SUBOPTIMAL SOLUTIONS OF CONTROL PROBLEMS FOR DISTRIBUTED PARAMETER

SYSTEMS

Andreas Hamel ^{∗}

Abstract

We consider a parabolic PDE in general form together with initial and first boundary conditions where the controls appear nonlinear in the lower order coefficients and the right hand side of the differential equation. It is very difficult to prove existence theorems for optimal solutions without convexity assumptions, but there are necessary optimality conditions in terms of a suitable defined Hamiltonian function. We prove the existence of minimizing sequences of admissible controls each of which satisfies a necessary condition with a slightly disturbed Hamiltonian. In the proof, we use Ekelands variational principle and the techniques for proving necessary optimality conditions.

Keywords: Optimal Control, Distributed Parameter Systems, Suboptimal Solutions, Ekelands Principle

AMS subject classification: 49K20, 49M05, 49J20

1 Introduction

Let M, N ≥ 1 be two fixed integers. Let be G ⊂ R^{N} an open and simple connected set with the compact closure \overline{G} = G ∪ ∂G and T > 0 fixed. We consider the system

\begin{aligned}L(u)z(x, t) & = & f(x, t, u(x, t)) & in & Q = G × (0, T), \\ z(x, t) & = & 0 & in & ∂G × [0, T], \\ z(x, 0) & = & z_{0}(x) & in & G \end{aligned}  (1)

where the differential operator L takes the following form:

∗Universität Halle-Wittenberg, Fachbereich Mathematik und Informatik, Theodor-Lieser-Str. 5, 06099 Halle(Saale), hamel@mathematik.uni-halle.d400.de

\begin{matrix}L(u)ϕ(x, t) = ϕ_{t}(x, t) − ∑_{i,j=1}^{N} a_{ij}(x, t)ϕ_{x_{i}x_{j}}(x, t) \\ − ∑_{i=1}^{N} b_{i}(x, t, u(x, t))ϕ_{x_{i}}(x, t) − c(x, t, u(x, t))ϕ(x, t). \end{matrix} (2)

We refer to the function u : \overline{Q} → R^{M} as the control function and z : \overline{Q} → R as the state function for the controlled system (1). If a control function u is given we denote by z(u) the state function belonging to the control u, i.e. the pair of functions (u, z(u)) fulfils the system (1) in a sense to be specified later. Furthermore, we consider a terminal functional of the form

J(u) = ∫_{G} j(x, z(u)(x, T))dx,  (3)

which is to be minimized over the class

U = \{u : \overline{Q} → R^{M}, u Lebesgue − measurable, u(x, t) ∈ U ⊂ R^{M} a.e. in Q\}.

The elements u ∈ U are called admissible controls. The assumptions on the data of the problem will be formulated later on. Problems of this type have been considered in a simple form by Lions [8] and a general theory, especially necessary optimality conditions and existence results for optimal controls, can be found in the book of Ahmed and Teo [1]. From the numerically point of view, the book of Teo and Wu [13] gives a good impression about several methods for constructing algorithms. In this paper, we apply the variational principle of Ekeland for proving the existence of a special type of minimizing sequences for the above control problem. The theorems 6 and 7 are the main results in this way. The idea is to give up the assumption on the existence of an optimal control and to look for necessary conditions for minimizing sequences in a similar form as the classical first order optimality conditions have. We call this type of conditions suboptimality conditions. Besides Ekeland in [3], Plotnikov and Sumin where the first who have investigated control problems in such a way. They proved in [10] a so-called disturbed minimum principle for a control system described by a hyperbolic PDE. Similar results for an abstract evolution equation setting can be found in a series of papers by Fattorini, cf. [4] and the literature therein. But the main goal in his papers is to establish necessary conditions for an optimal solution. Furthermore, we discuss the question of the computation of approximate solutions. We present an algorithm based on the Hamiltonian function which has been defined in order to formulate the suboptimality conditions.

2 Notation and Preliminary Results

2.1 Notation and assumptions

Upto slight changes, we use the conventions and notation of [1] and [13], respectively. We quote this books as standard references and do not prove results again contained in it. The symbol ∥·∥ without any subscript indicates one of the equivalent norms in a finite dimensional space R^{N}, R^{M}... For each p ∈ [1, ∞) let us denote by L_{p}(Q) the space of (equivalence classes of) all pth power Lebesgue-integrable functions ϕ : Q → R and by ∥.∥_{p,Q} the usual norm in this space. By L_{∞}(Q) we denote the space of all essentially bounded functions on the set Q with its norm ∥.∥_{∞,Q}. A similar notation will be used for vector valued functions. The

symbols ϕ_{t} = \frac{∂ϕ}{∂t}, ϕ_{x_{i}} = \frac{∂ϕ}{∂x_{i}}, ϕ_{x_{i}x_{j}} = \frac{∂^{2}ϕ}{∂x_{i}∂x_{j}} denote the generalized derivatives of the function ϕ. By W_{p}^{2,1}(Q) we denote the Banach space of all functions ϕ ∈ L_{p}(Q) which have a finite norm

∥ϕ∥_{p,Q}^{2,1} = ∥ϕ∥_{p,Q} + ∥ϕ_{t}∥_{p,Q} + ∑_{i=1}^{N} ∥ϕ_{x_{i}x_{j}} ∥_{p,Q} .

