Time Optimal Feedback Control for 3D Navier–Stokes–Voigt Equations

: In this article, we discuss a time optimal feedback control for asymmetrical 3D Navier– Stokes–Voigt equations. Firstly, we consider the existence of the admissible trajectories for the asymmetrical 3D Navier–Stokes–Voigt equations by using the well-known Cesari property and the Fillippove’s theorem. Secondly


Introduction
Let Ω be an open bounded domain in R 3 with C 1 boundary ∂Ω.For T > 0, let J = [0, T], Q = J × Ω, we consider the following 3D Navier-Stokes-Voigt equations:    z t − µ∆z − γ 2 ∆z + (z • ∇)z + ∇p = f , in Q, z(t, y) = 0, on (0, T) × ∂Q, z(0, y) = z 0 (y), in Q, (1) which is called the Navier-Stokes-Voigt equation.A model of motion of linear viscoelastic fluids was presented by Oskolkov in 1973 [1].Furthermore, Oskolkov studied the existence of time periodic solutions and no-slip Dirichlet boundary conditions for the Navier-Stokes-Voigt equation.From then on, the existence results and optimal control problem for the Navier-Stokes-Voigt equation have drawn great attention, for example, Sviridyuk [2] discussed the weakly compressible for the Navier-Stokes-Voigt equation.The long time dynamics and attractors were researched by [3,4].Anh-Nguyet [5] focused on an optimal control problem with quadratic objective functional for the Navier-Stokes-Voigt equation.
Over the past few decades, the optimal control of the Navier-Stokes equation has been extensively researched by a large number of authors.For example, the absence of state constraint for Navier-Stokes control systems has been discussed by [36][37][38].The optimal feedback control of Kelvin-Voigt fluid flows is presented in [39].The presence of state constraint for the control systems were investigated [40].Recently, Zeng [41] studied the feedback control for non-stationary 3D Navier-Stokes-Voigt Equations (3DNSVEs for short) by using monotone theory.
Since the concept of time optimal control was introduced by LaSalle [42] in 1960, the theories of time optimal control problems has caused widespread concern by many mathematics.For example, Berkovitz [7] and Warga [43] considered the time optimal control for functional equations, Barbu [6] studied the parabolic variational inequalities by using monotone theory.Fattorini [16,38] discussed the operational differential equation and viscous flows problem.Yong-Li gave the necessary and sufficient conditions for the time optimal control of distributed parameter equation [22], for more detail see the references therein.
In this paper, we consider the following 3DNSVEs: where z is a state function, Φ is a feedback multi-map, the control function v ∈ Φ.
The aim of this article is to consider the existence results of admissible trajectories and a time optimal control for 3D Navier-Stokes-Voigt systems.To achieve this aim, the existence result of admissible trajectories is discussed using the help of monotone theory and the well-known Fillippove's theorem.Furthermore, we investigate the existence of time optimal control for the 3D Navier-Stokes-Voigt systems by using optimal control theory.We note that our theory obtained in this article could be widely applied across numerous practical problems, such as static, quasistatic and dynamic frictional and frictionless contact problems.
The rest of this paper is structured as follows.In Section 2, some useful preliminaries and notations on the data are introduced.In Section 3, the existence of admissible trajectories is discussed.In Section 4, a time optimal control of feedback control for 3DNSVs is studied.Lastly, we use two examples to demonstrate our main theory.

