP corresponding

19 4 to the optimality system (4.3.8) is described by u tt u + Ψ(u) = ξ in (, T ) Ω, u Ω =, u(, ) = w(), u t (, ) = z() ; ξ tt ξ + [Ψ (u)] ξ = α 2 K (u) in (, T ) Ω, (2.3.8) ξ Ω =, ξ(, ) = ω(), ξ t (, ) = θ(), with unknown initial values ω and θ. Then the goal is to choose ω and θ such that the solution (u, ξ) of the IVP (4.3.9) satisfies the terminal conditions ξ(t, ) = γ 2 Φ 2(u t (T, )) and ξ t (T, ) = β 2 Φ (u(t, )). (2.3.9) A shooting method for solving (4.3.8) consists of the following main steps: choose initial guesses ω and θ; for m =, 2,, M solve for (u, ξ) from the IVP (4.3.8) update ω and θ. A criterion for updating (ω, θ) can be derived from the terminal conditions (4.3.2). A method for solving the nonlinear system (4.3.2) (as a system for the unknowns ω and θ) will yield an updating formula and here we invoke the well-known Newton s method to do so. Also, a discrete version of the algorithm must be used in actual implementations. For the ease of eposition we describe in details a Newton s-method-based shooting algorithm with finite difference discretizations in one space dimension. Algorithms in higher dimensions and their implementations will be briefly discussed subsequently and will form an prominent part of the proposed research.

20 Algorithm for D control problems In one dimension, we discretize the spatial interval [, X] into = < < 2 < 3 < < I+ = X with a uniform spacing h = X/(I + ) and we divide the time horizon [, T ] into = t < t 2 < t 3 < < t N = T with a uniform time step length δ = T/(N ). We use the central differencing scheme to approimate the initial value problem (4.3.9): u i = w i, u 2 i = w i + δz i, ξ i = ω i, ξ 2 i = ξ i + δθ i, i =, 2,, I ; u n+ i ξ n+ i = u n i + λu n i + 2( λ)u n i + λu n i+ δ 2 ξ n i δ 2 Ψ(u n i ), i =, 2,, I, = ξ n i + λξ n i + 2( λ)ξ n i + λξ n i+ + δ 2 α 2 K(un i ) δ 2 [Ψ (u n i )] ξ n i, i =, 2,, I ; (2.3.2) where λ = (δ/h) 2 (we also use the convention that u n = ξ n = u n I+ = ξn I+ =.) The gists of a discrete shooting method are to regard the discrete terminal conditions F 2i ξn i i =, 2,, I. ξ N i δ + β 2 Φ (u N i ) =, F 2i ξi N γ 2 Φ 2( un i u N i ) =, δ (2.3.2) By denoting q n ij = q n ij(ω, θ, ω 2, θ 2,, ω I, θ I ) = un i ω j, ρ n ij = ρ n ij(ω, θ, ω 2, θ 2,, ω I, θ I ) = ξn i ω j, r n ij = r n ij(ω, θ, ω 2, θ 2,, ω I, θ I ) = un i θ j, τ n ij = τ n ij(ω, θ, ω 2, θ 2,, ω I, θ I ) = ξn i θ j,

21 6 we may write Newton s iteration formula as (ω new, θ new, ω new 2, θ new 2,, ωi new, θi new ) T = (ω, θ, ω 2, θ 2,, ω I, θ I ) T [F (ω, θ, ω 2, θ 2,, ω I, θ I )] F (ω, θ, ω 2, θ 2,, ω I, θ I ) where the vector F and Jacobian matri J = F are defined by F 2i = ξn i ξ N i δ J 2i,2j = ρn ij ρ N ij δ J 2i,2j = ρ N ij γ 2δ Φ 2( un i u N i δ J 2i,2j = τij N γ 2δ Φ 2( un i u N i δ + β 2 Φ (u N i ), F 2i = ξi N γ 2 Φ 2( un i u N i ), δ + β 2 Φ (u N i )qij N, J 2i,2j = τ ij N τ N ij )(q N ij q N ij ), )(r N ij r N ij ). δ + β 2 Φ (u N i )rij N, Moreover, by differentiating (4.4.2) with respect to ω j and θ j we obtain the equations for determining q ij, r ij, ρ ij and τ ij : q ij =, q 2 ij =, r ij =, r 2 ij =, ρ ij = δ ij, ρ 2 ij = δ ij, τ ij =, τ 2 ij = δδ ij, i, j =, 2,, I ; q n+ ij r n+ ij ρ n+ ij τ n+ ij = q n ij = r n ij = ρ n ij + λq n i,j + 2( λ)q n ij + λq n i+,j δ 2 ρ n ij δ 2 Ψ (u n i )q n ij, i, j =, 2,, I, + λr n i,j + 2( λ)r n ij + λr n i+,j δ 2 τ n ij δ 2 Ψ (u n i )r n ij, i, j =, 2,, I, + λρ n i,j + 2( λ)ρ n ij + λρ n i+,j + δ 2 α 2 K (u n i )q N ij δ 2 [Ψ (u n i )] ρ n i δ 2 [Ψ (u n i )q N i ] ξ n i, i, j =, 2,, I ; = τ n ij + λτ n i,j + 2( λ)τ n ij + λτ n i+,j + δ 2 α 2 K (u n i )r N ij δ 2 [Ψ (u n i )] ρ n i δ 2 [Ψ (u n i )r N i ] ξ n i, i, j =, 2,, I ; (2.3.22)