We formulate the assumptions on the datas of the system (1), (2): (A0) The set U ⊂ R^{M} is compact and convex. (A1) There exists a constant β_{0} > 0 such that

|a_{ij}(x, t) − a_{ij}(\overline{x}, \overline{t})| ≤ β_{0} (∥x − \overline{x}∥ + |t − \overline{t}|)

for all i, j = 1, ..., N, and for all (x, t), (\overline{x}, \overline{t}) ∈ \overline{Q}. (A2) There exist real numbers β_{1}, β_{2} > 0 such that

β_{1} ∑_{i=1}^{N} ξ_{i}^{2} ≤ ∑_{i=1}^{N} a_{ij}(x, t)ξ_{i}ξ_{j} ≤ β_{2} ∑_{i=1}^{N} ξ_{i}^{2}

for all ξ ∈ R^{N} uniformly on \overline{Q}. (A3) The functions b_{i} : \overline{Q} × U → R, i = 1, ..., N, c : \overline{Q} × U → R, f : \overline{Q} × U → R are Lebesgue-mesurable and bounded on \overline{Q} × U, further there exists a constant β_{3} > 0 such that

|h(x, t, u) − h(x, t, v)| ≤ β_{3} ∥u − v∥ ∀ u, v ∈ U

almost everywhere in Q, where h ∈ {b_{1}, ..., b_{N}, c, f} (A4) The function z_{0} is twice continuously differentiable with compact support in G. We give the assumptions on the cost functional (3):

(A5) The function j : \overline{G} × R → R is Carathéodory and j(·, s) ∈ L_{1}(G) for all s ∈ R . (A6) There exists a Carathéodory function g : \overline{G} × R → R such that

j(x, s_{1}) ≤ j(x, s_{2}) + g(x, s_{2})(s_{1} − s_{2})

for almost all x ∈ G and for all s_{1}, s_{2} ∈ R with g(·, 0) ∈ L_{∞}(G). Furthermore, there exists a constant β_{4} > 0 such that

|g(x, s_{1}) − g(x, s_{2})| ≤ β_{4} |s_{1} − s_{2}|

for almost all x ∈ G and for all s_{1}, s_{2} ∈ R .

Definition 1 For each u ∈ U, a function z(u) : Q → R is called an almost everywhere solution of the system (1) belonging to the control u iff

(i) z(u) ∈ W_{p}^{2,1}(Q), p > \frac{3}{2}, (ii) z(u) satisfies the differential equation in (1) almost everywhere in Q, (iii) z(u) satisfies the initial condition and the boundary condition in (1) everywhere in G and ∂G × [0, T], respectively.

Lemma 1 Let the assumptions (A0)-(A4) be satisfied. Then, for each control u ∈ U there exists a unique almost everywhere solution z(u) of the system (1) belonging to the control u. Furthermore, there is a constant γ_{1} > 0 such that (i) ∥z(u)∥_{p,Q}^{2,1} ≤ γ_{1} [∥f(·, ·, u(·, ·))∥_{∞,Q} + ∥z_{0}∥_{W_{∞}^{2}(G)}] for all p ∈ (\frac{3}{2}, ∞],

(ii) ∥z(u)∥_{K^{1+μ, \frac{1}{2} (1+μ)}(\overline{Q})} ≤ γ_{1} [∥f(·, ·, u(·, ·))∥_{∞,Q} + ∥z_{0}∥_{2,G}] for all μ ∈ (0, 1).

