Multidomain Optimal Control of Variational Subpotential Mixed Evolution Inclusions

Multidomain variational optimal control of evolution mixed subpotential inclusions, are formulated and analyzed, on the basis of a perturbation conjugate duality convex analysis theory developed by the author. For Lagrangian optimality mixed conditions, fixed point existence results are demonstrated with an strongly monotone qualifying condition. Governing multidomain state systems correspond to primal evolution macro-hybrid mixed subpotential problems, whose solvability is similarly achieved. Innovative multidomain optimization existence results of primal, dual, Lagrangian mixed, as well as coupled pair state-control problems are established. Applications to underground macro-hybrid mixed control transport flow processes, illustrate the theory.


Introduction
Multidomain optimal control of primal evolution mixed variational subpotential inclusions, in a macro-hybrid mixed evolution real functional framework of primal and dual product reflexive Banach spaces, , are formulated and analyzed. All of this in relation with a spatial bounded domain Ω ⊂ d , B Gonzalo Alduncin alduncin@geofisica.unam.mx assumed to be a lower semicontinuous convex functional, whose integrand real functional components g 1  Optimal control minimization problem (O M H ) governed by primal evolution mixed multisystem (MH κ ), will be formulated and analyzed, on the basis of a perturbation conjugate duality convex analysis theory developed by the author [1], -in the spirit of Azé-Bolintinéanu's study on constrained convex parabolic control problems [2]. The governing state system MH κ will correspond to a macro-hybrid primal evolution mixed subpotential problem, whose existence result will be demonstrated as a resolvent primal fixed point characterization consequence, for regular primal initial conditions, with a primal operator strong monotonicity qualifying property [3]. On the other hand, a Lagrangian optimality mixed condition will be established, analyzing its macro-hybrid mixed solvability via a resolvent fixed point characterization too, with an objective subdifferential strong monotonicity ∂ J M H -qualifying condition.
As an innovative perturbation conjugate duality accomplishment of the present study, optimization multidomain macro-hybrid mixed existence results of primal, dual, Lagrangian mixed, as well as coupled pair state-control problems are established.
For some own applications of the original theory [1], to primal and dual variational evolution mechanical systems, we refer to [4][5][6][7][8]. Specifically, in paper [4] optimal control of nonlinear transport-flow mixed variational problems are studied; work [5] has to do with optimal control problems governed by primal and dual evolution macrohybrid mixed variational state inclusions, in reflexive Banach spaces, with applications to nonlinear constrained mechanical problems; optimal control of quasistatic elastoviscoplastic macro-hybrid mixed set-valued variational problems, are studied in [6], applying macro-hybrid variational formulations, as an strategy for multi-physics as well as parallel computing; paper [7] is concerned about optimality conditions of stationary macro-hybrid mixed variational inclusions, governing constrained optimal control problems, with applications to dual mechanical nonlinear distributed control steady diffusion processes, as well as to primal boundary static deformation systems. Lastly, macro-hybrid optimal control of dual evolution mixed variational transport flow processes, through elastoviscoplastic porous media, are studied in [8].
On the other hand, in contrast to our subpotential perturbation conjugate duality, operator variational approach, some optimal control studies based on diverse kinds of variational inequalities to be mentioned are the following [9][10][11][12][13][14]. In paper [9] existence and optimality conditions of optimal pairs are established for semilinear evolutionary variational inequality problems, with bilateral constraints; evolution implicit quasivariational inequalities of optimal control problems are analyzed in work [10], for quasi-static processes of elastic contact problems with friction between a body and a rigid foundation; paper [11] is concerned with the optimal control of systems modeled by differential inclusions, with anti-periodic conditions in Banach spaces, analyzing evolutionary hemi-variational inequalities for trajectory-control pair solutions; a class of subdifferential evolution inclusions are considered in study [12], with historydependent operators, establishing the existence of an optimal control for a dynamic frictional contact mechanical problem. Lastly, in paper [13] a result of optimal control to a minimization problem, applicable to problems of evolution differential inclusions is established, on the basis of quasi mixed equilibrium problems with a compact constraint set, bounded and unbounded; and in work [14] an optimal control problem for a differential quasivariational inequality is analyzed, applying the abstract results to a free boundary problem, for a viscoelastic body in a frictionless unilateral contact with a rigid foundation.
The present paper is organized as follows: In Sect. 2, the multidomain evolution mixed real functional framework, W M H × Y * M H × B * Γ , is introduced for the variational analysis; the primal evolution macro-hybrid mixed state system MH κ , governing optimal control problem O M H , is presented in Sect. 3, demonstrating its fixed point solvability result with a ∂ F-strong monotonicity qualifying condition; Sect. 4 has to do with the variational implementation of the multidomain transport flow mechanical underground processes, with intrinsic control constraints, that illustrate the theory of evolution macro-hybrid mixed state systems. Next Sect. 5 elaborates on the perturbation conjugate duality, multidomain evolution mixed optimal control theory, establishing the corresponding variational optimality mixed Lagrangian system, as a new optimization result; then in Sects. 5.1 and 5.2, the multidomain optimal control of the mechanical transport and flow state systems (of Sect. 4) are presented; in Sect. 6 the solvability analysis of the variational optimality macro-hybrid mixed Lagrangian system is proved, via a fixed point resolvent primal characterization; followed by some corresponding macro-hybrid mixed proximal penalty-duality algorithms. Then last Sect. 7 proceeds with the complementary innovating multidomain optimization analysis, to determine the primal, dual and Lagrangian mixed results, which provide the basis for concluding the central optimization result of the study: the solution existence of the own optimal control system O M H , as well as the corresponding variational solvability of the coupled macro-hybrid mixed pair state-control system MH κ -O M H . Section 8 states the conclusions of the paper at the end.