22 7 where δ ij is the Chronecker delta. Thus, we have the following Newton s-method-based shooting algorithm: Algorithm Newton method based shooting algorithm with Euler discretizations for distributed optimal control problems choose initial guesses ω i and θ i, i =, 2,, I; % set initial conditions for u and ξ for i =, 2,, I u i = w i, u 2 i = w i + δz i, ξ i = ω i, ξ 2 i = ξ i + δθ i ; % set initial conditions for q ij, r ij, ρ ij, τ ij for j =, 2,, I for i =, 2,, I q ij =, q 2 ij =, r ij =, r 2 ij =, ρ ij =, ρ 2 ij =, τ ij =, τ 2 ij = ; ρ jj =, ρ 2 ij =, τ 2 jj = δ, % Newton iterations for m =, 2,, M % solve for (u, ξ) for n = 2, 3,, N u n+ i ξ n+ i = u n i + λu n i + 2( λ)u n i + λu n i+ δ 2 ξ n i δ 2 Ψ(u n i ), = ξ n i +λξ n i +2( λ)ξ n i +λξ n i++δ 2 α 2 K(un i ) δ 2 [Ψ (u n i )] ξ n i ; % solve for q, r, ρ, τ for j =, 2,, I for n = 2, 3,, N for i = 2,, N q n+ ij = q n ij + λq n i,j + 2( λ)q n ij + λq n i+,j

23 8 r n+ ij ρ n+ ij τ n+ ij % evaluate F and F for i =, 2,, I F 2i = ξn i for j =, 2,, I δ 2 ρ n ij δ 2 Ψ (u n i )q n ij, = r n ij + λr n i,j + 2( λ)r n ij + λr n i+,j δ 2 τ n ij δ 2 Ψ (u n i )r n ij, = ρ n ij + λρ n i,j + 2( λ)ρ n ij + λρ n i+,j +δ 2 α 2 K (u n i )q N ij δ 2 [Ψ (u n i )] ρ n i δ 2 [Ψ (u n i )q N i ] ξ n i, = τ n ij + λτ n i,j + 2( λ)τ n ij + λτ n i+,j +δ 2 α 2 K (u n i )r N ij δ 2 [Ψ (u n i )] ρ n i δ 2 [Ψ (u n i )r N i ] ξ n i ; % (we need to build into the algorithm the following: q n = r n = ρ n = τ n =, q n I+ = rn I+ = ρn I+ = τ n I+ =.) ξn i δ + β 2 Φ (u N i ), F 2i = ξ N i ; J 2i,2j = ρn ij ρn ij + β δ 2 Φ (u N i )qij N, J 2i,2j = τ ij N N τij + β δ 2 Φ (u N i )rij N, J 2i,2j = ρ N ij γ 2δ Φ J 2i,2j = τ N ij γ 2δ Φ 2( un i un i δ 2( un i un i δ solve Jc = F by Gaussian eliminations; for i =, 2,, I ω new i = ω i + c 2i, θ new i = θ i + c 2i ; if ma i ω new i ω i + ma i θ new i otherwise, reset ω i = ω new i )(q N ij q N ij ), )(r N ij r N ij ); θ i < tol, stop; and θ i = θ new i, i =, 2,, I;

24 D Computational results We consider eamples of the following types : Seek the pair (u, f) that minimizes the cost functional J (u, f) = α 2 T Ω u U 2 d dt + β 2 Ω u(t, ) W () 2 d + 2 T Ω f 2 d dt subject to the wave equation u tt u + Ψ(u) = f in (, T ) Ω, u Ω =, u(, ) = g(), u t (, ) = h(). (2.3.23) Eample (linear case) Ψ(u) =, T =, Ω = [, ]. For given target functions, W () = sin(2π) sin(2πt ), U(t, ) = sin(2π) sin(2πt), (2.3.24) it can be verified that the eact optimal solution u and a corresponding Lagrange multiplier ξ are determined by (4.3.8). u(t, ) = sin(2π) sin(2πt), ξ(t, ) =. (2.3.25)

25 2 t = t = t = t= Figure 2. Optimal solution u and target W, U for t = /4 and = /2 : optimal solution u(t, ) : target functions W (), U(t, ) : eact optimal solution α =, β =

26 2 t = t = t = t= Figure 2.2 Optimal solution u and target W, U for t = /4 and = /2 :optimal solution u(t, ) : target functions W (), U(t, ) : eact optimal solution α =, β =

