aa r X i v : . [ m a t h . O C ] F e b OPTIMAL CONTROL OF PLASTICITY WITH INERTIA ∗ STEPHAN WALTHER † Abstract.
The paper is concerned with an optimal control problem governed by the equations of elasto plasticitywith linear kinematic hardening and the inertia term at small strain. The objective is to optimize the displacementfield and plastic strain by controlling volume forces. The idea given in [10] is used to transform the state equationinto an evolution variational inequality (EVI) involving a certain maximal monotone operator. Results from [27] arethen used to analyze the EVI. A regularization is obtained via the Yosida approximation of the maximal monotoneoperator, this approximation is smoothed further to derive optimality conditions for the smoothed optimal controlproblem.
Key words.
Optimal control of variational inequalities, plasticity with inertia, Yosida approximation, first-ordernecessary optimality conditions
AMS subject classifications. sec:1
1. Introduction.
We consider the following optimal control problem governed by theequations of elasto plasticity with linear kinematic hardening and the inertia term at smallstrain: {eq:optimization_problem_inertia} (1.1) ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ min 𝐽 ( 𝑢, . 𝑢, 𝑧, 𝑓 ) = Ψ( 𝑢, . 𝑢, 𝑧 ) + 𝛼 ‖ 𝑓 ‖ 𝔛 𝑐 , s.t. 𝜌 .. 𝑢 − div ℂ (∇ 𝑠 𝑢 − 𝑧 ) = 𝑓 , . 𝑧 ∈ 𝐴 ( ℂ ∇ 𝑠 𝑢 − ( ℂ + 𝔹 ) 𝑧 ) , ( 𝑢, . 𝑢, 𝑧 )(0) = ( 𝑢 , 𝑣 , 𝑧 ) ,𝑢 ∈ 𝐻 ( 𝐻 𝐷 (Ω; ℝ 𝑑 )) ∩ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) ,𝑧 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) ,𝑓 ∈ 𝔛 𝑐 . Herein, Ω ⊂ ℝ 𝑑 is the body under consideration with density 𝜌 , where 𝑑 ∈ ℕ is the dimension.Its boundary is split into two disjoints parts Γ 𝐷 and Γ 𝑁 . Furthermore, 𝑢 ∶ [0 , 𝑇 ] × Ω → ℝ 𝑑 is the displacement field and 𝑧 ∶ [0 , 𝑇 ] × Ω → ℝ 𝑑 × 𝑑𝑠 the plastic strain. The initial data ( 𝑢 , 𝑣 , 𝑧 ) is given and fixed. The volume force is given by 𝑓 ∶ [0 , 𝑇 ] × Ω → ℝ 𝑑 . Thetime derivative of, for instance, the plastic strain is denoted by . 𝑧 and the symmetric gradientby ∇ 𝑠 = ∕ (∇ + ∇ ⊤ ) . Moreover, ℂ is the elasticity tensor and 𝔹 the hardening Parameter.The flow rule is represented by the maximal monotone operator 𝐴 , in section 5 below we willchoose the von-Mises flow rule. The control space 𝔛 𝑐 is a subspace of 𝐻 ([0 , 𝑇 ]; 𝐿 (Ω; ℝ 𝑑 )) and 𝛼 > a fixed Tikhonov parameter. The precise definitions and assumptions are presentedin section 2 below. Note that the problem (1.1) is formulated only with the displacement andplastic strain as state variables. The stress is normally given by 𝜎 = ℂ (∇ 𝑠 𝑢 − 𝑧 ) (and can thusbe easily integrated in Ψ ), however, we eliminated it in (1.1) for convenience. Additionally, theDirichlet displacement and Neumann boundary forces (see the definition of the div -operatorin section 2) are set to zero, c.f. Remark 3.8 below. Regarding a detailed description andderivation of the plasticity model, we refer to [21, 19, 15]. ∗ Submitted to the editors DATE.
Funding:
This research was supported by the German Research Foundation (DFG) under grant num-ber ME 3281/9-1 within the priority program Non-smooth and Complementarity-based Distributed Parameter Sys-tems: Simulation and Hierarchical Optimization (SPP 1962). † This manuscript is for review purposes only.
S. WALTHER
Let us put our work into perspective. Optimal control problems governed by plastic-ity were consider in [14, 16, 22, 23, 24, 25, 27, 17, 18]. The articles [14, 16] are con-cerned with the static case of elasto plasticity, for further articles of the static case we re-fer to the references therein. For the time dependent quasi-static case we are only aware of[22, 23, 24, 25, 27, 17, 18]. The articles [22, 23, 24, 25] and the application in [27] are devotedto the case of elasto plasticity with linear kinematic hardening, whereas [17, 18] are concernedwith the case of perfect plasticity, that is, with no hardening. In contrast, we consider the caseof elasto plasticity with the inertia term (and linear kinematic hardening), that is, the secondtime derivative of the displacement (the accelaration) multiplied by the density 𝜌 is present inthe balance of momentum. Due to this inertia term, the equations are physically more reason-able than the quasi-static case (of course, the solution to both systems might not differ muchwhen the accelaration of the body is small, which is the reasoning when neglecting the inertiaterm). As said above, the application in [27] investigates quasi-static (homogenized) plasticitywith hardening. This application is analyzed by applying results concerned with an abstractoptimal control problem governed by an first-order evolution variational inequality (EVI) in-volving a maximal monotone operator. As we will see below, the state equation in (1.1) canalso be transformed into such an abstract EVI. Let us note that the existence of a solution wasalready proven in [10, Theorem 5.1] by using essentially the same transformation into an EVIas we will do. However, there it was transformed into a second order EVI and the maximalmonotonicity of the (slightly different) operator given therein was proven in another way. Incontrast, we consider a first order EVI and will provide the concrete form of the resolvent inProposition 3.11 (which will also be used later in subsection 4.2 to provide optimality condi-tions), the maximal monotonicity of our operator will then follow easily. Having transformedthe state equation, we can apply the results from [27]. We only have to heed two differencesbetween our EVI and the EVI analyzed in [27]. First, our maximal monotone operator doesnot fulfill some properties required in [27], second, the given data are more regular in timethan in [27], as we will elaborate on at the end of subsection 3.2. However, the better regular-ity in time will compensate the missing properties of our maximal monotone operator, so thatthe unique existence of a solution can still be shown (Theorem 3.15) and thus we can applyresults from [27]. There is a large list of literature on plasticity with inertia, we only refer to[1, 2, 8, 6, 7, 3] and the references therein. However, to best of the author’s knowledge, thereexists no contribution to optimal control of plasticity with the inertia term, except in [26]. Weemphasize that this paper is essentially based on [26, Part IV] and on the transformation ideafrom [10].The paper is organized as follows. After introducing our notation and standing assump-tions in section 2, we transform the state equation in (1.1) into a first-order EVI, prove theunique existence for given data and provide regularization and convergence results in sec-tion 3. Afterwards, in section 4, we analyze the optimal control problem (1.1), show theexistence of a global solution, provide an approximation result via a regularized problem andfinally present optimality conditions. sec:2
2. Notation and Standing Assumptions.
Notation.
Given two vector spaces 𝑋 and 𝑌 , we denote the space of linear and continu-ous functions from 𝑋 into 𝑌 by ( 𝑋, 𝑌 ) . If 𝑋 = 𝑌 , we simply write ( 𝑋 ) . The dual spaceof 𝑋 is denoted by 𝑋 ∗ = ( 𝑋, ℝ ) . If 𝐻 is a Hilbert space, we denote its scalar product by ( ⋅ , ⋅ ) 𝐻 . For the whole paper, we fix the final time 𝑇 > . For 𝑡 > we denote the Bochnerspace of square-integrable functions on the time interval [0 , 𝑡 ] by 𝐿 ([0 , 𝑡 ]; 𝑋 ) , the Bochner-Sobolev space by 𝐻 ([0 , 𝑡 ]; 𝑋 ) and the space of continuous functions by 𝐶 ([0 , 𝑡 ]; 𝑋 ) . Wefurthermore abbreviate 𝐿 ( 𝑋 ) ∶= 𝐿 ([0 , 𝑇 ]; 𝑋 ) , 𝐻 ( 𝑋 ) ∶= 𝐻 ([0 , 𝑇 ]; 𝑋 ) and 𝐶 ( 𝑋 ) ∶= 𝐶 ([0 , 𝑇 ]; 𝑋 ) . When 𝐺 ∈ ( 𝑋 ; 𝑌 ) is a linear and continuous operator, we can define an This manuscript is for review purposes only.