Multidomain Variational Evolution Functional Frameworks
For the multidomain evolution mixed functional setting of the analysis, we introduce a stationary mixed state real functional framework of primal and dual reflexive Banach spaces, V (Ω) and Y * (Ω), with topological duals V * (Ω) and Y (Ω), related to a spatial bounded domain Ω ⊂ d , d ∈ {1, 2, 3}, with a Lipschitz boundary ∂Ω. As corresponding Hilbert pivot spaces we shall have H (Ω) and Z * (Ω), such that , with continuous and dense embeddings. On the other hand, associated boundary primal and dual reflexive Banach trace spaces will be B(∂Ω) and its topological dual B * (∂Ω).
For a corresponding multidomain macro-hybrid mixed functional framework, we shall consider that the spatial domain Ω is decomposed in terms of disjoint and connected subdomains {Ω e }, Ω = E e=1 Ω e , with Lipschitz external and internal boundaries e = ∂Ω e ∩ ∂Ω and Γ e = ∂Ω e ∩ Ω, e = 1, 2, . . . , E, and interfaces Following our studies [15][16][17], we define primal and dual macro-hybrid local functional product spaces , as well as internal boundary functional product spaces B Γ ≡ E e=1 B(Γ e ) and its dual B * Γ ≡ E e=1 B * (Γ e ). Also, as proper trace operators, we consider the internal boundary primal and dual operators π Γ and δ * Γ , continuous and linear, assumed to satisfy the compatibility conditions [18] which are fundamental for macro-hybrid compositional duality results. Then, for the multidomain transmission constraints, to which decomposed

Primal Evolution Macro-hybrid Mixed State System
In this section, we present the multidomain variational primal evolution mixed governing κ-state multisystem, related to the macro-hybrid optimization problem O M H of the Introduction. In a general abstract sense [20], and in accordance with the evolution functional framework of previous Sect. 2, such a state multisystem has the variational form where B * ∈ L C M H , V * M H is the primal macro-hybrid coupling optimal control operator, linear and continuous, and λ * ∈ Q * ⊂ B * Γ is the dual macro-hybrid synchronizing internal boundary field implemented variationally by the conjugate indicator functional I Q * .
Here variational operator ∈ L V M H , Y M H corresponds to the primal linear continuous coupling operator with transpose T ∈ L Y * M H , V * M H . Also R − T and R , as well as R π Γ , denote range subspaces of the respective functionals. Further, the macro-hybrid primal subdifferential and right hand side term of the problem are defined by where the subdifferential ∂ϕ that models the primal intrinsic controlling distributed constraint, imposed to the system, will be assumed to satisfy the qualifying condition On the other hand, for such a macro-hybrid mixed state system, in particular v and y * stand as the primal Dirichlet (D-essential) and dual Neumman (N-natural) trace fields, prescribed on disjoint and complementary parts of the Ω-domain boundary, ∂Ω = ∂Ω D ∪ ∂Ω N , of the system under control. Further, regarding the dual subdifferentials ∂ G * , ∂ I Q * : Y * M H × B * Γ → 2 Y M H ×B Γ , they model variationally the dualized distributed primal constraint of the system, as well as the synchronization internal boundary transmission constraints due to the multidomain spatial decomposition. Recall that in this study all the subdifferential operators are assumed to be subpotential [21]; i.e, maximal monotone subdifferentials of proper convex semicontinuous functionals.
In order to determine the subpotential of primal subdifferential ∂ F, we incorporate the interior domain subpotential conditions, (cf. [21], Theorem 2.10), which guarantee the corresponding subdifferential subpotential sum relation Therefore, the desired primal subpotential characterization is concluded as follows.