Proof: This lemma is a consequence of theorem 2.1.3 and 2.1.4, respectively, in [1]. The proofs of this theorems can be found in [7]. The definition of the space K^{1+μ, \frac{1}{2} (1+μ)}(\overline{Q}) of Hölder continuous functions on \overline{Q} is given in [1]. In order to formulate the suboptimality conditions we consider the following adjoint system together with the system (1)

\begin{aligned}L^{∗}(u)p(x, t) & = & 0 & in & Q = G × (0, T), \\ p(x, t) & = & 0 & in & ∂G × [0, T], \\ p(x, T) & = & g(x, z(u)(x, T)) & in & G \end{aligned}  (4)

where the formal adjoint differential operator L^{∗} takes the following form

\begin{matrix}L^{∗}(u)ψ(x, t) = −ψ_{t}(x, t)− \\ − ∑_{i=1}^{N} ( ∑_{j=1}^{N} a_{ij}(x, t)ψ_{_{i}x_{j}}(x, t) + a_{i}(x, t, u(x, t))ψ(x, t))_{x_{i}} − \\ −c(x, t, u(x, t))ψ(x, t). \end{matrix}  (5)

The functions a_{i} are defined by

a_{i}(x, t, u) := ∑_{j=1}^{N} \frac{∂a_{ij}(x, t)}{∂x_{j}} − b_{i}(x, t, u), i = 1, ..., N.

This definition makes sense because of assumption (A1).

Definition 2 For each u ∈ U, a function p(u) : Q → R is called a weak solution of the system (4) belonging to the control u iff

(i) p(u) ∈V^{∘}_{2}^{1,0} (Q)

(ii) The integral identity

∫_{G} g(x, z(u)(x, T))η(x, T)dx =

= ∫_{Q}∫ p(u)(x, t)η_{t}(x, t)dxdt+

+ ∫_{Q}∫ ∑_{i=1}^{N} ( ∑_{j=1}^{N} a_{ij}(x, t)p(u)_{_{i}x_{j}}(x, t) + a_{i}(x, t, u(x, t))p(u)(x, t)) η_{x_{i}(x,t)}dxdt−

− ∫_{Q}∫ c(x, t, u(x, t))p(u)(x, t)η(x, t)dxdt

(6)

is valid for all η ∈W^{∘} _{2}^{1,1} (Q) with η(x, 0) = 0.

Lemma 2 Let the assumptions (A0)-(A6) be satisfied. Then, there exists for each u ∈ U a unique weak solution p(u) of the system (4), called the adjoint state, belonging to the control u. Further, there exists a constant γ_{2} > 0 such that

∥p(u)∥_{V_{2}^{1,0}(Q)} ≤ γ_{2} ∥g(·, z(u)(·, T))∥_{2,G} .  (7)

Proof: This is the statement of theorem 3.1.1 in [1]. The definitions of the

spaces V_{2}^{1,0}(Q), V^{∘}_{2}^{1,0} (Q) and W^{∘} _{2}^{1,1} (Q) can be found in this reference, too.

Remark 1 It can be proved (and this is important) that the adjoint functions are uniformly bounded in the L_{∞}-norm: There exists a constant γ_{3} > 0 such that

∥p(u)∥_{∞,Q} ≤ γ_{3}  ∀ u ∈ U.  (8)

See [1], remark 3.1.3. for the details._{□}

There is a relation between the state and the adjoint state.

Lemma 3 Let the assumptions (A0)-(A6) be satisfied. Let u, v ∈ U. Then, the following equality holds:

\begin{matrix}∫_{Q}∫ L(u) (z(u)(x, t) − z(v)(x, t)) p(u)(x, t)dxdt = \\ = ∫_{G} g(x, z(u)(x, T)) (z(u)(x, T) − z(v)(x, T)) dx. \end{matrix}

Proof: Because of z(u), z(v) ∈ W_{2}^{2,1}(Q) we have z(u) − z(v) ∈W^{∘} _{2}^{1,1} (Q) and because of the boundary condition in (1) we have (z(u) − z(v)) (x, 0) = 0. Therefore it is possible to use z(u) − z(v) as test function η in (6). Integration by parts and the definitions of L(u) and a_{i} yields the result.

3 The Suboptimality Theorem

We assume the inequalities

(A7) −∞ < inf_{u∈U} J(u) < +∞,

where the class U of admissible controls is as defined in section 1. The second inequality in (A7) simple means that there is at least one control function u ∈ U such that the state z(u) and the cost functional value J(u) are well defined and the first inequality gives the boundedness from below of the functional J on the set U. We define a metric on the set U by

d(u, v) = meas{(x, t) ∈ Q : u(x, t) ≠ v(x, t)},  (9)

where meas{E} stands for the Lebesgue-measure of the set E ⊂ R^{K} for an arbitrary integer K. We need the case K = N + 1.

Lemma 4 (U, d) is a complete metric space.

Proof: The proof is exactly the same as in the one-dimensional case, cf. [3]. It does not depend on the properties of the set U.

Lemma 5 Suppose that the assumptions (A0)-(A6) are satisfied. Then, the functional J : U → R is lower semicontinuous with respect to the metric d.