Some Notations, Definitions and Preliminaries
We set (X, • X ) as a Banach space and denote its dual space as X * .Furthermore, we us •, • X to denote duality pair between X and X * .For any T > 0, Let C(J, X) denote the Banach space of all continuous functions from J = [0, T] into X with the norm x C(J,X) = sup t∈J x(t) and L 2 ([0, T]; X) denote the Banach space of all square integrable functions from [0, T] into X with the norm . We denote the strong convergence as "→" and " " as the weak convergence.Now, we give the abstract framework for our main work.Defining two inner products as also, with the norms x 1 := (x, x) 1 and x 2 := (x, x) 2 .We set The closure of V 0 is denoted in (L 2 (Ω)) 3 (resp.in (H 1 0 (Ω)) 3 ) as H (resp. V).One can easily know that H (resp. V) is a Hilbert space with scalar products (•, •) 1 (resp.(•, •) 2 ).It follows that V ⊂ H ≡ H * ⊂ V * , here the embeddings are dense, continuous, and compact.Furthermore, we give the pairing between L 2 (J, V * ) and L 2 (J, V) as We define the Sobolev space as Clearly, Now, recalling some basic definitions and properties of multi-valued maps, which can be seen in the monograph [44].
Let P(X) be the set of all nonempty subsets of X, P f (X) is the set of all nonempty closed subsets of X. Defining the Hausdorff metric as follows Setting (Ω, Σ) be a measurable space and Y be a separable Banach space.A multifunction Γ : Let G, D be two Hausdorff topological spaces and Γ : G → P(D).Γ is said to be lower semicontinuous (l.s.c for short) at For more details, one can see the monograph [44].
Let F : Ω → P(X) be a multifunction.For 1 ≤ p ≤ +∞, we can define Besides the standard norm on L q (J, X) (here X is a separable, reflexive Banach space) for 1 < q < ∞, we also consider the so called weak norm Space L q (J, X) furnished with this norm will be denoted by L q ω (J, X).The following result establishes a relation between convergence in ω-L q (J, X) and convergence in L q ω (J, X).

Definition 1 ([45]). Let E and T be two metric spaces. A multifunction
where O (t) = {τ ∈ J : τ − t < }.The multifunction Γ is said to be pseudo-continuous on T if Γ is pseudo-continuous at every point t ∈ T.

Definition 2 ([45]
).Let X be a metric space and Y be a Banach space.Let Γ : X → P(Y) be a multifunction.Γ is said to possess the Cesari property at point x 0 ∈ X, if where coD is the closed convex hull of D, O δ (x) is the δ-neighborhood of x.If Γ has the Cesari property at every point x ∈ Q ⊂ X, then Γ has the Cesari property on Q.
Example 1.Let E = [0, 1] × R, let Z be a closed Cantor subset of [0, 1] whose measure is positive, and let Z = [0, 1]\Z.Then Z is the countable union of disjoint subintervals of [0, 1].Let σ(t) be a continuous function on Z .which tends to +∞ whenever t tends to an end of any interval component of Z .We define a multifunction as follows Then, U(t) has the Cesari property.
Now, let us denote the trilinear form g : x i (D i y j )z j ds whenever the integrals make sense.Let us give the property of g.
(iii) for any x, y, z ∈ V, Now, let us give the definition of a weak solution for the problem (2).

Now, we define a linear and continuous operator
By Lemma 1, one can know that B is a bounded mapping from H 1 (J, V) to L 2 (J, V * ), i.e., there is a positive constant c > 0 such that According to the work of [5], we have the following results for operators A and B.
Thanks to the properties of operators A and B, we can now give an equivalent formulate with Definition 3. Definition 4. The function z ∈ H 1 (J, V) is said to be a weak solution of the problem (2), if At the end of this section, we give a well-known priori estimate for the system (1) (see [41,47]).
To discuss the main results, we need the following definitions.
3 is called to be an admissible pair with z 0 ∈ H 1 (J, V) on J if z(•) is the weak solution for control system (2) and v(t) ∈ Φ(t, z(t)) a.e.t ∈ J.
where z(•) is the admissible trajectory and v(•) is the admissible control.
We set that Π : J → P(V) is a target trajectories set.Denoting the admissible control pair set as is an admissible pair with z 0 ∈ V}, and denoting the admissible trajectories set as Moreover, we denote the reachable set as for some t ≥ 0 to be the target admissible control pair set, and denote the target of time set by Now, we give our main problem as follows: inf J (z(•)). (5)