Lemma 2 Under functional compatibility condition
and moreover, due to the intrinsic control condition C ∂ϕ state primal subdifferential ∂ F turns out to be a strongly monotone maximal monotone operator.
Example 1 A primal intrinsic controlling distributed constraint may be modeled by a subpotential strongly monotone sum subdifferential with component ∂ ϕ of an obstacle type, subpotential subdiffferential [22]; and a linear potential strongly monotone component ∂ A.
Specifically, we may consider the following classical elliptic variational model problem related to a Hilbert space V , with topological dual and corresponding pivot spaces, V * and H; i.e. V ⊂ H ⊂ V * with embeddings being continuous and dense.

P
Given where G : V → ∪ {+∞} is a proper lower semicontinuous convex functional with effective domain the nonempty closed convex subset K ⊂ V . Then, as it is well known, a sufficient condition for variational existence and uniqueness solvability to problem P is precisely the qualifying monotonicity operator property result that naturally (cf. Theorem 2 Proof, below), can be validated via a fixed point characterization, in this case through the corresponding proximation operator P rox λG : V → V , for any real λ > 0, defined by Consequently it can be concluded that problem P states, in fact, a variational optimality condition of the minimization, optimization problem in K of the functional The solution of such a problem, which exists uniquely, is the proximity point of w relative to λG. Notice that when G = I K the solution then turns out to be the projection of w on K .

Solvability of Macro-hybrid Mixed State System(MH Ä )
For the existence analysis of subpotential primal evolution macro-hybrid mixed state problem (MH κ ), (cf. [3, Sect. 2]), we first establish its primal duality principle introducing the corresponding interior classical primal evolution compatibility condition under which the compositional result [23] holds true. Then, by convex dualization of the (MH κ )-dual inclusion, it follows that result that in conjunction with compositional dualization (8) leads to the evolution duality principle of the state multisystem.
Lastly, on the basis of Lemma 2 and Theorem 1, and the macro-hybrid monotonicity of the primal state operator the following solvability result can be established via a primal evolution resolvent fixed point existence analysis [3]. For completeness and due to the importance that this result plays in the context of our multidomain optimal control theory, we shall provide here the details of its validity.

Theorem 2 Under the functional compatibility condition C G,I Q , ,π Γ
, and the monotonicity primal operator qualifying condition C ∂ F , primal evolution macrohybridized state multisystem MH κ has a solution, with a unique primal component, for initial data as regular as Proof Taking into account regularity condition (9), we consider the auxiliary state primal evolution inclusion, with an homogeneous initial data, whose solvability, for s = v + v 0 , results to be equivalent to that of primal state problem (P M H κ ). Further, we note that, importantly, its time derivative operator in the sense is, in fact, a linear densely defined maximal monotone operator (cf. [24,Proposition 32.10]). Thereby, in accordance with [25], we can now apply a fixed-point subdifferential approach for the existence analysis. Indeed, due to the monotonicity of time derivative operator (10), qualifying condition C ∂ F results to be equivalent to the strong monotonicity condition of the primal combined operator Moreover, operator A is maximal monotone, since intD(d/dt) ∩ D(∂ F) = ∅. Then, for an m-linearly strongly monotone and a-Lipschitz operator M : augmented and exactly penalized with a parameter r > 0, for s * ∈ As, by Here J r which is a well defined 1/m-firm contraction [26]. Thereby, auxiliary state primal evolution inclusion problem ( P M H κ ) has the Mresolvent fixed-point problem characterization which has a unique solution by the Banach fixed-point theorem, due to the 1/m-firm contraction resolvent property that implies the contraction of fixed point operator with contraction parameter 1/m(a − r α) < 1, for r > (a − m)/α ≥ 0 . Now, we can conclude the multidomain state primal evolution mixed variational existence result of the theory, from the conjunction of Theorems 1 and 2.