Proof: Let {u^{n}}_{n=1,2,...} ⊂ U be a sequence of admissible controls which is convergent to the control u ∈ U with respect to the metric d, that means

lim_{n→∞} meas {(x, t) ∈ Q : u^{n}(x, t) ≠ u(x, t)} = 0  (10)

Then, we have by assumption (A6) and lemma 3 for every n = 1, 2, ...

J(u) = ∫_{G} j(x, z(u)(x, T))dx ≤

≤ ∫_{G} j(x, z(u^{n})(x, T))dx + ∫_{G} g(x, z(u^{n})(x, T)) (z(u)(x, T) − z(u^{n})(x, T)) dx =

= J(u^{n}) + ∫_{Q}∫ L(u^{n}) (z(u)(x, t) − z(u^{n})(x, t)) p(u^{n})(x, t)dxdt.

Furthermore, it holds for every n = 1, 2, ...

L(u^{n}) (z(u)(x, t) − z(u^{n})(x, t)) =
= L(u)z(u)(x, t) − L(u^{n})z(u^{n})(x, t) + (L(u^{n}) − L(u)) z(u)(x, t) =
= f(x, t, u(x, t)) − f(x, t, u^{n}(x, t))−

− ∑_{i=1}^{N} (b_{i}(x, t, u^{n}(x, t) − b_{i}(x, t, u(x, t)) z_{x_{i}}(u)(x, t)−

− (c(x, t, u^{n}(x, t) − c(x, t, u(x, t)) z(u)(x, t).

With the above inequality, we obtain for every n = 1, 2, ...

J(u) ≤ J(u^{n}) + ∫_{Q}∫ (f(x, t, u(x, t)) − f(x, t, u^{n}(x, t))) p(u^{n})(x, t)dxdt+

+ ∫_{Q}∫ (∑_{i=1}^{N} (b_{i}(x, t, u(x, t) − b_{i}(x, t, u^{n}(x, t)) z_{x_{i}}(u)(x, t)) p(u^{n})(x, t)dxdt+

+ ∫_{Q}∫ (c(x, t, u(x, t) − c(x, t, u^{n}(x, t)) z(u)(x, t)p(u^{n})(x, t)dxdt.

We define the sets Q_{n} := {(x, t) ∈ Q : u^{n}(x, t) ≠ u(x, t)}. Clearly, then lim_{n→∞} meas {Q_{n}} = 0. In the following, we estimate the integral terms of the last inequality. At first, we have

∫_{Q}∫ (f(x, t, u(x, t)) − f(x, t, u^{n}(x, t))) p(u^{n})(x, t)dxdt ≤ γ_{3}γ_{4}meas {Q_{n}} ,

where γ_{4} = 2ess sup_{(x,z,u)∈Q×U} |f(x, t, u)| and γ_{3} is the constant from remark

2.1. We set ϵ_{n}^{(1)} := γ_{3}γ_{4}meas {Q_{n}} and lim_{n→∞} ϵ_{n}^{(1)} = 0 holds. Furthermore, we have

\begin{matrix}∫_{Q}∫ (∑_{i=1}^{N} (b_{i}(x, t, u(x, t) − b_{i}(x, t, u^{n}(x, t)) z_{x_{i}}(u)(x, t)) p(u^{n})(x, t)dxdt ≤ \\ ≤ γ_{3}γ_{5}γ_{6}meas \{Q_{n}\} , \end{matrix}

where γ_{5}  :=  2 max_{i=1,...,N} ess sup_{(x,z,u)∈Q×U} |b_{i}(x, t, u)| and

γ_{6} := ∑_{i=1}^{N} sup_{(x,t)∈Q} |z_{x_{i}}(u)(x, t)|. γ_{6} is finite because of the fact that z(u) ∈

K^{1+μ, \frac{1}{2} (1+μ)}(\overline{Q}) for 0 < μ < 1. We set ϵ_{n}^{(2)} := γ_{3}γ_{5}γ_{6}meas \{Q_{n}\} ,and clearly lim_{n→∞} ϵ_{n}^{(2)} = 0 holds. In a similar way the last integral term can be estimated, and we get a third sequence {ϵ_{n}^{(3)}}_{n=1,2,...} with lim_{n→∞} ϵ_{n}^{(3)} = 0. Altogether,

we obtain for n = 1, 2, ...

J(u) ≤ J(u^{n}) + ϵ_{n}^{(1)} + ϵ_{n}^{(2)} + ϵ_{n}^{(3)}

with lim_{n→∞} (ϵ_{n}^{(1)} + ϵ_{n}^{(2)} + ϵ_{n}^{(3)}) = 0. We conclude that

J(u) ≤ lim inf_{n→∞} J(u^{n}),

and this proves the lemma.

Remark 2 In a similar way it is possible to prove the upper semicontinuity of the functional J (that means the lower semicontinuity of −J), and therefore the continuity of J on (U, d).