Existence Results for Admissible Trajectories
The aim of this section is to study the existence results for admissible trajectories of the system (2).To achieve this aim, we need the as follows. H(Φ):
Proof.For each l > 0, we assume that . The sequence {v i } is constructed as following.
To begin, we take v 0 ∈ Φ(0, x 0 ), using Theorem 1, that has a unique z l ∈ H 1 (J, V) given by z l (0) = z 0 and )), repeating this process to obtain z l on [ T l , 2T l ], etc.Using a similar approach, we denote as follows: Similar to the Theorem 1 and applying H(U)(i), we can find that there is a constant ϑ 0 > 0 such that z l H 1 (J,V) ≤ ϑ 0 .
Hence {z l } is boundness in H 1 (J, V).There then exists a subsequence of {z l }, denoting as {z l } again, such that z l z in H 1 (J, V), and 3 , which implies that x(0) = x 0 .Given the fact that the embedding H 1 (J, V) ⊂ (L 2 (Ω)) 3 is compact, then According to the fact that z k ∈ C(J, V) and V ∈ L 2 (Ω), for each δ > 0 there are positive constants l, k 0 such that By ( 6), for each δ > 0, there is a constant ζ 0 > 0 such that Combining Lemma 2, we have Moreover, by H(Φ)(i), there is η 1 , such that This means that the sequence {v l } l≥1 is bounded in (L 2 (Q)) 3 .Then, there exists a subsequence of {v l } l≥1 , that we also denote as {v l } l≥1 , such that Hence, z Furthermore, for l that is large enough, from the definition of u l (•), we have Secondly, applying (8) and Mazur Lemma, we set α il ≥ 0 and Then there is a subsequence of {λ l }, without loss of generality, such that Hence, from ( 7) and ( 9), for l that is large enough, we get Therefor, for any δ > 0, one can have Using H(Φ)ii), we get v(t) ∈ coΦ(t, z(t)), a.e.t ∈ J.
From the above work, we have (z, v) ∈ A (J, z 0 ).
In the end, setting {z l (•)} l≥1 ⊂ G (J, x 0 ) and Then, by the same way, we can get that {z l (•)} l≥1 is relatively compact in space C(J, V).Furthermore, there is a subsequence of {z By hypothesis H(Φ)(ii), one can obtain z(•) ∈ G (J, z 0 ).Hence, the admissible set G (J, z 0 ) is compact in space C(J, V).

Corollary 2.
Let hypothesis H(Φ) hold.Then for each z 0 ∈ V and 0 ≤ t ≤ T, the reachable set K (t; 0, z 0 ) is nonempty and compact in V.

Existence Results for Time Optimal Control
In the following section, we will study the existence of time optimal control for the 3DNSVs.
Theorem 4. Let the hypotheses H(Φ), H(Π) hold, then Problem (T) has at least one optimal solution.
Applying (11), for any δ > 0, and n large enough, we get Since Π(•) is pseudo-continuous, we know From ( 14) and ( 15), one can get Hence, the Problem (T) have at least one optimal solution.The proof is finished.

Application
In this section, we apply our main results to existence results for Clarke's subdifferential inclusions and a class of differential hemivariational inequalities.

Clarke's Subdifferential Systems
Let us recall the definition of the Clarke's subdifferential for a locally Lipschitz function j : K ⊂ L 2 (Ω) → R, where K is a nonempty subset of a Banach space L 2 (Ω) (one can see [48][49][50]).We denote by j 0 (x; y) the Clarke's generalized directional derivative of j at the point x ∈ K in the direction y ∈ L 2 (Ω), that is Recall also that the Clarke's subdifferential or generalized gradient of j at x ∈ K, denoted by ∂j(x), is a subset of L 2 (Ω) * given by ∂j(x) := {x * ∈ X * : j 0 (x; y) ≥ x * , y , ∀y ∈ L 2 (Ω)}.Consider the following Clarke's subdifferential inclusion: where j : [0, T] × L 2 (Ω) → R is a locally Lipschitz function with respect to the second variable with Y being a separable reflexive Banach space, ∂j(t, •) denotes the Clarke's subdifferential of j(t, •) for t ∈ [0, T] and γ : L 2 (Ω) → L 2 (Ω) is a linear, continuous and compact operator.
We have the following result.
Theorem 5.If (H j ) hold, then the system (16) has a solution.