Attractors of multivalued semi-flows generated by solutions of optimal control problems

In this paper we study the dynamical system generated by the solutions of optimal control problems. We obtain suitable conditions under which such systems generate multivalued semiprocesses. We prove the existence of uniform attractors for the multivalued semiprocess generated by the solutions of controlled reaction-diffusion equations and study its properties.

The possibility of applying the theory of global attractors to optimal control problems was mentioned at first in the monograph [42]. In [20] the existence of pullback attractors for the flow generated by controlled three-dimensional Navier-Stokes equations was proved. Moreover, the relationship between the obtained pullback attractor and the global attractor of the three-dimensional Navier-Stokes equations was established as well. As far as we know, there are no more results in the literature concerning existence of attractors for optimal control problems.
It is worth mentioning that optimal control problems are essentially nonautonomous in general, and therefore they generate processess rather than semigroups.
In the present paper we investigate general conditions under which optimal control problems generate dynamical semiprocesses, showing in several examples that this property is not true in general. Also, we prove the existence of the uniform attractor for a reaction-diffusion equation with additive control and quadratic optimality criterion.
2. Semiflows and semiprocesses generated by optimal control problems. Consider the following abstract optimal control problem where W (0, +∞) is some functional space of measurable functions, the closed set of admissible controls U 0 belongs to W (0, +∞), and f , g : R + × H → H are given functions. For every u(·) ∈ U 0 a solution of (1) will be a continuous function y (·) with values in the phase space H such that dy dt exists and (1) is satisfied for a.a. t ∈ (0, +∞).
A solution {y, u} of problem (1)-(4) is called an optimal pair. We make the following assumption: for any y 0 ∈ H there exists an optimal pair {y, u}.
Our aim is to investigate the asymptotic behavior of solutions of problem (1)-(4) by using the theory of attractors. For this reason we study the properties of the following map G : R + × H → P (H), G(t, y 0 ) = {y(t) | {y, u} is an optimal pair of problem (1)-(4}. In all the further arguments we will denote by P (H) (β(H)) the set of all non-empty (non-empty bounded) subsets of H.
Remark 1. We do not require uniqueness of solutions for problem (1)-(4). Thus, the map (6) is multivalued in general.
if problem (1)-(4) allows an autonomous optimal control in feedback form, i.e. there exists a map κ(·) (which does not depend on y 0 ) such that u(t) ≡ κ(y(t)) and y is a solution of the Cauchy problem    dy dt = f (y(t), κ(y(t))), It should be remarked that such optimal stabilization problems are difficult (and unresolvable in general case) even for linear-quadratic distributed systems [19]. Moreover, even in the simplest cases property (7) may fail to be true. We show this situation in the following examples.
where p(t) is the conjugate function. Therefore, we obtain that and then G(t, y 0 ) = (y 0 − 1)e λt + 1 − t, for any y 0 ∈ H.
This map does not satisfy (7), because On the other hand, we get These examples show that problem (1)-(4) is essentially non-autonomous in general. So, we need the notion of semiprocess [13], [7] to describe the long-time behavior of solutions of (1)- (4). For this reason we consider the following family of problems where τ ≥ 0, W (τ, +∞) is some functional space of measurable functions and the closed set of admissible controls U τ belongs to W (τ, +∞). As before, for every u(·) ∈ U τ we will say that y (·) is a solution of (11) if it is a continuous function with values in the phase space H such that dy dt exists and satisfies (11) a.e. on (τ, +∞). A solution {y, u} of problem (11)- (14) is called an optimal pair. We make the assumption that for any τ ≥ 0, y τ ∈ H there exists an optimal pair {y, u}.
We consider the following map associated to (11)- (14): where Does the map (16) generates a semiprocess, that is, is the relation true? Example 3 shows that the answer to the above question may be "not".
Example 3. Let us consider the problem: whose solutions generate the following map: Then for τ = 0, y τ = −1 we have and for s = ln 4, t ≥ s we obtain (17) is not fulfilled.
Nevertheless, for wide classes of optimal control problems relation (17) is true. In particular, for Example 2. Indeed, relation (17) for Example 2 can be easily obtained from the fact that y is an optimal solution if and only if it is a solution of the differential equationẏ = λ(y + t − 1) − 1. The Principle of Optimality [6] (the final part of the optimal process is optimal) plays a crucial role for (17). This principle holds for Example 2 but not for Example 1.
The following lemma gives us sufficient conditions for (17) to be fulfilled in terms of the coefficients of problem (11)- (14).
Suppose it is not true. Then there exists a solution { y, u} of problem (11)- (14) with initial conditions (s, y(s)) such that J s ( y, u) < J s (y, u).
As usual, denote by dist(B, C) = sup b∈B inf c∈C b − c the Hausdorff semidistance from the set B to the set C. The following definition seems to be natural for semiprocesses. In order to formulate the conditions for the existence of a uniform attractor of the m-semiprocess U , it is natural to consider the following map: Lemma 2. [20] G is a multivalued semiflow, i.e., for any t, s ≥ 0, x ∈ H we have G(s, x)).