27 22 Eample (linear case) Ψ(u) =, T =, Ω = [, ]. For given target functions, W () =, U(t, ) =..4.2 t = t = t = t= Figure 2.3 Optimal solution u and target W, U for t = / and = /5 : optimal solution u(t, ) : target functions W (), U(t, ) α =, β =

28 23 Eample (nonlinear case) Ψ(u) = u 3 u, T =, Ω = [, ]. For given target functions, W () = sin π cos πt U(t, ) = sin π cos πt. (2.3.26).8.2 t = t = t = t=..4 Figure 2.4 Optimal solution u and target W, U for t = / and = /5 : optimal solution u(t, ) : target functions W (), U(t, ) α =, β =

29 24 Eample (nonlinear case) Ψ(u) = e u, T =, Ω = [, ]. For given target functions, W () = sin π cos T U(t, ) = sin π cos t. (2.3.27) t =.24.2 t = t=..2 t = Figure 2.5 Optimal solution u and target W, U for t = / and = /5 : optimal solution u(t, ) : target functions W (), U(t, ) α =, β =

30 25 Eample (Sine-Gordon equation) Ψ(u) = sin u, T =, Ω = [, ]. For given target functions, W () = sin π cos T U(t, ) = sin π cos t. (2.3.28) t =.24.2 t = t = t=. Figure 2.6 Optimal solution u and target W, U for t = / and = /5 : optimal solution u(t, ) : target functions W (), U(t, ) α =, β =

31 Shooting methods in 2D control problems The basic idea for a shooting method in 2D is eactly the same as in D. In two dimensional case, we replace u by u. Lagrange multiplier rules provide the same optimality system of equations as in the previous section ecept the replacement part. Thus the IVP corresponding to the optimality system (4.3.8) is described by u tt u + Ψ(u) = ξ in (, T ) Ω, u Ω =, u(, ) = w(), u t (, ) = z() ; ξ tt ξ + [Ψ (u)] ξ = α 2 K (u) in (, T ) Ω, (2.4.29) ξ Ω =, ξ(, ) = ω(), ξ t (, ) = θ(), with unknown initial values ω and θ. Then the goal is to choose ω and θ such that the solution (u, ξ) of the IVP (2.4.29) satisfies the terminal conditions ξ(t, ) = γ 2 Φ 2(u t (T, )) and ξ t (T, ) = β 2 Φ (u(t, )). (2.4.3) 2.4. Algorithm for 2D control problems In two dimension, we discretize the spatial interval [, X], [, Y ] into = < < 2 < 3 < < I+ = X, = y < y < y 2 < y 3 < < y J+ = Y with a uniform spacing h = X/(I+), h y = Y/(J +) respectively, and we divide the time horizon [, T ] into = t < t 2 < t 3 < < t N = T with a uniform time step length δ = T/(N ). We use the central difference scheme to approimate the initial value problem (2.4.29): For i =, 2,, I, j =, 2,, J, u ij = w ij, u 2 ij = w ij + δz ij, ξ ij = ω ij, ξ 2 ij = ξ ij + δθ ij ; u n+ ij = u n ij + λ (u n i,j + u n i+,j) + 2( λ λ y )u n ij + λ y (u n i,j + u n i,j+) δ 2 ξ n ij δ 2 Ψ(u n ij), (2.4.3) ξ n+ ij = ξ n ij + λ (ξ n i,j + ξ n i+,j) + 2( λ λ y )ξ n ij + λ y (ξ n i,j + ξ n i,j+) + δ 2 α 2 K(un ij) δ 2 [Ψ (u n ij)] ξ n ij,

32 27 where λ = (δ/h ) 2, λ y = (δ/h y ) 2 (we also use the convention that u n ij = ξ n ij = if i = or I + or j = or j = J +.) The gists of a discrete shooting method are to regard the discrete terminal conditions: For i =, 2,, I, j =, 2,, J, F 2(ij) ξn ij ξ N ij δ + β 2 Φ (u N ij ) =, F 2(ij) ξij N γ 2 Φ 2( un ij u N ij ) =, (2.4.32) δ where (ij) is a reordering of the nodes with respect to X, Y ecept boundary points. Let ω() {ω, ω 2,, ω IJ } = {ω, ω 2,, ω IJ }, and θ() {θ, θ 2,, θ IJ } = {θ, θ 2,, θ IJ } where IJ = I J. By denoting q n (ij) k = q n (ij) k(ω, θ, ω 2, θ 2,, ω IJ, θ IJ ) = un ij ω k, r n (ij) k = r n (ij) k(ω, θ, ω 2, θ 2,, ω IJ, θ IJ ) = un ij θ k, ρ n (ij) k = ρ n (ij) k(ω, θ, ω 2, θ 2,, ω IJ, θ IJ ) = ξn ij ω k, τ n (ij) k = τ n (ij) k(ω, θ, ω 2, θ 2,, ω IJ, θ IJ ) = ξn ij θ k, we may write Newton s iteration formula as (ω new,θ new, ω new 2, θ new 2,, ω new IJ, θnew IJ )T = (ω, θ, ω 2, θ 2,, ω IJ, θ IJ ) T [F (ω, θ, ω 2, θ 2,, ω IJ, θ IJ )] F (ω, θ, ω 2, θ 2,, ω IJ, θ IJ ) where the vector F and Jacobian matri J = F are defined by F 2(ij) ξn ij ξ N ij δ + β 2 Φ (u N ij ), F 2(ij) ξij N γ 2 Φ 2( un ij u N ij ), δ J 2(ij),2k = ρn (ij) ρ N k (ij) k + β δ 2 Φ (u N ij )q(ij) N, k J 2(ij),2k = τ (ij) N τ N k (ij) k + β δ 2 Φ (u N ij )r(ij) N, k J 2(ij),2k = ρ N (ij) γ k 2δ Φ 2( un ij u N ij δ J 2(ij),2k = τ(ij) N γ k 2δ Φ 2( un ij u N ij δ )(q N (ij) k qn (ij) k ), )(r N (ij) k rn (ij) k ).

