On the Regularized Solutions of Optimal Control Problem in a Hyperbolic System

and Applied Analysis 3 Definition 2.1. The generalized solution of 1.2 will be defined as the function u ∈ H1 0 Ω , u x, 0 φ x which satisfies the following integral identity: ∫T


Introduction and Statement of the Problem
Optimal control problems for hyperbolic equations have been investigated by Lions in his famous book 1 .Lions examined the problems in detail when the control function is at the right hand side and in the boundary condition of the hyperbolic problem.Furthermore, when the control is in the boundaries 2-4 , in the coefficient 5, 6 , and at the right hand side of the equation 7, 8 , there have been some control problem studies for different types of cost functionals.As for the control of initial conditions, Lions mentioned the control of the initial velocity of the system in detail but stated briefly the control of initial status of the system solving the system in L 2 .
In this study, we consider the following problem of minimizing the cost functional: under the following condition: x, t ∈ Ω : 0, l × 0, T u x, 0 ϕ x , u t x, 0 ψ x , x ∈ 0, l u 0, t 0, u l, t 0, t ∈ 0, T .

1.2
Since the problem is usually ill posed for α 0, we use the parameter α > 0 as the regularization parameter which is the strong convexity constant, and this guaranties the uniqueness and stability of the regularized solution.The functional J α ϕ is called cost functional and the term ϕ 2 H 1 0 is called penalization term; its role is, on one hand, to avoid using "too large" controls in the minimization of J α ϕ and, on the other hand, to assure coercivity for J α ϕ .
Lions in 1 mentioned the observation of u x, T ; ϕ in L 2 0, l and u t x, T ; ϕ in H −1 0, l for the control ϕ ∈ L 2 0, l .Except this study, there is no investigation in the literature about the control of initial status of the hyperbolic system up to now.In this study, we investigate different targets.With the choice of the functional in 1.1 , we use u t x, T ; ϕ and u x x, T ; ϕ , which correspond to final velocity and force, respectively, for the control ϕ ∈ H 1 0 0, l .Since the Fréchet differential of the cost functional cannot be obtained with the usage of usual norm in H 1 0 , we get the differentiability with the only use of H 1 0 -Poincare norm.
The space H 1 0 0, l is a Hilbert subspace of H 1 0, l and the H 1 0 -Poincare inner product and the H 1 0 -Poincare norm are defined, respectively, as We search for Inf We organize this paper as follows.In Section 2, we establish the existence and the uniqueness of the optimal solution.In Section 3, we derive the necessary optimality condition.In Section 4, we construct an algorithm for the numerical approximation of the optimal solution according to steepest descent algorithm.In Section 5, we give symbolic representation for optimal solution by using this algorithm on some examples.

Existence and Uniqueness of the Optimal Solution
First we state the generalized solution of the hyperbolic problem 1.2 .
Definition 2.1.The generalized solution of 1.2 will be defined as the function u ∈ H 1 0 Ω , u x, 0 ϕ x which satisfies the following integral identity: for all v ∈ H 1 0 Ω with v x, T 0. To have this solution the following is needed: Theorem 2.2.Suppose that 2.2 holds, then 1.2 has a unique generalized solution and the following estimate is valid for the solution: Proof of this theorem can easily be obtained by Galerkin method used in [9].
The strategy to prove existence and uniqueness of this optimal control is to use the relationship between minimization of quadratic functionals and variational problems corresponding to symmetric bilinear forms.The key point is to write J α ϕ in the following way: Here is bilinear since the mapping ϕ → u ϕ − u 0 is linear and symmetric.Also, the difference function δu u x, t; ϕ − u x, t; 0 is the solution of the following problem: and for the solution of this problem the following estimates are valid: Hence, we write the following: and this implies the coercivity of π ϕ, ϕ .Since applying Cauchy-Schwartz inequality, we get for δu x, T ; ϕ u x, T ; ϕ − u x, T ; 0 and δu x, T ; η u x, T ; η − u x, T ; 0 .So, we obtain 12 for c 6 max{c 4 , c 5 , α}.Then π ϕ, η is continuous.The functional Lϕ in 2.4 is defined as

2.13
We can easily write that

2.16
Proof of this theorem can easily be obtained by showing the weak lower semicontinuity of J α ϕ as in [1].

Lagrange Multipliers and Optimality Condition
To derive the optimality condition, let us introduce the Lagrangian L u, ϕ, z t , given by

3.1
Notice that L is linear in z t , therefore constitutes the following Euler equation: So, we can state the following theorem in view of 10 .
Theorem 3.1.The control ϕ * and the state u * u ϕ * are optimal if there exists a multiplier z * t ∈ Φ ad such that z * and ϕ * satisfy the following optimality conditions: for ∀ϕ ∈ Φ ad .

An Iterative Algorithm and Its Convergence
Now, we can apply standard steepest descent iteration.Gradient of J α at any ϕ is given by It turns out that −∇J α ϕ plays the role of the steepest descent direction for J α .This suggests an iterative procedure to compute a sequence of controls {ϕ k } convergent to the optimal one.Select an initial control ϕ 0 .If ϕ k is known k ≥ 0 then ϕ k 1 is computed according to the following scheme.
1 Solve the state problem 1.2 in the sense 2.1 and get corresponding u k .
2 Knowing u k solve the adjoint problem 3.4 .
3 Using z k get the gradient ∇J α k .

Set
and select the relaxation parameter β k in order to assure that for sufficiently small β k > 0.
Concerning the choice of the relaxation parameter, there are several possibilities and these can be found in any optimization books.
One of the following can be taken as a stopping criterion to the iteration process: Lemma 4.1.The cost functional 1.1 is strongly convex with the strong convexity constant α.
From the following strongly convex functional definition: we can see that the cost functional 1.1 is strongly convex with the constant χ α.
So, we can give the following theorem which states the convergence of the minimizer to optimal solution.Theorem 4.2.Let ϕ * be optimum solution of the problem 1.1 -1.5 .Then the minimizer given in 4.2 satisfies the following inequality: Proof.If we take β 1/2 in the definition of the strongly convex functional, we write we find 4.9

Numerical Examples
Example 5.1.Let us consider the following problem of minimizing the cost functional: under the following condition:

5.2
Rewrite the functional as where

5.4
Choosing α 0.1 and starting the initial element ϕ 0 sin πx, then we get the minimizing sequence.Here the relaxation parameter β k 0.01 assures the inequality J 0.1 ϕ k 1 < J 0.1 ϕ k .
Example 5.2.We consider the following problem of minimizing the cost functional: We can rewrite the cost functional as

5.8
For Taking α 0.2 and the initial element ϕ 0 0, we obtain a minimizing sequence.In this example β 0.015 and stopping criteria 0.5 × 10 −6 are chosen.

Conclusions
In a hyperbolic problem, the initial condition u x, 0 ϕ x can be controlled from the targets u t x, T ; ϕ and u x x, T ; ϕ using H 1 0 -Poincare norm.The Lagrange multiplier is z t while the function z x, t is the solution of adjoint problem.The symbolic optimal control function is easily obtained in numerical examples.