Definition 2. The compact set A ⊂ H is called a uniform attractor for m-semiflow
and A is the minimal closed set satisfying such property.

Remark 2.
In this definition the semi-invariance property (A ⊆ G(t, A) for all t ≥ 0) in the classical definition of global attractor for m-semiflows (see [30], [32]) is replaced by the minimality condition.
It is easy to see that A is a uniform attractor of the m-semiprocess U if and only if A is a uniform attractor of the m-semiflow G which is defined in (22).
If τ ∈ (−∞, +∞), i.e. if U is a multivalued process, then there exists another way to describe the dynamics of U , which comes from the theory of pullback attractors (see [20], [7]).
In the present paper we consider the multivalued semiflow (22) as the basic dynamical object that is associated with the optimal control problem (11)- (14).
Then G has a uniform attractor A if and only if G is asymptotically compact, i.e., for any B ∈ β(H) and all ξ n ∈ G(t n , B) one has {ξ n } n≥1 is relatively compact in H. Moreover, If for any t > 0 G(t, ·) has closed graph, then that is, A is a global attractor of G.

Remark 3.
In terms of the m-semiprocess U the dissipativity condition (23) means that there is a set B 0 ∈ β(H) for which for any B ∈ β(H) there exists T = T (B) such that 3. Application to parabolic inclusions with additive control and quadratic quality criterion. Let V ⊂ H ⊂ V * be an evolution triple of Hilbert spaces with compact and dense embeddings, ·, · be the duality between V and V * , · , (·, ·) be the norm and the inner product in H, respectively, · V be the norm in V , and suppose that there exists α > 0 such that Let A : V → V * be a linear continuous self-adjoint operator such that fo some β > 0 we have Finally, let F : H → P (H) be such that F (y) is closed and convex in H, ∀y ∈ H, ∃c > 0 such that F (y) + := sup dist(F (y), F (y 0 )) → 0, y → y 0 , ∀y 0 ∈ H.
Proof. From (40) any solution of (33)-(34) satisfies where ε > 0 is such that Then So, when u(t) ≡ 0 ∈ U τ we have This implies by virtue of (43) that the set of admissible pairs in (33)-(36) is nonempty for any τ ≥ 0, y τ ∈ H. Let d ≥ 0 is the infimum value of the cost functional J τ . Let us choose a sequence of admissible pairs {y n , u n } such that Then for all T > τ , which by virtue of (44), (40), (29) leads to {y n } is bounded in W (τ, T ) and C([τ, T ]; H).
Then from the Compactness Lemma [34] we have that for some pair of functions {y, u} and for every T > τ up to a subsequence Then, passing to the limit in (38), (39) we obtain that {y, u} is a strong solution of problem (33), (34) and u ∈ U τ (see [25], [23]). Thus from (46) we have From (50) it follows that {y, u} is a solution of problem (33)- (36).
Then the multivalued semiflow G has the uniform attractor A.
In particular, y n (1) = ξ n → y(1) in H and the theorem is proved.
Unfortunately, whether the uniform attractor A is invariant with respect to the semiflow G or not is still an open question. However, we can say something about the properties of the set A using the semiflow generated by (33) with u ≡ 0.
Remark 4. It would be interesting to study the robustness of the global attractor when we vary the target function J τ (y, u), that is, if we consider functions J ε τ (y, u) converging to J τ in some sense as ε → 0.