33 28 Moreover, by differentiating (2.4.3) with respect to ω k and θ k we obtain the equations for determining q (ij) k, r (ij) k, ρ (ij) k and τ (ij) k: q (ij) k =, q2 (ij) k =, r (ij) k =, r2 (ij) k =, ρ (ij) k = δ (ij) k, ρ 2 (ij) k = δ (ij) k, i, j =, 2,, I, J ; τ (ij) k =, τ 2 (ij) k = δδ (ij) k, q n+ (ij) k = qn (ij) k + λ (q n (i,j) k + q n (i+,j) k) + 2( λ λ y )q n (ij) k + λ y (q n (i,j ) k + q n (i,j+) k) δ 2 ρ n (ij) k δ 2 Ψ (u n ij)q n (ij) k, i, j =, 2,, I, J, r n+ (ij) k = rn (ij) k + λ (r n (i,j) k + r n (i+,j) k) + 2( λ λ y )r n (ij) k + λ y (r n (i,j ) k + r n (i,j+) k) δ 2 τ n (ij) k (2.4.33) δ 2 Ψ (u n ij)r n (ij) k, i, j =, 2,, I, J, ρ n+ (ij) k = ρn (ij) k + λ (ρ n (i,j) k + ρ n (i+,j) k) + 2( λ λ y )ρ n (ij) k + λ y (ρ n (i,j ) k + ρ n (i,j+) k) + δ 2 α 2 K (u n ij)q N (ij) k δ 2 [Ψ (u n i )] ρ n (ij) k δ 2 [Ψ (u n i )q N (ij) k] ξ n ij, i, j =, 2,, I, J ; τ n+ n (ij) k = τ(ij) k + λ (τ(i,j) n k + τ(i+,j) n k) + 2( λ λ y )τ(ij) n k + λ y (τ n (i,j ) k + τ n (i,j+) k) + δ 2 α 2 K (u n ij)r N (ij) k δ 2 [Ψ (u n ij)] ρ n (ij) k δ 2 [Ψ (u n ij)r N (ij) k] ξ n ij, i, j =, 2,, I, J ; where δ ij is the Chronecker delta. Thus, we have the following Newton s-method-based shooting algorithm: Algorithm 2 Newton method based shooting algorithm with Euler discretizations for distributed optimal control problems choose initial guesses ω ij and θ ij, i, j =, 2,, I, J;

34 29 % set initial conditions for u and ξ for i, j =, 2,, I, J u ij = w ij, u 2 ij = w ij + δz ij, ξ ij = ω ij, ξ 2 ij = ξ ij + δθ ij ; % set initial conditions for q (ij) k, r (ij) k, ρ (ij) k, τ (ij) k for k =, 2,, (IJ) for i =, 2,, I for j =, 2,, J % Newton iterations for m =, 2,, M q (ij) k =, q2 (ij) k =, r (ij) k =, r2 (ij) k =, ρ (ij) k =, ρ2 (ij) k =, τ (ij) k =, τ 2 (ij) k = ; ρ kk =, ρ2 (ij) k =, τ 2 kk = δ, % solve for (u, ξ) for n = 2, 3,, N u n+ ij ξ n+ ij = u n ij + λ (u n i,j + u n i+,j) + 2( λ λ y )u n ij +λ y (u n i,j + u n i,j+), = ξ n ij +λ (ξi,j n +ξi+,j)+2( λ n λ y )ξij n +λ y (ξi,j +ξ n i,j+) n % solve for q, r, ρ, τ for j =, 2,, I +δ 2 α 2 K(un ij) δ 2 [Ψ (u n ij)] ξ n ij; for n = 2, 3,, N for i = 2,, N q n+ (ij) k = qn (ij) k + λ (q n (i,j) k + qn (i+,j) k ) +2( λ λ y )q n (ij) k + λ y(q n (i,j ) k + qn (i,j+) k ) δ 2 ρ n (ij) k δ2 Ψ (u n ij)q n (ij) k,