PTIMAL CONTROL OF PLASTICITY WITH INERTIA ( 𝐿 ( 𝑋 ); 𝐿 ( 𝑌 )) by 𝐺 ( 𝑢 )( 𝑡 ) ∶= 𝐺 ( 𝑢 ( 𝑡 )) for all 𝑢 ∈ 𝐿 ( 𝑋 ) and for almost all 𝑡 ∈ [0 , 𝑇 ] , we denote this operator also by 𝐺 , that is, 𝐺 ∈ ( 𝐿 ( 𝑋 ); 𝐿 ( 𝑌 )) , and analogfor Bochner-Sobolev spaces, i.e., 𝐺 ∈ ( 𝐻 ( 𝑋 ); 𝐻 ( 𝑌 )) . Given a coercive and symmet-ric operator 𝐺 ∈ ( 𝐻 ) in a Hilbert space 𝐻 , we denote its coercivity constant by 𝛾 𝐺 , i.e., ( 𝐺ℎ, ℎ ) 𝐻 ≥ 𝛾 𝐺 ‖ ℎ ‖ 𝐻 for all ℎ ∈ 𝐻 . With this operator we can define a new scalar product,which induces an equivalent norm, by 𝐻 × 𝐻 ∋ ( ℎ , ℎ ) ↦ ( 𝐺ℎ , ℎ ) 𝐻 ∈ ℝ . We denote theHilbert space equipped with this scalar product by 𝐻 𝐺 , that is ( ℎ , ℎ ) 𝐻 𝐺 = ( 𝐺ℎ , ℎ ) 𝐻 forall ℎ , ℎ ∈ 𝐻 . If 𝑝 ∈ [1 , ∞] , then we denote its conjugate exponent by 𝑝 ′ , that is 𝑝 + 𝑝 ′ = 1 .Throughout the paper, by 𝐿 𝑝 (Ω; 𝑀 ) we denote Lebesgue spaces with values in 𝑀 , where 𝑝 ∈ [1 , ∞] and 𝑀 is a finite dimensional space. By 𝑊 ,𝑝 (Ω; 𝑀 ) we denote Sobolev spacesand 𝑊 ,𝑝𝐷 (Ω; 𝑀 ) is the subspace containing functions which traces are zero on Γ 𝐷 . For thedual space of 𝑊 ,𝑝 ′ 𝐷 (Ω; 𝑀 ) we write 𝑊 −1 ,𝑝𝐷 (Ω; 𝑀 ) . Finally, by ℝ 𝑑 × 𝑑𝑠 , we denote the space ofsymmetric matrices and 𝑐, 𝐶 > are generic constants. Standing Assumptions.
The following standing assumptions are tacitly assumed for therest of the paper without mentioning them every time.
Domain.
The domain Ω ⊂ ℝ 𝑑 , 𝑑 ∈ ℕ , is bounded with Lipschitz boundary Γ . Theboundary consists of two disjoint measurable parts Γ 𝑁 and Γ 𝐷 such that Γ = Γ 𝑁 ∪ Γ 𝐷 . While Γ 𝑁 is a relatively open subset, Γ 𝐷 is a relatively closed subset of Γ with positive boundarymeasure. In addition, the set Ω ∪ Γ 𝑁 is regular in the sense of Gröger, cf. [11].Furthermore, the density of Ω is given by 𝜌 > . Coefficients.
The elasticity tensor and the hardening parameter satisfy ℂ , 𝔹 ∈ ( ℝ 𝑑 × 𝑑𝑠 ) and are symmetric and coercive, i.e., there exist constants 𝑐 > and 𝑏 > such that ( ℂ 𝜎, 𝜎 ) ℝ 𝑑 × 𝑑𝑠 ≥ 𝑐 ‖ 𝜎 ‖ ℝ 𝑑 × 𝑑𝑠 and ( 𝔹 𝜎, 𝜎 ) ℝ 𝑑 × 𝑑𝑠 ≥ 𝑏 ‖ 𝜎 ‖ ℝ 𝑑 × 𝑑𝑠 for all 𝜎 ∈ ℝ 𝑑 × 𝑑𝑠 .We abbreviate further 𝔻 ∶= 𝔹 ( ℂ + 𝔹 ) −1 ℂ ∈ ( ℝ 𝑑 × 𝑑𝑠 ) and 𝔼 ∶= ℂ ( ℂ + 𝔹 ) −1 ∈ ( ℝ 𝑑 × 𝑑𝑠 ) {eq:def_D_E_inertia} (2.1)and note that 𝔻 is symmetric and coercive, according to [10, Lemma 4.2]. Moreover, forinstance, we denote the adjoint of 𝔼 by 𝔼 ⊤ . Initial data.
We choose 𝑢 , 𝑣 ∈ 𝐻 𝐷 (Ω; ℝ 𝑑 ) and 𝑧 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) and define 𝑞 ∶= ℂ ∇ 𝑠 𝑢 − ( ℂ + 𝔹 ) 𝑧 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) . Moreover, we assume that ( 𝑢 , 𝑣 , 𝑞 ) is an element of 𝐷 ( ) , where 𝐷 ( ) is given in Definition 3.5. Operators.
Throughout the paper, ∇ 𝑠 ∶= (∇+∇ ⊤ ) ∶ 𝑊 ,𝑝 (Ω; ℝ 𝑑 ) → 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) de-notes the linearized strain. Its restriction to 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 × 𝑑𝑠 ) is denoted by the same symbol and,for the adjoint of this restriction, we write − div ∶= (∇ 𝑠 ) ∗ ∶ 𝐿 𝑝 ′ (Ω; ℝ 𝑑 × 𝑑𝑠 ) → 𝑊 −1 ,𝑝 ′ 𝐷 (Ω; ℝ 𝑑 ) .The operator 𝐴 ∶ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) → 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) is maximal monotone with domain 𝐷 ( 𝐴 ) .Furthermore, by 𝐴 𝜆 ∶ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) → 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , 𝜆 > , we denote the Yosida approxima-tion of 𝐴 and by 𝑅 𝜆 = ( 𝐼 + 𝜆𝐴 ) −1 the resolvent of 𝐴 , so that 𝐴 𝜆 = 𝜆 ( 𝐼 − 𝑅 𝜆 ) . Moreover, for ev-ery 𝜆 > the resolvent 𝑅 𝜆 can be expressed pointwise, that is, there exists ̃𝑅 𝜆 ∶ ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 such that 𝑅 𝜆 ( 𝜏 )( 𝑥 ) = ̃𝑅 𝜆 ( 𝜏 ( 𝑥 )) f.a.a. 𝑥 ∈ Ω and ∀ 𝜏 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) . {eq:resolvent_pointwise_inertia} (2.2)With a slight abuse of notation we denote also ̃𝑅 𝜆 by 𝑅 𝜆 . It is to be noted that this is the casefor the subdifferential of an indicator function of a pointwise defined set, where the resolventis simply the projection onto this set, this example will be considered in section 5 below. Forfurther reference on maximal monotone operators, we refer to [5], [28, Ch. 32], [4, Ch. 55],and [20, Ch. 55]. This manuscript is for review purposes only.
S. WALTHER
Optimization Problem. By 𝐽 ∶ 𝐿 ( ) × 𝔛 𝑐 → ℝ , 𝐽 ( 𝑢, 𝑣, 𝑧, 𝑓 ) ∶= Ψ( 𝑢, 𝑣, 𝑧 ) + 𝛼 ‖ 𝑓 ‖ 𝔛 𝑐 we denote the objective function, where is given in Definition 3.5 and the control space 𝔛 𝑐 is a Hilbert space and embedded into 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) . We assume that Ψ ∶ 𝐿 ( ) → ℝ is weakly lower semicontinuous, continuous and bounded from below and that the Tikhonovparameter 𝛼 is a positive constant. sec:inertia_se
3. State Equation.
We begin our investigation with the state equation. At first we givethe definition of a solution and then transform the state equation into an EVI with a new(maximal monotone) operator . In subsection 3.2 we prove the existence of a solution byshowing that the operator is maximal monotone, then we can apply [4, Theorem 55.A].Finally, in subsection 3.3 we can use some results in [27] to obtain convergence results.The formal strong formulation of the state equation reads eq:state_equation_inertia 𝜌 .. 𝑢 − ∇ ⋅ ℂ (∇ 𝑠 𝑢 − 𝑧 ) = 𝑓 in Ω , {eq:state_equation_a_inertia}{eq:state_equation_a_inertia} (3.1a) 𝜈 ⋅ ℂ (∇ 𝑠 𝑢 − 𝑧 ) = 0 on Γ 𝑁 , {eq:state_equation_b_inertia}{eq:state_equation_b_inertia} (3.1b) 𝑢 = 0 on Γ 𝐷 , {eq:state_equation_c_inertia}{eq:state_equation_c_inertia} (3.1c) . 𝑧 ∈ 𝐴 ( ℂ ∇ 𝑠 𝑢 − ( ℂ + 𝔹 ) 𝑧 ) in Ω , {eq:state_equation_e_inertia}{eq:state_equation_e_inertia} (3.1d) ( 𝑢, . 𝑢, 𝑧 )(0) = ( 𝑢 , 𝑣 , 𝑧 ) in Ω . (3.1e)Note that we have assumed in the standing assumptions above that the density is constant in Ω . It is possible to consider a density which has a spatial dependency (that is, a function from Ω to (0 , ∞) ), one has then in particular to verify that the operator 𝑄 , given in Definition 3.5,is well defined, that is, the multiplication of 𝜌 (and also ∕ 𝜌 ) with a Sobolev function is againa Sobolev function. However, for simplicity we assume that 𝜌 is constant.We impose the following assumption for the rest of this section.A SSUMPTION
Let 𝑓 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) begiven. sec:inertia_se_dat Let us begin with the definition of a solution tothe state equation (3.1).D
EFINITION
We call 𝑢 ∈ 𝐻 ( 𝐻 𝐷 (Ω; ℝ 𝑑 )) ∩ def:state_equation_inertia 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) and 𝑧 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) solution of (3.1) if 𝜌 .. 𝑢 − div ℂ (∇ 𝑠 𝑢 − 𝑧 ) = 𝑓 , . 𝑧 ∈ 𝐴 ( ℂ ∇ 𝑠 𝑢 − ( ℂ + 𝔹 ) 𝑧 ) , ( 𝑢, . 𝑢, 𝑧 )(0) = ( 𝑢 , 𝑣 , 𝑧 ) {eq:state_equation_def_inertia} (3.2) holds. Before we can transform the state equation into an EVI we need to reformulate it, to thisend we introduce the followingD
EFINITION 𝑧 to 𝑞 mapping). We define def:QQ_and_ZZ_inertia 𝔔 ∶ 𝐻 (Ω; ℝ 𝑑 ) × 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) → 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , ( 𝑢, 𝑧 ) ↦ ℂ ∇ 𝑠 𝑢 − ( ℂ + 𝔹 ) 𝑧 and its inverse (for fixed 𝑢 ) ℨ ∶ 𝐻 (Ω; ℝ 𝑑 ) × 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) → 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , ( 𝑢, 𝑞 ) ↦ ( ℂ + 𝔹 ) −1 ( ℂ ∇ 𝑠 𝑢 − 𝑞 ) . This manuscript is for review purposes only.