Let {u^{n}}_{n=1,2,...} ⊂ U be a minimizing sequence for the problem (1)-(3), i.e.

lim_{n→∞} J(u^{n}) = lim_{n→∞} ∫_{G} j(x, z(u^{n})(x, T)) dx = inf_{u∈U} J(u).  (11)

Then there exists a sequence {α_{n}}_{n=1,2,...} ⊂ R_{+}\ {0} with lim_{n→∞} α_{n} = 0 such that

J(u^{n}) ≤ inf_{u∈U} J(u) + α_{n}.  (12)

Theorem 6 Suppose that the assumptions (A0)-(A7) are satisfied. Let be {u^{n}}_{n=1,2,...} ⊂ U a minimizing sequence for the problem (1)-(3) which satisfies (12). Then, there exists another minimizing sequence {v^{n}}_{n=1,2,...} ⊂ U which satisfies the conditions

(i) J(v^{n}) ≤ J(u^{n}), (ii) d(u^{n}, v^{n}) ≤ \sqrt{α_{n}}, (iii) J(v^{n}) < J(u) + \sqrt{α_{n}}d(u, v^{n}) ∀ u ∈ U.

Proof: We apply Ekelands variational principle [3] (theorem 1.1) to the functional J on the complete metric space (U, d) with ε = α_{n} and λ = \sqrt{α_{n}}. We exploit the condition (iii) to obtain more informations on the sequence {v^{n}}_{n=1,2,...}. This is possible with the same procedure as in the proof of the necessary optimality conditions for the problem (1)-(3) as used in [1]. For every element of the sequence {v^{n}}_{n=1,2,...} we introduce a sequence of variations in the following way. For every n = 1, 2, ... we choose a regular point (x_{0}^{n}, t_{0}^{n}) ∈ Q for all functions involved in the differential operator L(v^{n}) and for the right hand side f(·, ·, v^{n}(·, ·)). Then, we consider sequences of hypercubes \left\{{ \atop G_{k}^{n}}\right\} k = 1, 2, ... with the centre x_{0}^{n} ∈ G and sequences of intervals {I_{k}^{n}}_{k=1,2,...} with the centre t_{0}^{n} ∈

(0, T) such that both lim_{k→∞} meas \left\{{ \atop G_{k}^{n}}\right\} = 0 in R^{N} and lim_{k→∞} meas \{I_{k}^{n}\} =

0 in R. Let be G_{k}^{n} := G̃_{k}^{n} ∩G and Q_{k}^{n} := G_{k}^{n} ×I_{k}^{n}. Then lim_{k→∞} meas {Q_{k}^{n}} = 0 in R^{N+1}. Next, we choose an arbitrary w ∈ U and define the functions

u^{n,k}(x, t) := (1 − χ_{k}^{n}(x, t)) v^{n}(x, t) + χ_{k}^{n}(x, t)w,

where the characteristic functions χ_{k}^{n} of the sets Q_{k}^{n} are defined by χ_{k}^{n}(x, t) := 1 for (x, t) ∈ Q_{k}^{n} and χ_{k}^{n}(x, t) := 0 for (x, t) ∉ Q_{k}^{n}. With this definition, we have d(u^{n,k}, v^{n}) ≤ meas {Q_{k}^{n}} and from (iii) in theorem 6 it follows that

\frac{1}{meas \{Q_{k}^{n}\}} (J(v^{n}) − J(u^{n,k})) < \sqrt{α_{n}}.

Now, we are in the same position as in the proof of theorem 3.1.3. in [1] with one exception: the right hand side of the last inequality is not zero, but positive. Nevertheless, on the left hand side we can pass to the limit for k → ∞ in the same way as in [1] and get the following result. For the reason of abbreviation we define the function

H(z, u, p, x.t) := p · [∑_{i=1}^{n} b_{i}(x, t, u)z_{x_{i}} + c(x, t, u)z + f(x, t, u)] .  (13)

Theorem 7 Suppose that the assumptions (A0)-(A7) are satisfied and let {v^{n}}_{n=1,2,...} be the sequence from theorem 6. Then, the following inequalities are valid:

\begin{matrix}H(z(v^{n})(x, t), v^{n}(x, t), p(v^{n})(x, t), x.t) ≤ \\ ≤ H(z(v^{n})(x, t), w, p(v^{n})(x, t), x.t) + \sqrt{α_{n}}  ∀ w ∈ U \end{matrix}  (14)

for almost (x, t) ∈ Q and for all n = 1, 2, ....

Remark 3 We note that it is unknown whether it is possible to pass to the limit for k → ∞ in the proof of theorem 6 in the case of more general cost functionals. See [1], chap. 3, § 3.4 for the details.