Multidomain Primal Evolution Macro-hybrid Mixed Transport Flow Mechanical System
In this section, we apply the multidomain evolution variational theory developed above, to mechanical Darcian transport flow processes in the subsurface, with intrinsic mass concentration and pressure controlling constraints, demonstrating its versatility.

Multidomain Macro-hybrid Mixed Variational Transport Local State System
For a transport model with implemented intrinsic mass concentration control mechanisms, applicable to underground compressible flow mechanical systems, in the setting of spatial multidomain decompositions and macro-hybrid mixed real functional frameworks of Sect. 2, we shall consider the following one.
Let } is the diffusion-dispersion tensor (cf., e.g., [27]). Further, let {s * α e } be the local distributed intrinsic mass concentration control scalar fields, and let { f * e } denote a given contaminant local source. Then related to instantaneous material connected local parts P e t (x) of each subdomain Ω e , e = 1, 2, . . . , E, surrounding any point x ∈ Ω e at a time t ∈]0, T [, the Reynolds transport theorem establishes that, for local unit P e t -n e boundary normal vectors, the following local mass balances must hold true (cf. e.g., [28] and by the divergence theorem applied to (13)-boundary term, and a continuous integral localization, the next local evolution mixed concentration-flux multidomain physical transport models are obtained: where [∂ϕ τ e ] stands as a subpotential maximal monotone subdifferential of primal distributed local control mechanisms, for the intrinsic concentration control,-the constraint of the transport processes,-with inverse control mechanisms [∂ϕ * τ e ] (ϕ * τ e , the local conjugate functional of ϕ τ e ) [22].
Here we are considering the classical macro-hybrid primal evolution transport formulation, whose continuity local transmission internal boundary constraints correspond to the dual normal diffusion-dispersion fluxes and the primal concentrations: across the internal boundary interfaces Γ ek = Γ e ∩ Γ k , 1 ≤ e < k ≤ E .

Multidomain Macro-hybrid Mixed Variational Transport Local State System
Related to [16], the variational primal evolution formulation of localized macro-hybrid mixed constrained τ -transport model (13)-(15), as a state system governing the optimal control minimization problem (O M H ) of the theory, is given as follows, adopting a simplified boldface generic notation of the previous sections.
In order to apply primal evolution macro-hybrid mixed fixed point existence Theorem 2, we assume that the transport local concentration initial data satisfies the regularity condition Theorem 3 Under Theorem 2 corresponding transport interior compatibility local condition, (C (D * −1 w * ,∂ I Q * τ ) ), and qualifying strong monotonicity condition (C ∂ F τ , primal evolution transport state system (MH κ τ ) attains a multidomain macro-hybrid mixed solution with a unique primal concentration solution component.
For this transport underground mechanical system, we recall that the primal constraints of intrinsic concentration control mechanisms ∂ϕ τ must be defined by strongly monotone maximal monotone operators [22].

Multidomain Primal Evolution Macro-hybrid Mixed Compressible Darcian Flow Mechanical System
For a compressible flow component of the underground mechanical process, we shall consider a constrained driving flow in the spatial multidomain, with a macro-hybrid decomposition Ω = E e=1 {Ω e } ⊂ n of the porous medium in the sense of Sect. 2. Let {u * e } = {φ e u * a e } be considered as the dual local fluid velocity vector field, defined in terms of the local porosity {φ e } and the average fluid velocity {u * a e } fields. We shall assume that the fluid is characterized by the local compressibility parameters Thereby, by a continuous integral localization, and applying the expanded time differentiation the local equations system of compressible Darcian flow constitutivity and mass balance turns out to be: ϑ ρ e ∂ p e ∂t + div w * e = −s * e + q * e ρ e , μ e (ρ e K e ) −1 w * e = −gr ad p e + ρ e g, s * e ∈ ∂ϕ f e ( p e ) ⇔ p e ∈ ϕ f e (s * e ), with given local flow physical parameters: {μ e } and {K e } of dynamic viscosity and invertible symmetric intrinsic permeability tensor. Here, for variational convenience, the mass flux rate field w * e = ρ e u * e is utilized as the dual dependent variable, instead of the common compressible fluid velocity u * e . Further, g denotes the gravity acceleration vector field.
Flow system (21) constitutes a local primal evolution mixed pressure-velocity constrained model, with a local primal pressure intrinsic control mechanism ∂ϕ e , a subpotential maximal monotone subdifferential, with inverse ∂ϕ * e , subdifferential of the conjugate functional ϕ * e [22]. Regarding the interaction of the flow system with its underground exterior, we shall consider, for e = 1, 2, . . . , E, prescribed local normal mass flux rates { w * n e } as Neumann natural boundary conditions and local pressures { p e } as Dirichlet essential boundary ones; i.e., w * e · n e = w * n e on ∂Ω N e and p e = p e on ∂Ω D e . Particularly, in the case of pure local Neumann boundary conditions, ∂Ω D e = ∅, we assume that for conservation of mass the local compatibility conditions ∂Ω e u * n e d∂Ω e = Ω e q * e dΩ e are satisfied. On the other hand, in relation with the spatial nonoverlapping decompositions of the underground porous medium, the continuity transmission internal boundary constraints are stated by the dual mass flux rate and primal pressure mechanical constraints: across the underground multidomain interfaces Γ ek = Γ e ∩ Γ k , 1 ≤ e < k ≤ E.