PTIMAL CONTROL OF PLASTICITY WITH INERTIA 𝔻 and 𝔼 given in the standing assumptionsabove, that is, 𝔻 = 𝔹 ( ℂ + 𝔹 ) −1 ℂ and 𝔼 = ℂ ( ℂ + 𝔹 ) −1 .L EMMA 𝑧 to 𝑞 ). We consider lem:transformation_of_z_to_q_inertia 𝜌 .. 𝑢 − div( 𝔻 ∇ 𝑠 𝑢 + 𝔼 𝑞 ) = 𝑓 , ( ℂ + 𝔹 ) −1 . 𝑞 + 𝐴 ( 𝑞 ) − 𝔼 ⊤ ∇ 𝑠 . 𝑢 ∋ 0 , ( 𝑢, . 𝑢, 𝑞 )(0) = ( 𝑢 , 𝑣 , 𝑞 ) = ( 𝑢 , 𝑣 , 𝔔 ( 𝑢 , 𝑧 )) {eq:state_equation_transformed_inertia} (3.3) for functions 𝑢 ∈ 𝐻 ( 𝐻 𝐷 (Ω; ℝ 𝑑 )) ∩ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) , 𝑞 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) . Recall that 𝔼 ⊤ is the adjoint of 𝔼 . Then the following holds:When ( 𝑢, 𝑧 ) is a solution of (3.1) , then ( 𝑢, 𝑞 ) = ( 𝑢, 𝔔 ( 𝑢, 𝑧 )) solves (3.3) . Vice versa, when ( 𝑢, 𝑞 ) solves (3.3) , then ( 𝑢, 𝑧 ) = ( 𝑢, ℨ ( 𝑢, 𝑞 )) is a solution of (3.1) .Proof. Both implications can be immediately obtained by using the definition of 𝔔 and ℨ and inserting 𝑧 in (3.2) and 𝑞 in (3.3), respectively (note that ℂ − ℂ ( ℂ + 𝔹 ) −1 ℂ = ( 𝐼 − ℂ ( ℂ + 𝔹 ) −1 ) ℂ = 𝔹 ( ℂ + 𝔹 ) −1 ℂ = 𝔻 ).We are now in the position to introduce the EVI, respectively the operator .D EFINITION ). For 𝑝 ∈ [1 , ∞] we set def:AA_and_spaces_inertia 𝑝 ∶= 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 ) × 𝐿 (Ω; ℝ 𝑑 ) × 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) and ∶= . The scalar product on is defined by ( ( 𝑢 , 𝑣 , 𝑞 ) , ( 𝑢 , 𝑣 , 𝑞 ) ) ∶= ( 𝔻 ∇ 𝑠 𝑢 , ∇ 𝑠 𝑢 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ( 𝑣 , 𝑣 ) 𝐿 (Ω; ℝ 𝑑 ) + ( 𝑞 , 𝑞 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) (recall that 𝔻 is symmetric and coercive). We define ∶ 𝐷 ( ) → , ( 𝑢, 𝑣, 𝑞 ) ↦ ⎛⎜⎜⎝ − 𝑣 − div( 𝔻 ∇ 𝑠 𝑢 + 𝔼 𝑞 ) 𝐴 ( 𝑞 ) − 𝔼 ⊤ ∇ 𝑠 𝑣 ⎞⎟⎟⎠ with the domain 𝐷 ( ) ∶= {( 𝑢, 𝑣, 𝑞 ) ∈ 𝐻 𝐷 (Ω; ℝ 𝑑 ) × 𝐻 𝐷 (Ω; ℝ 𝑑 ) × 𝐷 ( 𝐴 ) ∶ div( 𝔻 ∇ 𝑠 𝑢 + 𝔼 𝑞 ) ∈ 𝐿 (Ω; ℝ 𝑑 )} . Moreover, we set 𝑅 ∶ 𝐿 (Ω; ℝ 𝑑 ) → ∞ , 𝑓 ↦ (0 , 𝑓 , and 𝑄 ∶= ( 𝐼, ( ∕ 𝜌 ) 𝐼, ℂ + 𝔹 ) . L EMMA
The tuple ( 𝑢, 𝑞 ) solves (3.3) if and only if lem:transformation_into_an_EVI_inertia ( 𝑢, 𝑣, 𝑞 ) = ( 𝑢, . 𝑢, 𝑞 ) ∈ 𝐻 ( ) is a solution of 𝑄 −1 ( . 𝑢, . 𝑣, . 𝑞 ) + ( 𝑢, 𝑣, 𝑞 ) ∋ 𝑅𝑓 , ( 𝑢, 𝑣, 𝑞 )(0) = ( 𝑢 , 𝑣 , 𝑞 ) . {eq:state_equation_as_EVI_inertia} (3.4) This manuscript is for review purposes only.
S. WALTHER
Proof.
This follows immediately from the definition of . Remark 𝐴 , aswe did in the case of elasto plasticity. However, our approach is different, due to the trans-formation into an EVI we can regularize the operator , this is our method in subsection 3.3and section 4. We also mention that the fact that 𝑣 = . 𝑢 will be lost after the regularization(cf. Corollary 4.5 and Definition 4.7) and that we will transform our objective function inDefinition 4.1, so that we obtain an optimal control problem with respect to the state ( 𝑢, 𝑣, 𝑞 ) in (4.1). The optimality conditions given in Theorem 4.14 below are then also formulated forthis transformed problem. Remark rem:Neumann_and_Dirichlet_inertia cuss some issues with possible surface forces and Dirichlet displacements. Regarding surfaceforces, they are currently equal to zero and contained in the domain 𝐷 ( ) by the requirement div( 𝔻 ∇ 𝑠 𝑢 + 𝔼 𝑞 ) ∈ 𝐿 (Ω; ℝ 𝑑 ) . Allowing now surface forces which are time dependent, thedomain, and thus itself, would also depend on the time.An approach for Dirichlet displacements would be to exchange the displacement with a“new” displacement minus the Dirichlet displacement, then one could still define the domain 𝐷 ( ) as a subset of 𝐻 𝐷 (Ω; ℝ 𝑑 ) × 𝐻 𝐷 (Ω; ℝ 𝑑 ) × 𝐷 ( 𝐴 ) . However, this would again make thedomain and the operator itself time dependent (the Dirichlet displacement would occur alsoin the operator).In both cases one could still show that the arising operator is maximal monotone for afixed time, but for different points in time the monotonicity would be perturbed by the timedependent functions. Having now a closer look at Theorem 3.15 below, respectively [4, The-orem 55.A], we see that a comparison of two different points in time is used to derive a prioriestimates. Following this proof, the time depend functions would occur and a straightforwardadaption is not possible.At this juncture, let us also elaborate on the underlying spaces of the operator . Onemight try to exchange 𝐿 ( ℝ ; Ω) with a negative Sobolev space in the definition of to allowsurface forces. However, with this definition of , for instance, the proof of Lemma 3.12(which is used to show the monotonicity of ) would not be valid anymore. Thus, our choiceof seems reasonable. sec:inertia_se_eoas We prove now the existence of a solution to (3.1) by usingan existence result for EVIs involving a maximal monotone operator given in [4, Theorem55.A], thus we need to show that is maximal monotone. Since the monotonicity of canbe easily obtained (cf. Lemma 3.12), it remains to prove that the resolvent exists (cf. the proofof Proposition 3.13). For this it is sufficient to show the existence of a solution to (3.9) in thecase 𝑝 = 2 . However, since the existence and Lipschitz continuity for 𝑝 > is needed toderive optimality conditions in subsection 4.2, we already provide the following corollary forlater needed results.C OROLLARY