Remark 4 We refer to (14) as suboptimality condition.

Remark 5 Suppose additionally the existence of an optimal control u^{0} ∈ U , i.e. it holds

J(u^{0}) = inf_{u∈U} J(u) ≤ inf_{u∈U} J(u) + α_{n} ∀ n = 1, 2, ...

Then, we can consider the special minimizing sequence {u^{n}}_{n=1,2,...} with u^{n} = u^{0} for all n and apply theorem 6 and 7. The sequence {v^{n}}_{n=1,2,...} has to be a sequence of optimal solutions which converges to u^{0} with respect to the metric d according to (i) and (ii) of theorem 6. From (14) we obtain the necessary optimality condition for u^{0} as it has been formulated in [1] (theorem 3.1.3).

Corollary 8 Suppose that the hypotheses of theorem 6 are satisfied. Let α > 0 be given. Then, there exists a control function u_{α} ∈ U such that

(i) J(u_{α}) ≤ inf_{u∈U} J(u) + α (ii) H(z(u_{α})(x, t), u_{α}(x, t), p(u_{α})(x, t), x.t) ≤

≤ H(z(u_{α})(x, t), w, p(u_{α})(x, t), x.t) + \sqrt{α} ∀ w ∈ U

for almost (x, t) ∈ Q and for all n = 1, 2, ....

The above theorems gives a justification for using the function H defined in (13) - usually called Hamiltonian - for constructing algorithms even in cases in which an optimal control does not exist or it is unknown whether it exists or not.

4 An Algorithm

4.1 The idea of the algorithm

In the history of control theory it is an old idea to use a suitable Hamiltonian function for constructing algorithms in the following way: If a pair of control and state function is known then a new control function will be determined in minimizing the Hamiltonian with respect to the control variable only. This method appears in the literature as the method of successive aproximation or the strong variation method, see [6] and [9] for the case of controlled ordinary differential equation and [12], [1] and [13] for problems of the form considered in this paper. But it is very hard and requires sophisticated realization strategies to prove convergence results for algorithms of this typ. For this reason we introduce a disturbed Hamiltonian and get some kind of stability result (theorem 4.1) already with a simple structured algorithm. The idea of this procedure goes back to Y. Sakawa and Y. Shindo [11].

The algorithm: Step 1: Let {λ_{n}}_{n=1,2,...} , λ_{n} > 0, be given. Choose an initial control function u^{0} ∈ U. Compute z(u^{0}) and J(u^{0}). Set n := 0. Step 2: Compute p(u^{n}). Step 3: Compute u^{n+1} pointwise almost everywhere in Q as solution of

H(z(u^{n})(x, t), w, p(u^{n})(x, t), x, t) + λ_{n+1} ∥w − u^{n}(x, t)∥ → min_{w∈U} .  (15)

Step 4: Compute z(u^{n+1}) and J(u^{n+1}).

Step 5: Stop if a stopping criterion is fulfilled. Otherwise go to step 2 with n := n + 1.

Remark 6 As it has been observed in [5] it may be impossible to solve the optimization problem in step 3 exactly almost everywhere in Q or in every discretization point, respectively. In this cases, step 3 can be substituted by the following step 3*. The idea behind this is to solve the optimization problems (15) not exactly but approximately using a suitable algorithm.

Step 3*: Determine the new control function u^{n+1} ∈ U such that the following inequality is fulfilled almost everywhere in Q:

\begin{matrix}H(z(u^{n})(x, t), u^{n+1}(x, t), p(u^{n})(x, t), x, t) + λ_{n+1} ∥u^{n+1}(x, t) − u^{n}(x, t)∥ ≤ \\ ≤ H(z(u^{n})(x, t), u^{n}(x, t), p(u^{n})(x, t), x, t). \end{matrix}

(16)

Remark 7 If there is a measurable control function u^{n+1} which is a solution of (15) then it satisfies the inequality (16). But it is a priori not clear whether the problem (15) has a measurable solution or not. Furthermore, there is always a measurable control function u^{n+1} which satisfies (16) because this inequality is true for u^{n+1} := u^{n}.

4.2 Properties of the sequence of controls generated by the given algorithm

The following theorem is concerned with sequences of control functions generated by the algorithm using step 3* instead of step 3. It has been proved in [5] for the first time.