35 3 % evaluate F and F for i, j =, 2,, I, J r n+ (ij) k = rn (ij) k + λ (r n (i,j) k + rn (i+,j) k ) +2( λ λ y )r n (ij) k + λ y(r n (i,j ) k + rn (i,j+) k ) δ 2 τ n (ij) k δ2 Ψ (u n ij)r n (ij) k, ρ n+ (ij) k = ρn (ij) + λ (ρ n (i,j) k + ρn (i+,j) k ) +2( λ λ y )ρ n (ij) k + λ y(ρ n (i,j ) k + ρn (i,j+) k ) +δ 2 α 2 K (u n ij)q N (ij) k δ2 [Ψ (u n i )] ρ n ij δ 2 [Ψ (u n i )q N ij ] ξ n ij τ n+ n (ij) k = τ(ij) k + λ (τ(i,j) n k + τ (i+,j) n k ) +2( λ λ y )τ n (ij) k + λ y(τ n (i,j ) k + τ n (i,j+) k ) +δ 2 α 2 K (u n ij)r N (ij) k δ2 [Ψ (u n ij)] ρ n ij δ 2 [Ψ (u n ij)r N i ] ξ n ij % (we need to build into the algorithm the following: q n = r n = ρ n = τ n =, q n I+ = rn I+ = ρn I+ = τ n I+ =.) F 2(ij) ξn ij ξn ij + β δ 2 Φ (u N ij ), F 2(ij) ξij N ; for j =, 2,, I J 2(ij),2k = ρn (ij) k ρn (ij) k δ + β 2 Φ (u N ij )q(ij) N k, J 2(ij),2k = τ (ij) N N τ k (ij) k + β δ 2 Φ (u N ij )r(ij) N k, J 2(ij),2k = ρ N (ij) k γ 2δ Φ J 2(ij),2k = τ N (ij) k γ 2δ Φ 2( un ij un ij )(q N δ (ij) k 2( un ij un ij )(r N δ (ij) k solve Jc = F by Gaussian eliminations; for i, j =, 2,, I, J ω new ij = ω ij + c 2(ij), θ new ij = θ ij + c 2(ij) ; if ma ij ω new ij ω ij + ma ij θ new ij otherwise, reset ω ij = ω new ij θ ij < tol, stop; q N (ij) k ), r N (ij) k ); and θ ij = θ new ij, i, j =, 2,, I, J;

36 D computational results For the following eamples, we provide several graphs with fied time and y-coordinates in order to observe the computational results easily, so called snap-shot. For fied time, we present the three graphs with y-coordinates.25,.5, and.75 from left to right. Eample (Linear case) Ψ(u) =, T =, Ω = [, ] [, ]. For given target functions, W (, y) = ( )y(y ) cos, U(t,, y) = ( )y(y ) cos t. (2.4.34) t = t = t = t = Figure 2.7 Optimal solution u and target W, U for t = /36, = /6 and y = /6 : optimal solution u(t, ) : target functions W (), U(t, ) α =, β =

37 t = t = t = t = Figure 2.8 Optimal solution u and targets W, U for t = /36, = /6 and y = /6 : optimal solution u(t, ) : target functions W (), U(t, ) α =, β =

38 33 Eample (Nonlinear case) Ψ(u) = u 3 u, T =, Ω = [, ] [, ]. For given target functions, W (, y) = cos π sin π sin πy U(t,, y) = cos πt sin π sin πy. (2.4.35) t = t = t = t = Figure 2.9 Optimal solution u and target W, U for t = /36, = /6 and y = /6 : optimal solution u(t, ) : target functions W (), U(t, ) α =, β =

39 34 3 Shooting Methods for Numerical Solutions of Eact Controllability Problems Constrained by Linear and Nonlinear Wave Equations with Local Distributed Controls Numerical solutions of eact controllability problems for linear and semilinear wave equations with local distributed controls are studied. In chapter, we introduced the optimality system of equations and the corresponding algorithm for shooting method. For the nonhomogeneous term in the state equation, we multiply a characteristic function which will render the problems local controllability cases. The algorithm for these problems is a simple modification of the algorithm in chapter 2, and so we will skip the section of the algorithm. The convergence properties of the numerical shooting method in the contet of eact controllability are illustrated through computational eperiments. 3. Eact controllability problems for the wave equations We will study numerical methods for optimal control and eact controllability problems with local distributed controls associated with the linear and nonlinear wave equations. As before, our concern is to investigate the relevancy and applicability of high performance computing (HPC) for these problems. As an prototype eample of optimal control problems for the wave equations with local distributed controls, we consider the following distributed optimal control problem: choose a control f and a corresponding u such that the pair (f, u) minimizes the cost