Let 𝜆 > and 𝑝 ∈ [2 , 𝑝 ] , where 𝑝 cor:W1r_existence_inertia is from [13, Theorem 1.1], with 𝑑 ≥ − 𝑑𝑝 . We assume that there exist 𝑚, 𝑀, 𝐷 ∈ ℝ , 𝐷 ≥ < 𝑚 ≤ 𝑀 , such that the family of functions { 𝑏 𝜎 ∶ Ω × ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 } 𝜎 ∈ ℝ 𝑑 × 𝑑𝑠 has theThis manuscript is for review purposes only. PTIMAL CONTROL OF PLASTICITY WITH INERTIA following properties: 𝑏 ( ⋅ ,
0) ∈ 𝐿 ∞ (Ω; ℝ 𝑑 × 𝑑𝑠 ) , {eq:W1r_existence_inertia_1}{eq:W1r_existence_inertia_1} (3.5) 𝑏 𝜎 ( ⋅ , 𝜏 ) is measurable , {eq:W1r_existence_inertia_2}{eq:W1r_existence_inertia_2} (3.6) ( 𝑏 𝜎 ( 𝑥, 𝜏 ) − 𝑏 𝜎 ( 𝑥, 𝜏 )) ∶ ( 𝜏 − 𝜏 ) + 𝐷 ( | 𝜎 − 𝜎 | + | 𝜏 − 𝜏 | ) | 𝜎 − 𝜎 | ≥ 𝑚 | 𝜏 − 𝜏 | , {eq:W1r_existence_inertia_3}{eq:W1r_existence_inertia_3} (3.7) | 𝑏 𝜎 ( 𝑥, 𝜏 ) − 𝑏 𝜎 ( 𝑥, 𝜏 ) | ≤ 𝑀 (| 𝜏 − 𝜏 | + | 𝜎 − 𝜎 |) {eq:W1r_existence_inertia_4} (3.8) for almost all 𝑥 ∈ Ω and all 𝜎, 𝜎, 𝜏, 𝜏 ∈ ℝ 𝑑 × 𝑑𝑠 .Then for every 𝜑 ∈ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) and 𝐿 ∈ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) there exists a unique solution 𝑢 ∈ 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 ) of − div 𝑏 𝜑 ( ⋅ , ∇ 𝑠 𝑢 ) + 𝑢𝜆 = 𝐿. Moreover, there exists a constant 𝐶 such that the inequality ‖ 𝑢 − 𝑢 ‖ 𝑊 ,𝑝 (Ω; ℝ 𝑑 ) ≤ 𝐶 (‖ 𝜑 − 𝜑 ‖ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ‖ 𝐿 − 𝐿 ‖ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) ) holds for all 𝜑 , 𝜑 ∈ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) and 𝐿 , 𝐿 ∈ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) , where 𝑢 and 𝑢 are thesolutions with respect to ( 𝜑 , 𝐿 ) and ( 𝜑 , 𝐿 ) .Proof. Note that 𝑏 𝜎 ( ⋅ , 𝜏 ) ∈ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) holds for all 𝜏, 𝜎 ∈ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) (and in fact forall 𝑝 ∈ [1 , ∞] ), which follows from (3.5), (3.6) and (3.8) (taking into account that a pointwiselimit of measurable functions is also measurable, see [22, Corollary 3.1.5]).Let us at first consider the case 𝑝 = 2 . Then the existence of a solution follows from theBrowder-Minty theorem, Korn’s inequality and the Poincaré inequality. In order to verify theinequality, let 𝜑 , 𝜑 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , 𝐿 , 𝐿 ∈ 𝐻 −1 (Ω; ℝ 𝑑 ) and 𝑢 , 𝑢 ∈ 𝐻 𝐷 (Ω; ℝ 𝑑 ) thecorresponding solutions. Then we obtain ⟨ 𝐿 − 𝐿 , 𝑢 − 𝑢 ⟩ = ( 𝑏 𝜑 ( ⋅ ) ( ⋅ , ∇ 𝑠 𝑢 ( ⋅ )) − 𝑏 𝜑 ( ⋅ ) ( ⋅ , ∇ 𝑠 𝑢 ( ⋅ )) , ∇ 𝑠 ( 𝑢 − 𝑢 ) ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ‖‖‖‖ 𝑢 − 𝑢 𝜆 ‖‖‖‖ 𝐿 (Ω; ℝ 𝑑 ) ≥ 𝑚 ‖ ∇ 𝑠 ( 𝑢 − 𝑢 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) − 𝐷 ‖ 𝜑 − 𝜑 ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) − 𝐷 ˆ Ω | ∇ 𝑠 ( 𝑢 − 𝑢 ) | | 𝜑 − 𝜑 | + 1 𝜆 ‖ 𝑢 − 𝑢 ‖ 𝐿 (Ω; ℝ 𝑑 ) , hence, the asserted inequality is fulfilled.For the general case let now 𝜑 ∈ 𝐿 ∞ (Ω; ℝ 𝑑 × 𝑑𝑠 ) and 𝐿 ∈ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) , we define 𝑏 ∶Ω × ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 by 𝑏 ( 𝑥, 𝜏 ) ∶= 𝑏 𝜑 ( 𝑥 ) ( 𝑥, 𝜏 ) and 𝐿 𝑢 ∈ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) by ⟨ 𝐿 𝑢 , 𝑣 ⟩ ∶= ⟨ 𝐿, 𝑣 ⟩ − 1 𝜆 ( 𝑢, 𝑣 ) 𝐿 (Ω; ℝ 𝑑 ) , where 𝑢 ∈ 𝐻 𝐷 (Ω; ℝ 𝑑 ) ↪ 𝐿 𝑞 (Ω; ℝ 𝑑 ) , with 𝑑 = − 𝑑𝑞 when 𝑑 > and 𝑞 = ∞ otherwise, isthe solution in the case 𝑝 = 2 and 𝑣 ∈ 𝑊 ,𝑝 ′ (Ω; ℝ 𝑑 ) ↪ 𝐿 𝑞 ′ (Ω; ℝ 𝑑 ) (note that 𝑑𝑝 ′ + 𝑑𝑞 ′ = This manuscript is for review purposes only.
S. WALTHER 𝑑𝑝 − 𝑑𝑞 = 2 + 𝑑𝑝 − 𝑑 ≥ when 𝑑 > and 𝑑𝑝 ′ + 𝑑𝑞 ′ = 1 − 𝑑𝑝 ′ + 𝑑 ≥ ≥ otherwise).We can now apply [13, Theorem 1.1] (here we need 𝜑 ∈ 𝐿 ∞ (Ω; ℝ 𝑑 × 𝑑𝑠 ) to satisfy [13, (1.6a)],the other requirements in [13, Assumption 1.5] are obviously fulfilled due to (3.5)–(3.8)) toobtain 𝑢 ∈ 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 ) such that ( 𝑏 ( ⋅ , ∇ 𝑠 𝑢 ) , ∇ 𝑠 𝑣 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) = ⟨ 𝐿 𝑢 , 𝑣 ⟩ , that is, ( 𝑏 𝜑 ( ⋅ , ∇ 𝑠 𝑢 ) , ∇ 𝑠 𝑣 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + 1 𝜆 ( 𝑢, 𝑣 ) 𝐿 (Ω; ℝ 𝑑 ) = ⟨ 𝐿, 𝑣 ⟩ , holds for all 𝑣 ∈ 𝑊 ,𝑝 ′ 𝐷 (Ω; ℝ 𝑑 ) , we get in particular 𝑢 = 𝑢 ∈ 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 ) since 𝑢 is the uniquesolution of the equation above for all 𝑣 ∈ 𝐻 𝐷 (Ω; ℝ 𝑑 ) .To prove the asserted inequality let 𝜑 , 𝜑 ∈ 𝐿 ∞ (Ω; ℝ 𝑑 × 𝑑𝑠 ) , 𝐿 , 𝐿 ∈ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) and 𝑢 , 𝑢 ∈ 𝑊 ,𝑝 (Ω; ℝ 𝑑 ) the corresponding solutions and define 𝐿 𝑢 , 𝐿 𝑢 as before. Having acloser look at the proof of [13, Theorem 1.1], respectively [11, Theorem 1], one can see thatthere exists a constant 𝑐 > , depending only on 𝑝, 𝑚 and 𝑀 (thus not on 𝐿 , 𝐿 , 𝜑 , 𝜑 ), suchthat ‖ 𝑢 − 𝑢 ‖ 𝑊 ,𝑝 (Ω; ℝ 𝑑 ) ≤ 𝑐 ‖ 𝐴 ( 𝑢 ) − 𝐴 ( 𝑢 ) − 𝐿 𝑢 + 𝐿 𝑢 ‖ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) , where 𝐴 𝑖 ∶ 𝑊 ,𝑝 (Ω; ℝ 𝑑 ) → 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) is defined by ⟨ 𝐴 𝑖 ( 𝑣 ) , 𝑣 ⟩ ∶= ( 𝑏 𝜑 𝑖 ( ⋅ , ∇ 𝑠 𝑣 ) , ∇ 𝑠 𝑣 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) for all 𝑣 ∈ 𝑊 ,𝑝 (Ω; ℝ 𝑑 ) , 𝑣 ∈ 𝑊 ,𝑝 ′ (Ω; ℝ 𝑑 ) and for 𝑖 ∈ {1 , . We finally obtain ‖ 𝑢 − 𝑢 ‖ 𝑊 ,𝑝 (Ω; ℝ 𝑑 ) ≤ 𝑐 (‖ 𝐴 ( 𝑢 ) − 𝐴 ( 𝑢 ) ‖ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) + ‖ 𝐿 𝑢 − 𝐿 𝑢 ‖ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) ) ≤ 𝑐 ( 𝑀 ‖ 𝜑 − 𝜑 ‖ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ‖ 𝐿 − 𝐿 ‖ 𝑊 −1 ,𝑝𝐷 (Ω; ℝ 𝑑 ) + 𝐶𝜆 ‖ 𝑢 − 𝑢 ‖ 𝐻 (Ω; ℝ 𝑑 ) ) , where we have used again the embeddings 𝐻 (Ω; ℝ 𝑑 ) ↪ 𝐿 𝑞 (Ω; ℝ 𝑑 ) and 𝑊 ,𝑝 ′ (Ω; ℝ 𝑑 ) ↪ 𝐿 𝑞 ′ (Ω; ℝ 𝑑 ) . Taking into account that the assertion is already proven in the case 𝑝 = 2 , we seethat the desired inequality holds.One can now obtain the result for all 𝜑 , 𝜑 ∈ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) by an approximation (usingthe just proven inequality to see that the corresponding sequence 𝑢 𝑛 is a Cauchy sequence).The operator 𝑅 in the following proposition will later be the resolvent, or a smoothedversion of the resolvent, of 𝐴 and should not be confused with 𝑅 from Definition 3.5.P ROPOSITION 𝑅 ). Let 𝜆 > and 𝑝 ≥ as in Corollary prop:TR_lipschitz_inertia and ℎ = ( ℎ , ℎ , ℎ ) ∈ 𝑝 . Moreover, let 𝑅 ∶ ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 be Lipschitz continuous andmonotone. Then there exists a unique solution 𝑢 ∈ 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 ) of − div( 𝔻 ∇ 𝑠 𝑢 + 𝔼 𝑅 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑢 − ℎ ) + ℎ )) = ℎ 𝜆 + ℎ − 𝑢𝜆 . {eq:TR_def_inertia} (3.9) We denote the solution operator of this equation by 𝑅 ∶ 𝑝 → 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 ) , that is, 𝑅 ( ℎ ) = 𝑢 . Furthermore, 𝑅 is Lipschitz continuous. Note that the dependency of 𝑅 on 𝜆 and 𝑝 will always be clear from the context.This manuscript is for review purposes only. PTIMAL CONTROL OF PLASTICITY WITH INERTIA Proof.
For all 𝜎 ∈ ℝ 𝑑 × 𝑑𝑠 we define 𝑏 𝜎 ∶ Ω × ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 by 𝑏 𝜎 ( 𝑥, 𝜏 ) ∶= 𝔻 𝜏 + 𝔼 𝑅 ( 𝔼 ⊤ 𝜏 + 𝜎 ) , then the assertion follows from Corollary 3.9 (with 𝜑 ∶= − 𝔼 ⊤ ∇ 𝑠 ℎ + ℎ for a given ℎ ∈ 𝑝 ),let us only prove that (3.7) is fulfilled, the other requirements can be easily checked. To thisend let 𝜎, 𝜎, 𝜏, 𝜏 ∈ ℝ 𝑑 × 𝑑𝑠 , then ( 𝑏 𝜎 ( 𝑥, 𝜏 ) − 𝑏 𝜎 ( 𝑥, 𝜏 )) ∶ ( 𝜏 − 𝜏 ) ≥ 𝛾 𝔻 | 𝜏 − 𝜏 | + ( 𝑅 ( 𝔼 ⊤ 𝜏 + 𝜎 ) − 𝑅 ( 𝔼 ⊤ 𝜏 + 𝜎 ) ) ∶ ( 𝔼 ⊤ ( 𝜏 − 𝜏 ) + ( 𝜎 − 𝜎 ) ) − ( 𝑅 ( 𝔼 ⊤ 𝜏 + 𝜎 ) − 𝑅 ( 𝔼 ⊤ 𝜏 + 𝜎 ) ) ∶ ( 𝜎 − 𝜎 ) ≥ 𝛾 𝔻 | 𝜏 − 𝜏 | − 𝐿 𝑅 | 𝜎 − 𝜎 | − 𝐿 𝑅 ‖ 𝔼 ‖ | 𝜏 − 𝜏 | | 𝜎 − 𝜎 | holds, where 𝐿 𝑅 is the Lipschitz constant of 𝑅 .Note that 𝑅 𝜆 ∶ ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 fulfills the requirements in Proposition 3.10 since 𝑅 𝜆 ∶ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) → 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) is Lipschitz continuous and also monotone (cf. [4, Proposition55.1 (ii) and Proposition 55.2 (a)]) and due to (2.2) these properties carry over to 𝑅 𝜆 ∶ ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 .Let us also mention that 𝑅 in Proposition 3.10 does not have to be monotone, the inequal-ity ( 𝑅 ( 𝑎 ) − 𝑅 ( 𝑏 )) ∶ ( 𝑎 − 𝑏 ) ≥ − 𝜀 | 𝑎 − 𝑏 | for 𝑎, 𝑏 ∈ ℝ 𝑑 × 𝑑𝑠 with 𝜀 < 𝛾 𝔻 ∕ ‖ 𝔼 ⊤ ‖ would be sufficient.We can now prove the existence of the resolvent of , from which we can then derive themaximal monotonicity of in Proposition 3.13 below.P ROPOSITION ). For every 𝜆 > and ℎ = prop:concrete_form_of_yosida_inertia ( ℎ , ℎ , ℎ ) ∈ , the tuple ⎛⎜⎜⎝ 𝑢𝑣𝑞 ⎞⎟⎟⎠ = ⎛⎜⎜⎝ 𝑅 𝜆 ( ℎ ) 𝜆 ( 𝑅 𝜆 ( ℎ ) − ℎ ) 𝑅 𝜆 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑅 𝜆 ( ℎ ) − ℎ ) + ℎ ) ⎞⎟⎟⎠ is contained in 𝐷 ( ) and the unique solution of ( 𝑢, 𝑣, 𝑞 ) + 𝜆 ( 𝑢, 𝑣, 𝑞 ) ∋ ℎ .Proof. Using the definition of 𝑅 𝜆 we get − 𝜆 div( 𝔻 ∇ 𝑠 𝑢 + 𝔼 𝑞 ) = ℎ − 𝑣, which is the second row in ( 𝑢, 𝑣, 𝑞 ) + 𝜆 ( 𝑢, 𝑣, 𝑞 ) ∋ ℎ and we also get ( 𝑢, 𝑣, 𝑞 ) ∈ 𝐷 ( ) (notethat 𝑟𝑔 ( 𝑅 𝜆 ) ⊂ 𝐷 ( 𝐴 ) ). That the first and last row in ( 𝑢, 𝑣, 𝑞 ) + 𝜆 ( 𝑢, 𝑣, 𝑞 ) ∋ ℎ is also fulfilledfollows immediately from the definitions of 𝑢 , 𝑣 and 𝑞 .Furthermore, when ( 𝑢, 𝑣, 𝑞 ) is a solution of ( 𝑢, 𝑣, 𝑞 ) + 𝜆 ( 𝑢, 𝑣, 𝑞 ) ∋ ℎ , then one verifiesanalog that ( 𝑢, 𝑣, 𝑞 ) must have the claimed form, therefore the uniqueness follows from theuniqueness of a solution to (3.9).L EMMA ). The equation lem:AA_to_A_in_scalarproduct_inertia ( ( 𝑢 , 𝑣 , 𝑞 ) − ( 𝑢 , 𝑣 , 𝑞 ) , ( 𝑢 , 𝑣 , 𝑞 ) − ( 𝑢 , 𝑣 , 𝑞 ) ) = ( 𝐴 ( 𝑞 ) − 𝐴 ( 𝑞 ) , 𝑞 − 𝑞 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) holds for all ( 𝑢 , 𝑣 , 𝑞 ) , ( 𝑢 , 𝑣 , 𝑞 ) ∈ 𝐷 ( ) .This manuscript is for review purposes only. S. WALTHER
Proof.