Multidomain Primal Evolution Macro-hybrid Mixed Variational Compressible Darcian Flow State System
As for the previous transport case, on the basis of [16], the variational primal evolution formulation of localized macro-hybrid mixed pressure-flux rate constrained f -model (18)- (22), as a state system governing the optimal control minimization problem (O M H ) of the theory, turns out to be the following variational version of the Darcian flow underground process, (utilizing once again the simplified boldface generic notation). ( Here dual unknown λ * f = δ * Γ f w * ∈ Q * τ ⊂ B * Γ τ stands as the flow macro-hybrid synchronizing internal boundary field, (cf. (3)). Also macro-hybrid primal pressure flow subdifferential and right hand side term correspond to where the local essential Dirichlet π f p = p and Neumann δ * f w * = w * , prescribed external boundary trace fields, indicator functional I p implementing the primal Dirichlet condition.
Regarding the abstract primal evolution macro-hybrid mixed state problem (MH κ ) of Sect. 3, as operator identifications: the primal coupling operator ; and the primal subdifferential ∂ F f = ∂ϕ f , the dual pressure maximal monotone control mechanism. Also, corresponding dual subdifferential ∂ G * f is given by the operator μ(ρ K ) −1 : In order to apply primal evolution macro-hybrid mixed fixed point existence Theorem 2, we assume that the subsurface flow local pressure initial data satisfies the regularity Hence, a primal compressible flow solvability result is as follows.

Theorem 4
Under dual corresponding interior local compatibility condition C (∂G * ,∂ I Q * ) and qualifying strong monotonicity condition (C ∂ F f of Theorem 2, primal state evolution compressible Darcian flow constrained problem (MH κ f ) has a macro-hybrid mixed solution with a unique primal pressure component.
Lastly, for this compressible Darcian flow case, we emphasize that the primal constraints of intrinsic pressure control mechanisms ∂ϕ f should be modeled by strongly monotone maximal monotone operators [22].

Perturbation Conjugate Duality Evolution Optimal Control
In this section we present the multidomain, macro-hybrid mixed perturbation conjugate duality optimal control theory, an extension of the evolution mixed set-value variational theory [1].
Specifically, once the perturbation theory is stated through the corresponding minimization primal, maximization dual and Lagrangian minimax optimization problems, the corresponding multidomain macro-hybrid mixed optimality condition will be established, whose existence analysis via a primal resolvent stationary type fixed point characterization, will be treated in the next section, Sect. 6, performing then the whole optimization analysis of the theory in the last Sect. 7.
In accordance with the perturbation conjugate duality evolution mixed method [1], from the (MH κ )-primal evolution state inclusion, the primal macro-hybrid mixed state-control operator T M H : whose closed convex constraint domain turns out to be Then, assuming that there is a closed subspace Q * M H ⊂ V * M H of macro-hybrid perturbations such that and introducing the closed convex subset with an associated marginal or infimal value convex functional μ * M H : Thereby, the primal, dual and Lagrangian mixed optimization problems of the theory, in a multidomain macro-hybrid perturbation sense, following the duality theory [23] (cf. [1]), can be defined as follows: Firstly, the optimal control with a perturbed primal evolution macro-hybrid mixed optimal control problem given by which results to be the conjugate of marginal functional (29); i.e, π M H = μ M H . Then the perturbation dual maximization problem is expressed as And thirdly, the perturbed convex-concave dual macro-hybrid mixed Lagrangian where Next, we lastly determine a new multidomain macro-hybrid mixed optimality condition for this perturbed Lagrangian problem (cf. [8]).   [26]. Thereby, the primal optimality problem (P M H ) is characterized by the M M H -resolvent fixed point problem, and by the Banach fixed-point theorem the desired existence result is concluded, due to the fact that the 1/m-firm contraction resolvent property implies the contraction of fixed point operator F r v * : Therefore, from the composition duality principle of Theorem 6, and the primal existence result of Theorem 7, the macro-hybrid mixed optimality variational solvability is finally achieved.