40 35 functional J (u, f) = α 2 = α 2 T + 2 T T + 2 K(u) d dt + β Φ (u(t, )) d + γ Ω 2 Ω 2 f 2 d dt Ω T K(u) d dt + β Ω 2 χ Ω f 2 d dt Ω Ω Φ (u(t, )) d + γ 2 Ω Ω Φ 2 (u t (T, )) d Φ 2 (u t (T, )) d (3..) subject to the wave equation u tt u + Ψ(u) = χ Ω f in (, T ) Ω, u Ω =, u(, ) = w(), u t (, ) = z(). (3..2) Here Ω is a bounded spatial domain in R d (d = or 2 or 3) with a boundary Ω and Ω Ω; u is dubbed the state, and g is the distributed control. Also, K, Φ and Ψ are C mappings (for instance, we may choose K(u) = (u U) 2, Ψ(u) =, Ψ(u) = u 3 u and Ψ(u) = sin u, Φ (u) = (u(t, ) W ) 2, Φ 2 (u) = (u t (T, ) Z) 2, where U, W, Z is a target function.) Using Lagrange multiplier rules one finds the following optimality system of equations that the optimal solution (f, u) must satisfy: u tt u + Ψ(u) = χ Ω f in (, T ) Ω u Ω =, u(, ) = w(), u t (, ) = z() ; ξ tt ξ + [Ψ (u)] ξ = α 2 K (u) in Q ξ Ω =, ξ(t, ) = γ 2 Φ 2(u t (T, )), ξ t (T, ) = β 2 Φ (u(t, )) ; χ Ω f + χ Ω ξ = in Q.

41 36 This system may be simplified as u tt u + Ψ(u) = χ Ω ξ in (, T ) Ω u Ω =, u(, ) = w(), u t (, ) = z() ; ξ tt ξ + [Ψ (u)] ξ = α 2 K (u) in (, T ) Ω (3..3) ξ Ω =, ξ(t, ) = γ 2 Φ 2(u t (T, )), ξ t (T, ) = β 2 Φ (u(t, )). Such control problems are classical ones in the control theory literature; see, e.g., [5] for the linear case and [6] for the nonlinear case regarding the eistence of optimal solutions as well as the eistence of a Lagrange multiplier ξ satisfying the optimality system of equations. However, numerical methods for finding discrete (e.g., finite element and/or finite difference) solutions of the optimality system are largely limited to gradient type methods which are sequential in nature and generally require many iterations for convergence. The optimality system involves boundary conditions at t = and t = T and thus cannot be solved by marching in time. Direct solutions of the discrete optimality system, of course, are bound to be epensive computationally in 2 or 3 spatial dimensions since the problem is (d + ) dimensional (where d is the spatial dimensions.) 3.2 The solution of the eact local controllability problem as the limit of optimal control solutions The eact distributed local control problem we consider is to seek a distributed local control f L 2 ((, T ) Ω ) where Ω Ω and a corresponding state u such that the

42 37 following system of equations hold: u tt u + Ψ(u) = χ Ω f in Q (, T ) Ω, u t= = w and u t t= = z in Ω, u t=t = W and u t t=t = Z in Ω, u Ω = in (, T ). (3.2.4) Under suitable assumptions on f and through the use of Lagrange multiplier rules, the corresponding optimal local control problem: minimize (3..) with respect to the control f subject to (3..2). (3.2.5) In this section we establish the equivalence between the limit of optimal solutions and the minimum distributed L 2 norm eact local controller. We will show that if α, β and γ, then the corresponding optimal solution (û αβγ, f αβγ ) converges weakly to the minimum distributed L 2 norm solution of the eact distributed local controllability problem (3.2.4). The same is also true in the discrete case. Let Q = (, T ) Ω. Theorem Assume that the eact distributed local controllability problem (3.2.4) admits a unique minimum distributed L 2 norm solution (u e, f e ). Assume that for every (α, β, γ) R + R + R + (where R + is the set of all positive real numbers,) there eists a solution (u αβγ, f αβγ ) to the optimal local control problem (3.2.5). Then f αβγ L 2 (Q ) f e L 2 (Q ) (α, β, γ) R + R + R +. (3.2.6) Assume, in addition, that for a sequence {(α n, β n, γ n )} satisfying α n, β n

43 38 and γ n, Then u αn β n γ n u in L 2 (Q) and Ψ(u αn β n γ n ) Ψ(u) in L 2 (, T ; [H 2 (Ω) H (Ω)] ). (3.2.7) f αnβ nγ n f e in L 2 (Q ) and u αnβ nγ n u e in L 2 (Q) as n. (3.2.8) Furthermore, if (3.2.7) holds for every sequence {(α n, β n, γ n )} satisfying α n, β n and γ n, then f αβγ f e in L 2 (Q ) and u αβγ u e in L 2 (Q) as α, β, γ. (3.2.9) Proof. Since (u αβγ, f αβγ ) is an optimal solution, we have that α 2 u αβγ(t ) U L 2 (Q) + β 2 u αβγ(t ) W L 2 (Ω) + γ 2 tu αβγ (T ) Z H (Ω) + 2 f αβγ L 2 (Q ) = J (u αβγ, f αβγ ) J (u e, f e ) = 2 f e L 2 (Q ) so that (3.2.6) holds, u αβγ t=t W in L 2 (Ω) and ( t u αβγ ) t=t Z in H (Ω) as α, β, γ. (3.2.) Let {(α n, β n, γ n )} be the sequence in (3.2.7). Estimate (3.2.6) implies that a subsequence of {(α n, β n, γ n )}, denoted by the same, satisfies f αn β n γ n f in L 2 (Q ) and f L 2 (Q ) f e L 2 (Q ). (3.2.)