Using the definition of and the scalar product in we obtain ( ( 𝑢 , 𝑣 , 𝑞 ) , ( 𝑢 , 𝑣 , 𝑞 ) − ( 𝑢 , 𝑣 , 𝑞 ) ) = − ( 𝔻 ∇ 𝑠 𝑣 , ∇ 𝑠 ( 𝑢 − 𝑢 ) ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) − ( div( 𝔻 ∇ 𝑠 𝑢 + 𝔼 𝑞 ) , 𝑣 − 𝑣 ) 𝐿 (Ω; ℝ 𝑑 ) + ( 𝐴 ( 𝑞 ) − 𝔼 ⊤ ∇ 𝑠 𝑣 , 𝑞 − 𝑞 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) = ( 𝔻 ∇ 𝑠 𝑣 , ∇ 𝑠 𝑢 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) − ( 𝔻 ∇ 𝑠 𝑢 , ∇ 𝑠 𝑣 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) − ( 𝔼 ⊤ ∇ 𝑠 𝑣 , 𝑞 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ( 𝔼 ⊤ ∇ 𝑠 𝑣 , 𝑞 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ( 𝐴 ( 𝑞 ) , 𝑞 − 𝑞 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , evaluating now ( ( 𝑢 , 𝑣 , 𝑞 ) , ( 𝑢 , 𝑣 , 𝑞 ) − ( 𝑢 , 𝑣 , 𝑞 ) ) and taking the difference yields theassertion.P ROPOSITION is maximal monotone). The operator ∶ → is maximal prop:AA_maximal_monotone_inertia monotone.Proof. The monotonicity of follows immediately from Lemma 3.12 and the monotonic-ity of 𝐴 .To prove that is maximal monotone, it is, according to [4, Proposition 55.1 (B)], suffi-cient that 𝑅 ( 𝐼 + ) = , that is, we have to show that for every ( ℎ , ℎ , ℎ ) ∈ there exists ( 𝑢, 𝑣, 𝑞 ) ∈ 𝐷 ( ) such that ( 𝑢, 𝑣, 𝑞 ) + ( 𝑢, 𝑣, 𝑞 ) ∋ ( ℎ , ℎ , ℎ ) . This follows from Proposi-tion 3.11 with 𝜆 = 1 .In what follows it is convenient to give the integration operator a name.D EFINITION
We define ∶ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) → def:integration_operator_inertia 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) by ( 𝑓 )( 𝑡 ) ∶= ´ 𝑡 𝑓 ( 𝑠 ) 𝑑𝑠 for all 𝑓 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) . Moreover, we ab-breviate 𝜌 ∶= ∕ 𝜌 . As usual, we denote the operators with different inverse images andranges with the same symbol, for instance ∶ 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) → 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) . T HEOREM
There exists a unique thm:existence_of_a_solution_to_the_state_equation_inertia solution ( 𝑢, 𝑣, 𝑞 ) ∈ 𝐻 ( ) of (3.4) . Moreover, the inequality ‖ ( . 𝑢, . 𝑣, . 𝑞 ) ‖ 𝐿 ( ) ≤ 𝐶 (1 + ‖ 𝑓 ‖ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) ) holds, where the constant 𝐶 does not depend on 𝑓 .Proof. The tuple ( 𝑢, 𝑣, 𝑞 ) is a solution of (3.4) if and only if 𝑤 ∶= ( 𝑢, 𝑣, 𝑞 ) solves . 𝑤 + ̃ ( 𝑤 ) ∋ 1 𝜌 𝑅𝑓 , 𝑤 (0) = ( 𝑢 , 𝑣 , 𝑞 ) , with ̃ ∶= 𝑄 . One easily verifies that ̃ is a maximal monotone operator with respect to 𝑄 −1 (that is, the space equipped with the scalar product ( 𝑄 −1 ⋅ , ⋅ ) 𝑄 −1 ), thus we can apply[4, Theorem 55.A] to obtain a solution 𝑤 ∈ 𝐻 ( ) . Moreover, as can be seen in the proof of[4, Theorem 55.A] we get √ 𝛾 𝑄 −1 ‖ . 𝑤 𝜆 ‖ 𝐶 ( ) ≤ ‖ . 𝑤 𝜆 ‖ 𝐶 ( 𝑄 −1 ) ≤ 𝐶 (1 + 1 𝜌 ‖ 𝑅𝑓 ‖ 𝐻 ( ) )) = 𝐶 (1 + 1 𝜌 ‖ 𝑓 ‖ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) ) for all 𝜆 > , where 𝑤 𝜆 is the solution of . 𝑤 𝜆 + ̃ 𝜆 ( 𝑤 𝜆 ) = 1 𝜌 𝑅𝑓 , 𝑤 𝜆 (0) = ( 𝑢 , 𝑣 , 𝑞 ) . Since 𝑤 𝜆 ⇀ 𝑤 in 𝐻 ( ) , we obtain the desired inequality. This manuscript is for review purposes only.
PTIMAL CONTROL OF PLASTICITY WITH INERTIA Remark is not a subdifferential). Let us show that the maximal monotone oper- rem:AA_is_not_a_subdifferential_inertia ator ∶ → is not a subdifferential, that is, there exists no proper, convex and lowersemicontinuous function Φ ∶ → (−∞ , ∞] such that ( 𝑢, 𝑣, 𝑞 ) = 𝜕 Φ( 𝑢, 𝑣, 𝑞 ) = {( 𝑐, 𝑑, 𝑒 ) ∈ ∶ Φ( ̂𝑢, ̂𝑣, ̂𝑞 ) ≥ Φ( 𝑢, 𝑣, 𝑞 )+ (( 𝑐, 𝑑, 𝑒 ) , ( ̂𝑢 − 𝑢, ̂𝑣 − 𝑣, ̂𝑞 − 𝑞 )) ∀( ̂𝑢, ̂𝑣, ̂𝑞 ) ∈ } holds for all ( 𝑢, 𝑣, 𝑞 ) ∈ . In fact, there exists even not any function Φ ∶ → (−∞ , ∞] suchthat the equation above holds, which can be seen as follows:Let us assume that such a Φ exists and recall that ( 𝑢 , 𝑣 , 𝑞 ) ∈ 𝐷 ( ) . Then, usingLemma 3.12 with ( 𝑢 , 𝑣 , 𝑞 ) = ( 𝑢 + 𝑢 , 𝑣, 𝑞 ) and ( 𝑢 , 𝑣 , 𝑞 ) = ( 𝑢 , , 𝑞 ) , Φ( 𝑢 , , 𝑞 ) ≥ Φ( 𝑢 + 𝑢 , 𝑣, 𝑞 ) − ( ( 𝑢 + 𝑢 , 𝑣, 𝑞 ) , ( 𝑢, 𝑣, ) = Φ( 𝑢 + 𝑢 , 𝑣, 𝑞 ) − ( ( 𝑢 , , 𝑞 ) , ( 𝑢, 𝑣, ) ≥ Φ( 𝑢 , , 𝑞 ) + ( ( 𝑢 , , 𝑞 ) , ( 𝑢, 𝑣, ) − ( ( 𝑢 , , 𝑞 ) , ( 𝑢, 𝑣, ) = Φ( 𝑢 , , 𝑞 ) holds for all ( 𝑢, 𝑣 ) such that ( 𝑢 + 𝑢 , 𝑣, 𝑞 ) ∈ 𝐷 ( ) , hence, ( ( 𝑢 , , 𝑞 ) , ( ̂𝑢 − 𝑢, ̂𝑣 − 𝑣, ) = Φ( ̂𝑢 + 𝑢 , ̂𝑣, 𝑞 ) − Φ( 𝑢 + 𝑢 , 𝑣, 𝑞 ) ≥ ( ( 𝑢 + 𝑢 , 𝑣, 𝑞 ) , ( ̂𝑢 − 𝑢, ̂𝑣 − 𝑣, ) which gives ≥ ( 𝔻 ∇ 𝑠 𝑢, ∇ 𝑠 ̂𝑣 ) − ( 𝔻 ∇ 𝑠 ̂𝑢, ∇ 𝑠 𝑣 ) for all ( ̂𝑢, ̂𝑣 ) , ( 𝑢, 𝑣 ) such that ( 𝑢 + 𝑢 , 𝑣, 𝑞 ) , ( ̂𝑢 + 𝑢 , ̂𝑣, 𝑞 ) ∈ 𝐷 ( ) . Choosing now an arbitrary 𝑢 ∈ 𝐶 ∞ 𝑐 (Ω; ℝ 𝑑 ) , 𝑢 ≠ , ̂𝑣 = 𝑢 and ̂𝑢 = 𝑣 = 0 , we obtain the desired contradiction.In light of Remark 3.16, the case of plasticity with inertia essentially differs from theEVI analyzed in [27] in two aspects. First, we have more regularity in time as explained afterLemma 3.6. Second, we lose a certain boundedness of the maximal monotone operator, whichwas assumed in [27, Sect. 2], and it is not a subdifferential.It is also to be noted that Remark 3.16 is independent of the operator 𝐴 . sec:inertia_se_racr As already pointed out earlier, (3.4)can be transformed into the EVI which was analyzed in [27], namely . 𝑝 ∈ ( 𝑅 𝜌 𝑓 − 𝑄𝑝 ) , 𝑝 (0) = 𝑝 , {eq:state_equation_as_EVI_inertiap} (3.10)where we set 𝑝 ∶= − 𝑄 −1 ( 𝑢 , 𝑣 , 𝑞 ) . We can observe that when ( 𝑢, 𝑣, 𝑞 ) is a solution of (3.4),then 𝑝 ∶= 𝑅 𝑓 − 𝑄 −1 ( 𝑢, 𝑣, 𝑞 ) is a solution of (3.10) and when 𝑝 is a solution of (3.10), then ( 𝑢, 𝑣, 𝑞 ) ∶= 𝑅 𝜌 𝑓 − 𝑄𝑝 is a solution of (3.4).Thanks to this transformation we can use several results from [27] (namely Lemma 3.7,Lemma 3.8 and Proposition 3.5), with which the derivation of convergence results will be aneasy task. Let us emphasize that the maximal monotone operator therein was assumed to havea closed domain and that 𝐴 ∶ 𝐷 ( 𝐴 ) → , ℎ ↦ arg min 𝑣 ∈ 𝐴 ( ℎ ) ‖ 𝑣 ‖ is bounded on bounded sets. Clearly, both assumptions are not fulfilled for , however, theywere only needed to prove [27, Theorem 3.3], which can be replaced by Theorem 3.15, cf.also [27, Remark 3.13]. Therefore we can still apply the above mentioned results. This manuscript is for review purposes only. S. WALTHER T HEOREM