Theorem 9 Let (A0)-(A7) be satisfied. If the sequence {u^{n}}_{n=1,2,...} has been determined by the algorithm using step 3* instead of step 3 then, there exists a number γ_{7} > 0 such that

(i) J(u^{n+1}) − J(u^{n}) ≤ −(λ_{n+1} − γ_{7}) ∥u^{n+1} − u^{n}∥_{1,Q} , n = 0, 1, 2, ... If additionally λ_{n} ≥ λ_{0} > γ_{7} for n = 1, 2, ..., then (ii) the sequence {J(u^{n})}_{n=0,1,...} decreases and hence converges to a number J^{∗} ≥ inf_{u∈U} J(u). (iii) lim_{n→∞} ∥u^{n+1} − u^{n}∥_{1,Q} = 0.

Proof: The most difficult step is to prove the inequalities (i). Assume we have done this, then (ii) follows directly. Further, we conclude from (i) that

J(u^{n+1}) − J(u^{n}) ≤ −(λ_{0} − γ_{7}) ∥u^{n+1} − u^{n}∥_{1,Q} , n = 0, 1, 2, ...

Adding up these inequalities from n = 0 to n = l − 1 we get

J(u^{l}) − J(u^{0}) ≤ −(λ_{0} − γ_{7}) ∑_{n=0}^{l−1} ∥u^{n+1} − u^{n}∥_{1,Q} .

Hence

∑_{n=0}^{l−1} ∥u^{n+1} − u^{n}∥_{1,Q} ≤ \frac{1}{(λ_{0} − γ_{7})} (J(u^{0}) − J(u^{l})) ,

and passing to the limit l → ∞ we get (iii). Lets prove (i). By assumption (A6) and lemma 3 we have

J(u^{n+1}) − J(u^{n}) ≤

≤ ∫_{G} (j(x, z(u^{n+1})(x, T)) − j(x, z(u^{n})(x, T))) dx

≤ ∫_{G} g(x, z(u^{n})(x, T)) (z(u^{n−1})(x, T) − z(u^{n})(x, T)) dx =

= ∫_{Q}∫ L(u^{n}) (z(u^{n+1})(x, t) − z(u^{n})(x, t)) p(u^{n})(x, t)dxdt.

Some algebraic manipulations lead to

L(u^{n}) (z(u^{n+1})(x, t) − z(u^{n})(x, t)) = = f(x, t, u^{n+1}(x, t)) − f(x, t, u^{n}(x, t))+

+ ∑_{i=1}^{N} (b_{i}(x, t, u^{n+1}(x, t)) − b_{i}(x, t, u^{n}(x, t))) z(u^{n})_{x_{i}}(x, t)+

+ (c(x, t, u^{n+1}(x, t)) − c(x, t, u^{n}(x, t))) z(u^{n})(x, t)+

+ ∑_{i=1}^{N} (b_{i}(x, t, u^{n+1}(x, t)) − b_{i}(x, t, u^{n}(x, t))) (z(u^{n+1})_{x_{i}}(x, t) − z(u^{n})_{x_{i}}(x, t)) +

+ (c(x, t, u^{n+1}(x, t)) − c(x, t, u^{n}(x, t))) (z(u^{n+1})(x, t) − z(u^{n})(x, t))

For the reason of abbreviation, we mostly drop the arguments x and t in the following. The definition (13) of the function H yields from the two above relations

J(u^{n+1}) − J(u^{n}) ≤

≤ ∫_{Q}∫ (H(z(u^{n}), u^{n+1}, p(u^{n}), x, t) − H(z(u^{n}), u^{n}, p(u^{n}), x, t)) dxdt+

+ ∫_{Q}∫ [∑_{i=1}^{N} (b_{i}(x, t, u^{n+1}) − b_{i}(x, t, u^{n})) (z(u^{n+1})_{x_{i}} − z(u^{n})_{x_{i}})] p(u^{n})dxdt+

+ ∫_{Q}∫ (c(x, t, u^{n+1}) − c(x, t, u^{n})) (z(u^{n+1}) − z(u^{n})) p(u^{n})dxdt ≤

≤ −λ_{n+1} ∫_{Q}∫ ∥u^{n+1}(x, t) − u^{n}(x, t)∥ dxdt+

+ ∫_{Q}∫ [∑_{i=1}^{N} (b_{i}(x, t, u^{n+1}) − b_{i}(x, t, u^{n})) (z(u^{n+1})_{x_{i}} − z(u^{n})_{x_{i}})] p(u^{n})dxdt+

+ ∫_{Q}∫ (c(x, t, u^{n+1}) − c(x, t, u^{n})) (z(u^{n+1}) − z(u^{n})) p(u^{n})dxdt.