44 39 (u αβγ, f αβγ ) satisfies the initial value problem in the weak form: T T u αβγ (v tt v ) d dt + [Ψ(u αβγ ) χ Ω f αβγ ]v d dt Ω Ω + (v t u αβγ ) t=t d v t= z d (u αβγ t v) t=t d Ω Ω Ω + (w t v) t= d = v C 2 ([, T ]; H 2 (Ω) H(Ω)) Ω (3.2.2) Passing to the limit in (3.2.2) as α, β, γ and using relations (3.2.) and (3.2.) we obtain: T T u(v tt v ) d dt + [Ψ(u) χ Ω f]v d dt Ω Ω + v t=t Z() d v t= z d W ()( t v) t=t d Ω Ω Ω + (w t v) t= d = v C 2 ([, T ]; H 2 (Ω) H(Ω)). Ω The last relation and (3.2.) imply that (u, f) is a minimum boundary L 2 norm solution to the eact control problem (3.2.4). Hence, u = u e and f = f e so that (3.2.8) and (3.2.9) follows from (3.2.7) and (3.2.). Remark If the wave equation is linear, i.e., Ψ =, then assumption (3.2.7) is redundant and (3.2.9) is guaranteed to hold. Indeed, (3.2.2) implies the boundedness of { u αβγ L 2 (Q)} which in turn yields (3.2.7). The uniqueness of a solution for the linear wave equation implies (3.2.7) holds for an arbitrary sequence {(α n, β n, γ n )}. Theorem Assume that i) for every (α, β, γ) R + R + R + there eists a solution (u αβγ, f αβγ ) to the optimal control problem (3.2.5);

45 4 ii) the limit terminal conditions hold: u αβγ t=t W in L 2 (Ω) and ( t u αβγ ) t=t Z in H (Ω) as α, β, γ ; (3.2.3) iii) the optimal solution (u αβγ, f αβγ ) satisfies the weak limit conditions as α, β, γ : f αβγ f in L 2 (Q ), u αβγ u in L 2 (Q), (3.2.4) and Ψ(u αβγ ) Ψ(u) in L 2 (, T ; [H 2 (Ω) H (Ω)] ) (3.2.5) for some f L 2 (Q ) and u L 2 (Q). Then (u, f) is a solution to the eact boundary controllability problem (3.2.4) with f satisfying the minimum boundary L 2 norm property. Furthermore, if the solution to (3.2.4) admits a unique solution (u e, f e ), then f αβγ f e in L 2 (Q ) and u αβγ u e in L 2 (Q) as α, β, γ. (3.2.6) Proof. (u αβγ, f αβγ ) satisfies (3.2.2). Passing to the limit in that equation as α, β, γ and using relations (3.2.3), (3.2.4) and (3.2.5) we obtain: T T u(v tt v ) d dt + [Ψ(u) χ Ω f]v d dt Ω Ω + v t=t Z() d v t= z d W ()( t v) t=t d Ω Ω Ω + (w t v) t= d = v C 2 ([, T ]; H 2 (Ω) H(Ω)). Ω This implies that (u, f) is a solution to the eact boundary controllability problem (3.2.4).

46 4 To prove that f satisfies the minimum boundary L 2 norm property, we proceeds as follows. Let (u e, f e ) denotes a eact minimum boundary L 2 norm solution to the controllability problem (3.2.4). Since (u αβγ, f αβγ ) is an optimal solution, we have that α 2 u αβγ U 2 L 2 (Q) + β 2 u αβγ W 2 L 2 (Ω) + γ 2 tu αβγ Z 2 H (Ω) + 2 f αβγ 2 L 2 (Q ) = J (u αβγ, f αβγ ) J (u e, f e ) = 2 f e 2 L 2 (Q ) so that f αβγ 2 L 2 (Q ) f e 2 L 2 (Q ). Passing to the limit in the last estimate we obtain f 2 L 2 (Q ) f e 2 L 2 (Q ). (3.2.7) Hence we conclude that (u, f) is a minimum boundary L 2 norm solution to the eact boundary controllability problem (3.2.4). Furthermore, if the eact controllability problem (3.2.4) admits a unique minimum boundary L 2 norm solution (u e, f e ), then (u, f) = (u e, f e ) and (3.2.6) follows from assumption (3.2.4). Remark If the wave equation is linear, i.e., Ψ =, then assumptions i) and (3.2.5) are redundant. Remark Assumptions ii) and iii) hold if f αβγ and u αβγ converges pointwise as α, β, γ. Remark A practical implication of Theorem is that one can prove the eact controllability for semilinear wave equations by eamining the behavior of a sequence of optimal solutions (recall that eact controllability was proved only for some special classes of semilinear wave equations.) If we have found a sequence of optimal