Let 𝑓 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) , { 𝑓 𝑛 } 𝑛 ∈ ℕ ⊂ thm:weak_convergence_of_the_state_inertia 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) such that 𝑓 𝑛 ⇀ 𝑓 in 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) and 𝑓 𝑛 → 𝐹 in 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) .Moreover, let ( 𝑢, 𝑣, 𝑞 ) ∈ 𝐻 ( ) be the solution of (3.4) and ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) ∈ 𝐻 ( ) , for every 𝑛 ∈ ℕ , either the solution of 𝑄 −1 ( . 𝑢 𝑛 , . 𝑣 𝑛 , . 𝑞 𝑛 ) + ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) ∋ 𝑅𝑓 𝑛 , ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 )(0) = ( 𝑢 , 𝑣 , 𝑞 ) or 𝑄 −1 ( . 𝑢 𝑛 , . 𝑣 𝑛 , . 𝑞 𝑛 ) + 𝜆 𝑛 ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) = 𝑅𝑓 𝑛 , ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 )(0) = ( 𝑢 , 𝑣 , 𝑞 ) , where { 𝜆 𝑛 } 𝑛 ∈ ℕ ⊂ (0 , ∞) , 𝜆 𝑛 ↘ .Then ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) ⇀ ( 𝑢, 𝑣, 𝑞 ) in 𝐻 ( ) and ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) → ( 𝑢, 𝑣, 𝑞 ) in 𝐶 ( 𝐻 (Ω; ℝ 𝑑 )) × 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) × 𝐶 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) . If additionally 𝑓 𝑛 → 𝑓 in 𝐶 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) , then 𝑣 𝑛 → 𝑣 in 𝐶 ( 𝐿 (Ω; ℝ 𝑑 )) .Proof. The function 𝑝 ∶= 𝑅 𝑓 − 𝑄 −1 ( 𝑢, 𝑣, 𝑞 ) ∈ 𝐻 is the unique solution of (3.10)and 𝑝 𝑛 ∶= 𝑅 𝑓 𝑛 − 𝑄 −1 ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) ∈ 𝐻 either the unique solution of . 𝑝 𝑛 ∈ ( 𝑅 𝜌 𝑓 𝑛 − 𝑄𝑝 𝑛 ) , 𝑝 𝑛 (0) = 𝑝 . or . 𝑝 𝑛 = 𝜆 𝑛 ( 𝑅 𝜌 𝑓 𝑛 − 𝑄𝑝 𝑛 ) , 𝑝 𝑛 (0) = 𝑝 . Thanks to Theorem 3.15, ( . 𝑢 𝑛 , . 𝑣 𝑛 , . 𝑞 𝑛 ) is bounded in 𝐿 ( ) . We can now apply [27, Lemma3.7] with 𝐴 𝑛 = and 𝐴 𝑛 = 𝜆 𝑛 (note that we can choose 𝐴 𝑛 = 𝜆 𝑛 according to [27,Lemma 3.8]) to obtain the desired result. Note that the convergence in [27, Lemma 3.7] thenmeans 𝑅 𝑓 𝑛 − 𝑄 −1 ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) → 𝑅 𝑓 − 𝑄 −1 ( 𝑢, 𝑣, 𝑞 ) in 𝐶 ( ) , so that the convergence ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) → ( 𝑢, 𝑣, 𝑞 ) in 𝐶 ( 𝐻 (Ω; ℝ 𝑑 )) × 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) × 𝐶 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) follows from thefact that the range of 𝑅 is a subset of {0} × 𝐿 (Ω; ℝ 𝑑 ) × {0} .P ROPOSITION
Let 𝑓 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) and prop:strong_convergence_for_fixed_forces_inertia ( 𝑢, 𝑣, 𝑞 ) ∈ 𝐻 ( ) be the solution of (3.4) and ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) ∈ 𝐻 ( ) , for every 𝑛 ∈ ℕ , thesolution of 𝑄 −1 ( . 𝑢 𝑛 , . 𝑣 𝑛 , . 𝑞 𝑛 ) + 𝜆 𝑛 ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) = 𝑅𝑓 , ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 )(0) = ( 𝑢 , 𝑣 , 𝑞 ) . where { 𝜆 𝑛 } 𝑛 ∈ ℕ ⊂ (0 , ∞) , 𝜆 𝑛 ↘ .Then ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 ) → ( 𝑢, 𝑣, 𝑞 ) in 𝐻 ( ) .Proof. We can argue as in the proof of Theorem 3.17, the assertion follows then directlyfrom [27, Proposition 3.5]. sec:inertia_oc
4. Optimal Control.
Also in this section we will make use of [27]. Since the smoothedoperator 𝑠 , given in Definition 4.7, possesses the required properties for 𝐴 𝑠 in [27, Assump-tion 5.1 (ii)] with = (as we will see in Definition 4.7 and Proposition 4.10 below), wecan apply the finding concerned with the differentiability of the solution operator associatedwith the EVI therein. Before we give the details in subsection 4.2, we tend to the existenceand approximation of optimal controls. This manuscript is for review purposes only.
PTIMAL CONTROL OF PLASTICITY WITH INERTIA sec:inertia_oc_eaaooc Let us now consider the op-timal control problem (1.1). Note that we assumed in section 2 that Ψ is defined on 𝐿 ( ) and not on 𝐻 ( ) , which excludes for example evaluations at certain points in time. Similaras in [27], we could also consider an objective function on 𝐻 ( ) , then we would only obtaina (possible) weak solution of the adjoint state in Theorem 4.14, see also [27, Theorem 5.12].We decided to define Ψ on 𝐿 ( ) only for simplicity and to keep the discussion concise.Since we have transformed our state equation (3.1) into (3.3) by introducing the new vari-able 𝑞 , it is reasonable to do the same with the optimal control problem. To this end, we needthe followingD EFINITION
We define def:transformed_objective_function Ψ 𝑧 ∶ 𝐿 ( ) → ℝ , ( 𝑢, 𝑣, 𝑞 ) ↦ Ψ( 𝑢, 𝑣, ℨ ( 𝑢, 𝑞 )) and the transformed objective function 𝐽 𝑧 ∶ 𝐿 ( ) × 𝔛 𝑐 → ℝ , ( 𝑢, 𝑣, 𝑞, 𝑓 ) ↦ Ψ 𝑧 ( 𝑢, 𝑣, 𝑞 ) + 𝛼 ‖ 𝑓 ‖ 𝔛 𝑐 Using the Definition above and the transformation of the state equation into (3.4), weobtain the equivalence of (1.1) and {eq:optimization_problem_inertia_q} (4.1) ⎧⎪⎪⎨⎪⎪⎩ min 𝐽 𝑧 ( 𝑢, 𝑣, 𝑞, 𝑓 ) = Ψ 𝑧 ( 𝑢, 𝑣, 𝑞 ) + 𝛼 ‖ 𝑓 ‖ 𝔛 𝑐 , s.t. 𝑄 −1 ( . 𝑢, . 𝑣, . 𝑞 ) + ( 𝑢, 𝑣, 𝑞 ) ∋ 𝑅𝑓 , ( 𝑢, 𝑣, 𝑞 )(0) = ( 𝑢 , 𝑣 , 𝑞 ) , ( 𝑢, 𝑣, 𝑞 ) ∈ 𝐻 ( 𝐻 𝐷 (Ω; ℝ 𝑑 ) × 𝐿 (Ω; ℝ 𝑑 ) × 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) ,𝑓 ∈ 𝔛 𝑐 . Let us now select a sequence { 𝜆 𝑛 } 𝑛 ∈ ℕ ⊂ (0 , ∞) such that 𝜆 𝑛 ↘ . We consider theregularized optimization problem {eq:optimization_problem_inertia_n} (4.2) ⎧⎪⎪⎨⎪⎪⎩ min 𝐽 𝑧 ( 𝑢, 𝑣, 𝑞, 𝑓 ) = Ψ 𝑧 ( 𝑢, 𝑣, 𝑞 ) + 𝛼 ‖ 𝑓 ‖ 𝔛 𝑐 , s.t. 𝑄 −1 ( . 𝑢, . 𝑣, . 𝑞 ) + 𝜆 𝑛 ( 𝑢, 𝑣, 𝑞 ) = 𝑅𝑓 , ( 𝑢, 𝑣, 𝑞 )(0) = ( 𝑢 , 𝑣 , 𝑞 )( 𝑢, 𝑣, 𝑞 ) ∈ 𝐻 ( 𝐻 𝐷 (Ω; ℝ 𝑑 ) × 𝐿 (Ω; ℝ 𝑑 ) × 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) ,𝑓 ∈ 𝔛 𝑐 . T HEOREM
