ESTIMATE FOR OPTIMALITY OF DISTRIBUTED PARAMETER CONTROL PROBLEMS VIA DUALITY

Sharp error estimates for optimality are established for a class of distributed parameter control problems that include elliptic, parabolic, hyperbolic systems with impulsive control and boundary control. The estimates are obtained by constructing manageable dual problems via the extremum principle.


Introduction, Notation and Definitions
We develop a method that generates computable error estimates for optimality as well as dual problems for the optimal control of distributed systems described in the classic work of J.L. Lions [4] and [5]. The approach that we use relies on establishing manageable dual problems as opposed to formidable convex conjugate dual problems. Our approach is elementary in that the quadratic nature of the cost functionals is exploited. The resulting cost functional of the dual system is more explicit than those given in [4] and [5]. Furthermore, we can bypass the requirement of the system equation to be an isomorphism when a conjugate function method is taken in developing a duality theory and our framework permits constrained control sets. The basic idea of our approach is the extremum-principle, which is developed in finite dimensional space in [8] and [1], and here, we successfully extend it to infinite-dimensional problems. Other duality studies for distributed systems can be found in [2,3,6,7].
In the following three sections, we treat error estimates and duality theorems on systems governed by elliptic, parabolic, and hyperbolic equations, respectively. Illustrative examples are given including those involving impulsive control and boundary control. In fact, we believe that the method developed here can cope with almost all situations studied in [4].
We begin with real Hilbert spaces V, H, ell and . Assume that the injection V C_ H is continuous and V is dense in H. Identify H with its dual, denote by V' the dual of V and we may write 178 W.L. CHAN and S.P. YUNG VCHCV'. Both the dual pairing of V and V' as well as other Hilbert spaces and their duals are denoted by 1 (-,-). We write (.,.)v for the inner product of the Hilbert space V, I I I I v for nd drop V when it is clear from the context which space we are referring to. Inner products and norms of other Hilbert spaces are denoted similarly. We denote by (X,Y) the space of continuous linear mappings between the topological vector spaces X and Y.

Elliptic Systems
Assume that we are given operators B (,V') and A (V,V') such that the bilinear form (Au, v) on V is coercive.
For a given f V' and a control u , we are interested in the system given by Ayf + Bu, y Y (2.1) with state y: -y(u). We are also given an observation equation and a cost functional J(u, y) (Cy(u), Cy(u))5 + (Nu, u)q (2.3) where C C (V, 5) and N is a Hermitian positive definite operator on q.t. Let q.l, ad (the set of admissible controls) be a given closed convex subset of . We are interested in the optimal control problem of finding u 0 and Y0: -y(uo) such thatthey satisfy (2.1) and d(u0, Y0) inf{J(u, y): u e q-Lad}. (2.4) We shall develop a dual problem for this optimal control problem and use it to obtain error estimates for optimality. We now introduce a dual problem.
Let A be the canonical isomorphism of 5 onto its dual :', C* C (5', V') be the adjoint of C. Then for , C V we have (C*A:C, > <A:C, C> (C, C)5.
(2.6) Let A* (Y,Y') be the adjoint of A. The state '-(y) Y of the dual system is defined and given by A* C*ACy ' Y.
To prove (2.18), we apply Lernma 2.1 to both sides of the following inequality We get J(uo, Yo) J (v, y) <_ J(u, y) J (v, y).
Putting v u completes the proof.
Example 2.4: Let f be a bounded open set in Nn such that its closure f is a compact manifold with boundary F, which is an (n-1)-dimensional smooth manifold. The Euclidean norm of Nn is denoted by and the inner product in n is denoted by ordinary multiplication for brevity. Set V U(12), V'-H-1(12), U-L2(12). Let A be the second order elliptic operator Furthermore, the error estimates of Theorem 2.3 hold for this problem.

Parabolic Systems
We continue to use the notation of Section 2 and introduce additional notation.
If V is a Hilbert space, we write L2(0, T; V) for the space (of equivalence classes) of functions The cost functional J(u,y) of the system (3.5) and the dual cost functional J (v,y of the system (3.6) satisfy for all u, v E qlad, all y satisfying (3.5), all y e V, and all dual state " given by (3.6).
The optimal control u o and the corresponding Yo are characterized by (3.5),(3.6), (3.7) with " taken to be y. Furthermore, we have supJ (v,)-infJ(u,y) J (uo, Yo) where the infimum is taken over all u Cad and y satisfying (3.5), and the supremum is over all (',' satisfying (3.6) and v e Ckl.ad.
If we put Yo" Y(Uo), " (Y) and take u in Cad v in Uad( ), then we have the fol- The optimal control problem is to minimize the cost given by J(u, y) i (] y(x, t)[ 2 + (Nu)(x, t)u(x, t))dx dr. Q We define the dual problem to be maximizing the dual cost functional The techniques we have used can also be applied to construct the dual problem to the problem of impulsive control of linear evolution problems. We use on in [5] Chapter 2 as an example.
Example 3.4: Suppose that the regularity conditions in [4], p. 182 are imposed on , which is a domain in Rn, with n _< 3. Let A be the elliptic operator described in Example 2.4. Unless otherwise specified, we shall continue to use the notation of Example 3.3. Let b be given in .
We denote by 5(x-b) the Dirac mass at the point b. Let the cost functional be T J(u, y) / y(x, T; u) 2dx + k / / u(x, t) 2dx dt (3.16) where k is a given positive real number and y y(x, t; u) is the state of the system given by By defining the dual cost functional to be " (v, / "ff (x, T; we may conclude, thorough proving parallel results of Theorem 3.1 and Theorem 3.2, and the problem of maximizing J with (x, T) and given by (3.19), is a dual problem to that of minimizing J with u E q-Lad for the system (3.17). Corresponding error estimates may be established similarly.

Hyperbolic Systems
We continue to use the notation of Section 2. where N is as in Section 2 satisfying (2.13). The problem is to minimize J(u, y) for u C %Lad. I] Cy I I 2 (Nv, v) + 2(f + By, 2(y(0), p + inf 2(A/I1B* +Nv, w).

w E Ckl, ad
The proofs of the following lemma and theorem are the same as those in Section 3.