The last two terms in this inequality will be estimated. We use the inequality (8) in remark 1, assumption (A3) and get

∫_{Q}∫ [∑_{i=1}^{N} (b_{i}(x, t, u^{n+1}) − b_{i}(x, t, u^{n})) (z(u^{n+1})_{x_{i}} − z(u^{n})_{x_{i}})] p(u^{n})dxdt ≤

≤ γ_{3}β_{3} ∫_{Q}∫ (∥u^{n+1} − u^{n}∥ ∑_{i=1}^{N} |z(u^{n+1})_{x_{i}} − z(u^{n})_{x_{i}} |) dxdt ≤

≤ γ_{3}β_{3}γ_{8} ∫_{Q}∫ ∥u^{n+1}(x, t) − u^{n}(x, t)∥ dxdt.

The constant γ_{8} comes from the inequality

∑_{i=1}^{N} ess sup_{(x,t)∈Q} |z(u^{n+1})_{x_{i}}(x, t) − z(u^{n})_{x_{i}}(x, t)| ≤ γ_{8}

and this constant is finite because of the fact that the set of states which belongs to any admissible control is uniformly bounded in the space of Hölder continuous functions K^{1+μ, \frac{1}{2} (1+μ)}(\overline{Q}). In a similar way we conclude that

∫_{Q}∫ (c(x, t, u^{n+1}) − c(x, t, u^{n})) (z(u^{n+1}) − z(u^{n})) p(u^{n})dxdt ≤

≤ γ_{3}β_{3} ∫_{Q}∫ ∥u^{n+1} − u^{n}∥ · |z(u^{n+1}) − z(u^{n})| dxdt ≤

≤ γ_{3}β_{3}γ_{9} ∫_{Q}∫ ∥u^{n+1}(x, t) − u^{n}(x, t)∥ dxdt

with a constant γ_{9} > 0 which is finite according the same argument as it has been used for γ_{8}. If we collect all terms we get

J(u^{n+1}) − J(u^{n}) ≤ (−λ_{n+1} + γ_{3}β_{3}(γ_{8} + γ_{9})) ∫_{Q}∫ ∥u^{n+1}(x, t) − u^{n}(x, t)∥ dxdt

and with γ_{7} = γ_{3}β_{3}(γ_{8} + γ_{9}) the desired result (i). For testing the indicated algorithm there has been used the examples from [1], p. 189 and from [13], pp. 118ff. The results are not worse in comparison to the algorithm used by Teo and his coworkers but the algorithm presented here is much simpler. The numerical results and a more sophisticated description of the computational realization can be found in [2].

5 Concluding Remark

In view of corollary 3.5 we state the following problem: For a given α > 0, determine a control function u_{α} ∈ U which satisfies the assertions of this corollary. Further, we look for an algorithm which yields after a finite number of steps such an u_{α}, at least for some subclasses of the general problem. It is clear that the algorithm which has been proposed in section 4 has not this property.

References

[1] N. U. Ahmed, K. L. Teo, Optimal Control of Distributed Parameter Systems (North Holland New York · Oxford 1981)

[2] H. Benker, A. Hamel, S. Schleiff, ”Computational Solution of Parabolic Coefficient Control Problems”, in preparation

[3] I. Ekeland, ”On the Variational Principle”, J. Math. Anal. Appl. 47 (1974) 324-353

[4] H. O. Fattorini, H. Frankowska, ”Necessary Conditions for Infinite-Dimensional Control Problems”, Math. Control Signals Systems 4 (1991) 41-67

[5] A. Hamel, Anwendungen des Variationsprinzips von Ekeland in der Optimalen Steuerung (Doct. Diss., Halle (Saale) 1996)

[6] I. A. Krylov, F. L. Chernousko ”On the Method of Successive Approximation for Solutions of Optimal Control Problems”, J. Comp. Math. and Mathem. Phys. 11 (1) (1962, in Russian)

[7] O. A. Ladyshenskaja, V. A. Solonikov, N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type (AMS, Providence 1968)

[8] J. M. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer-Verlag Berlin Heidelberg New York 1971)

[9] D. Q. Mayne/E. Polak ”First-Order Strong Variation Algorithms for Optimal Control”, J. Optim. Theory Appl. 16 (3/4) (1975) 277-301

[10] V. I. Plotnikov, M. I. Sumin ”The Construction of Minimizing Sequences in Problems of the Control of Systems with Distributed Parameters”, U.S.S.R.Comput. Math. Phys. 22 (1) 49-57 (1982)

[11] Y. Sakawa, Y. Shindo, ”On Global Convergence of an Algorithm for Optimal Control”, IEEE Transactions on Automatic Control AC-25 (6) (1980) 1149-1153

[12] K. L. Teo, D. W. Reid, ”First-Order Strong Variation Algorithm for Optimal Control Problems Involving Parabolic Systems”, Numer. Funct. Anal. Optimiz. 5 (2) (1982) 141-171

[13] K. L. Teo, Z. S. Wu, Computational Methods for Optimizing Distributed Systems (Academic Press 1984)