Macro-hybrid Mixed Proximal Penalty-Duality Algorithms
In this subsection we introduce two-and three-field proximal penalty-duality algorithms, of a stationary type, for the resolution of multidomain macro-hybrid primal mixed optimality problem (MOC M H ) [26,29,30], which follow via preconditioned and augmented variational reformulations, with exact penalizations, characterized in terms of resolvent and proximation operators. Such preconditioned iterative penalty-duallity procedures have demonstrated to be of the most effective schemes in mechanics, for parallel computing, of multidomain mixed variational inclusions associated to optimization problems.

Two-field Algorithm
In accordance with [1, Sect. 5.1.1], in terms of a linear symmetric M : Q → Q * preconditioner, continuous and m-strongly monotone, with inverse M −1 : Q * → Q, and an exact penalization parameter r > 0, a primal augmented regularization of Consequently, a natural macro-hybrid primal mixed proximal-point iterative threefield algorithm is given by A relevant interpretation, in the context of evolutionary systems, of these stationary variational proximal penalty-duality algorithms (cf. [30], Sect. 5), is that variational proximation time marching schemes under appropriate maximal monotonicity conditions, evolve as their step times m → ∞ to corresponding stationary states limits. Thereby, optimal states (v, (y * , λ * ), κ) ∈ M M H of macro-hybrid mixed optimality problem (MOC M H ) may be regarded as stationary states limits, for instance of a primal semi-implicit Euler time marching scheme, like ALG0 analyzed in ( [30], Sect. 5.1).
For the convergence analysis of these optimality proximal-point, macro-hybrid mixed penalty-duality iterative algorithms, we refer to [31,Sect. 4], where additionally some important algorithmic implementation variational results are treated.
Lastly, at this algorithmic stage of the study, we state that some alternative modern important local optimization splitting methods for multidomain structured monotone inclusions, are the following [32][33][34][35][36][37]. Specifically, in paper [32] a primal-dual parallel proximal splitting method is proposed and analyzed for domain decomposition of linear and nonlinear PDE's problems, via local coupling subdomains minimization energy interface functions; several results concerning the solution of monotone inclusion problems by splitting methods are outlined in study [33], for optimality conditions related to convex optimization problems where the sum of maximally monotone operators is involved, presenting some important contributions of their algorithmic realizations; paper [34] is concerned with structured coupled monotone inclusions in Hilbert spaces, analyzing the asymptotic behavior of a general primal-dual splitting solving algorithm, most steps of which can be executed in parallel. In paper [35] two different primal-dual splitting algorithms for solving structured monotone inclusions are proposed: the preconditioned forward-backward splitting algorithm and the forward-backward-half-forward splitting algorithm, both being calculated with a simple framework where single-valued operators are processed via explicit steps and set-valued operators are computed by their resolvents; rapidly convergent forwardbackward algorithms for computing zeroes of the sum of finitely many maximally monotone operators, are developed in paper [36], incorporating an inertial term, a

Conclusions
Once the evolution macro-hybrid mixed, variational reflexive Banach functional framework of the study was stated, the solvability of the multidomain subpotential primal evolution mixed state governing system, was demonstrated as a resolvent fixed point characterization result, under a strongly monotone qualifying condition. Then the analysis has been illustrated, in the sequel, through the variational implementation of an underground transport flow mechanical process, with intrinsic control constraints. Next the perturbation conjugate duality optimal control, of the multidomain evolution mixed theory was presented, establishing its own Lagrangian mixed variational optimality condition, as a new result, showing its applicability to the multidomain optimal control of the mechanical transport flow process. Then the mixed optimality fixed point solvability was proved, concluding with the optimality two-and three-field macro-hybrid mixed proximal penalty-duality algorithms. Finally, an innovating optimization existence analysis was performed, determining the solvability of the minimization optimal control problem of the theory, as well as the corresponding variational solution existence of the coupled macro-hybrid mixed pair state-control system.
Funding The authors declare that no funds or grants, were received during the preparation of this manuscript.

Declarations
Competing Interests The authors declare that they have no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.