47 42 control solutions {(u αn β n γ n, f αn β n γ n )} where α n, β n, γ n and this sequence appears to satisfy the convergence assumptions ii) and iii), then we can confidently conclude that the underlying semilinear wave equation is eactly controllable and the optimal solution (u αnβnγ n, f αnβnγ n ) when n is large provides a good approimation to the minimum boundary L 2 norm eact controller (u e, f e ). 3.3 Computational results We consider eamples of the following types : Seek the pair (u, f) that minimizes J (u, f) = α 2 T + γ 2 Ω Ω u U 2 d dt + β u(t, ) W () 2 d 2 Ω u t (T, ) Z() 2 d + T f 2 d dt 2 Ω (3.3.8) subject to the wave equation u tt u + Ψ(u) = χ Ω f in (, T ) Ω, u Ω =, (3.3.9) u(, ) = g(), u t (, ) = h(), where Ω Ω Eact controllability problems with linear cases Eample (full domain control) Ψ(u) =, T =, Ω = Ω = [, ]. For given target functions, W () =, Z() = 2π sin(2π), U(t, ) = sin(2π) sin(2πt). (3.3.2)

48 43.5 t= β=γ= β=γ= β=γ= β=γ=.5.5 Figure 3. Optimal solution u and target W for t = /8 and = /4, : optimal solution u(t, ) : target function W () α =, β = γ =,, 9 β=γ= t= β=γ= β=γ= 9.5 Figure 3.2 Optimal solution u t and target Z for t = /8 and = /4, : optimal solution u t (T, ) : target function Z() α =, β = γ =,,

49 Figure 3.3 Optimal solution u and target W for t = /8 and = /4 : optimal solution u(t, ) : target function W () α =, β = γ = 8 t= 8.5 Figure 3.4 Optimal solution u t and target Z for t = /8 and = /4 : optimal solution u t (T, ) : target function Z() α =, β = γ =

50 t Figure 3.5 Optimal solution u for t = /8 and = /4 : optimal solution u(t, ) α =, β = γ =.5 t=.5 Figure 3.6 Optimal solution u and target W for t = /4 and = /4 : optimal solution u(t, ) : target function W () α =, β = γ =

51 46 8 t= 8 Figure 3.7 Optimal solution u t and target Z for t = /4 and = /4 : optimal solution u t (T, ) : target function Z() α =, β = γ =.5 t.5 Figure 3.8 Optimal solution u for t = /4 and = /4 : optimal solution u(t, ) α =, β = γ =

52 47 Eample (linear case and local domain control) Ψ(u) =, T =, Ω = [, ], Ω = [,.] [.9, ]. Suppose we have the same target functions as (3.3.2)..5 t= β=γ=.5.5 Figure 3.9 Optimal solution u for t = /8 and = /4 : optimal solution u(t, ) : target function W () α =, β = γ =

53 48 8 β=γ= t= β=γ= 8.5 Figure 3. Optimal solution u for t = /8 and = /4, : optimal solution u t (T, ) : target function Z() α =, β = γ =, Figure 3. Optimal solution u for t = /8 and = /4 : optimal solution u(t, ) : target function W () α =, β = γ =

54 49 8 t= 8.5 Figure 3.2 Optimal solution u for t = /8 and = /4 : optimal solution u t (T, ) : target function Z() α =, β = γ =.5.5 t Figure 3.3 Optimal solution u for t = /8 and = /4 : optimal solution u(t, ) α =, β = γ =

55 5 Eample (linear case and local domain control) Ψ(u) =, T =, Ω = [, ], Ω = [.45,.55]. Suppose we have the same target functions as (3.3.2)..5 t=.5.5 Figure 3.4 Optimal solution u for t = /8 and = /4 : optimal solution u(t, ) : target function W () α =, β = γ =

56 5 8 t= 8.5 Figure 3.5 Optimal solution u for t = /8 and = /4 : optimal solution u t (T, ) : target function Z() α =, β = γ = t Figure 3.6 Optimal solution u for t = /8 and = /4 : optimal solution u(t, ) α =, β = γ =

57 52 Eample (linear case and local domain control) Ψ(u) =, T = 2, Ω = [, ], Ω = [.45,.55]. Suppose we have the same target functions as (3.3.2)..5 t=2.5.5 Figure 3.7 Optimal solution u for t = /8 and = /4 : optimal solution u(t, ) : target function W () α =, β = γ =

58 53 8 t=2 8.5 Figure 3.8 Optimal solution u for t = /8 and = /4 : optimal solution u t (T, ) : target function Z() α =, β = γ = t 2 Figure 3.9 Optimal solution u for t = /8 and = /4 : optimal solution u(t, ) α =, β = γ =

59 54 Eample (linear case and local domain control) Ψ(u) =, T =, Ω = [, ], Ω = [.495,.55]. Suppose we have the same target functions as (3.3.2) Figure 3.2 Optimal solution u for t = /4 and = /4 : optimal solution u(t, ) : target function W () α =, β = γ =