Suppose that the con- thm:existence_and_approximation_of_optimal_solutions_inertia trol space 𝔛 𝑐 is such that ∶ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) → 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) with ( 𝑓 )( 𝑡 ) = ´ 𝑡 𝑓 ( 𝑠 ) 𝑑𝑠 is compact from 𝔛 𝑐 into 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) .Then there exists a global solution of (4.1) (and thus of (1.1) ) and of (4.2) for every 𝑛 ∈ ℕ .Moreover, let ( 𝑢 𝑛 , 𝑣 𝑛 , 𝑞 𝑛 , 𝑓 𝑛 ) 𝑛 ∈ ℕ be a sequence of global solution of (4.2) . Then there existsa weak accumulation point ( 𝑢, 𝑣, 𝑞, 𝑓 ) and every weak accumulation point is a global solu-tion of (4.1) . The subsequence of states which converges weakly towards ( 𝑢, 𝑣, 𝑞 ) in 𝐻 ( ) ,converges also strongly in 𝐶 ( 𝐻 (Ω; ℝ 𝑑 )) × 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) × 𝐶 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) and, when is compact from 𝔛 𝑐 into 𝐶 ( 𝐿 (Ω; ℝ 𝑑 )) , then the subsequence of 𝑣 𝑛 converges also strongly in 𝐶 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) . Moreover, the subsequence of controls converges strongly to 𝑓 in 𝔛 𝑐 .Proof. The existence of a global solution to (4.1) follows from the standard direct methodof the calculus of variations using Theorem 3.17 and the assumed compactness of , the proofis for instance analog to the proof of [27, Theorem 4.2]. The existence of a global solution This manuscript is for review purposes only. S. WALTHER to (4.2) follows easily using the Lipschitz continuity of 𝜆 𝑛 (which implies the Lipschitzcontinuity of the corresponding solution operator).The convergence result can also be obtained by standard arguments using again Theo-rem 3.17 and Proposition 3.18, the proof is again analog to [27, Theorem 4.5 & Corollary4.6]. Note that the strong convergence of the states in 𝐶 ( 𝐻 (Ω; ℝ 𝑑 )) × 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) × 𝐶 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) , and also of 𝑣 𝑛 in 𝐶 ( 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) when is compact from 𝔛 𝑐 into 𝐶 ( 𝐿 (Ω; ℝ 𝑑 )) , follows directly from Theorem 3.17.A strong convergence result of the states (in 𝐻 ( ) ) is not provided in the Theorem above.In [27, Corollary 4.6] we were able to prove the strong convergence either when the associatedmaximal monotone operator is a subdifferential, which is here not the case (Remark 3.16), orwhen it can be deduced from the weak convergence and the convergence of the evaluationsof Ψ . Since we supposed that Ψ is defined on 𝐿 ( ) , this cannot be the case. However,as elaborated on at the beginning of this subsection, it is possible for instance to consider adifferent Ψ defined on 𝐻 ( ) such that this property holds.Let us shortly interrupt the discussion and give two examples for the control space 𝔛 𝑐 .E XAMPLE
In order to satisfy the assumption on 𝔛 𝑐 in Theorem , ex:control_space_inertia we can use the lemma of Lions-Aubin (cf. [20, III. Proposition 1.3]) and for instance choose 𝔛 𝑐 = 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) ∩ 𝐿 ( 𝐻 (Ω; ℝ 𝑑 )) or 𝔛 𝑐 = { 𝑓 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) ∶ 𝑓 ∈ 𝐿 ( 𝐻 (Ω; ℝ 𝑑 ))} with corresponding norms. Having dealt with the existence and approximation of optimal solutions we turn to theoptimality condition for a further smoothed problem. sec:inertia_oc_oc
In order to derive first order optimality conditions wesmoothen at first the optimal control problem further. Then we prove the differentiability ofthe smoothed solution operator and can after that finally present our main result, the optimalityconditions for the smoothed optimization problem.We impose the following assumptions for the rest of this subsection.A
SSUMPTION assu:standing_assumptions_for_sec42assu:item:Rs_inertia (i) Let 𝑅 𝑠 ∶ ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 be monotone, Lipschitz continuous and Fréchet differen-tiable. assu:item:phat_and_p_inertia (ii) We fix < ̂𝑝 < 𝑝 < 𝑝 , where 𝑝 is from Corollary (respectively [13, Theorem1.1]), such that 𝑑 ≥ − 𝑑𝑝 . assu:item:initial_conditions_regularity_inertia (iii) Let the initial data ( 𝑢 , 𝑣 , 𝑞 ) be an element of 𝑝 , where 𝑝 is given in Item (ii) . Thanks to Proposition 3.11, we can give the precise form of the resolvent and Yosidaapproximation of in the followingC OROLLARY
Let 𝜆 > and denote the resolvent of cor:concrete_form_of_yosida_inertia by 𝜆 . Then 𝜆 ( ℎ ) = ⎛⎜⎜⎝ 𝑅 𝜆 ( ℎ ) 𝜆 𝑅 𝜆 ( ℎ ) − ℎ 𝜆 𝑅 𝜆 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑅 𝜆 ( ℎ ) − ℎ ) + ℎ ) ⎞⎟⎟⎠ This manuscript is for review purposes only.
PTIMAL CONTROL OF PLASTICITY WITH INERTIA so that 𝜆 ( ℎ ) = 1 𝜆 ⎛⎜⎜⎝ ℎ − 𝑅 𝜆 ( ℎ ) ℎ − 𝜆 𝑅 𝜆 ( ℎ ) + ℎ 𝜆 ℎ − 𝑅 𝜆 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑅 𝜆 ( ℎ ) − ℎ ) + ℎ ) ⎞⎟⎟⎠ for every ℎ = ( ℎ , ℎ , ℎ ) ∈ . The Yosida approximation 𝜆 is in view of Proposition 3.10 Lipschitz continuous from 𝑝 to 𝑝 , where 𝑝 is given in Assumption 4.4 Item (ii). Therefore the state equation in (4.2) admitsa solution in 𝑝 (note that 𝑅 maps into ∞ ). However, since this regularity is not present in(1.1), we did not use it. In contrast, the same is true for the smoothed Yosida approximation,which is given below in Definition 4.7 (see Definition 4.11), but here this additional regularitywill be used to prove the differentiability of the smoothed solution operator in Proposition 4.12.In order to smoothen the Yosida approximation, respectively the resolvent, of , wesmoothen the resolvent of 𝐴 and then define the smoothed resolvent for analog to 𝜆 .We denote this smoothed resolvent of 𝐴 by 𝑅 𝑠 ∶ ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 (which indicates that the re-solvent of 𝐴 can be expressed pointwise), from the properties given in Assumption 4.4 Item (i)one can easily derive the following inequalities, which will be useful when proving the differ-entiability of 𝑅 𝑠 in Lemma 4.9 below.L EMMA 𝑅 ′ 𝑠 ). There exists a constant 𝐶 such that | 𝑅 ′ 𝑠 ( 𝜎 ) 𝜏 | ≤ 𝐶 | 𝜏 | and lem:Rs_derivative_estimate_inertia ≤ 𝑅 ′ 𝑠 ( 𝜎 ) 𝜏 ∶ 𝜏 holds for all 𝜎, 𝜏 ∈ ℝ 𝑑 × 𝑑𝑠 . Moreover, the same is true for 𝑅 ′ 𝑠 ( ⋅ ) ∗ .Proof. Let 𝜎, 𝜏 ∈ ℝ 𝑑 × 𝑑𝑠 be arbitrary. The Lipschitz continuity and Fréchet differentiabilityof 𝑅 𝑠 gives ||| 𝑟 ( 𝑡𝜏 ) 𝑡 + 𝑅 ′ 𝑠 ( 𝜎 ) 𝜏 ||| = | 𝑅 𝑠 ( 𝜎 + 𝑡𝜏 ) − 𝑅 𝑠 ( 𝜎 ) | 𝑡 ≤ 𝐿 | 𝜏 | for all 𝑡 ∈ ℝ ⧵ {0} , where 𝑟 is the remainder term of 𝑅 𝑠 . The limit 𝑡 → yields the firstassertion.The second claim follows using the monotonicity, ≤ 𝑅 𝑠 ( 𝜎 + 𝑡𝜏 ) − 𝑅 𝑠 ( 𝜎 ) 𝑡 ∶ 𝜏 → 𝑅 ′ 𝑠 ( 𝜎 ) 𝜏 ∶ 𝜏 as ≠ 𝑡 → .Now, by definition we have 𝑅 ′ 𝑠 ( 𝜎 ) 𝜏 ∶ 𝜂 = 𝜏 ∶ 𝑅 ′ 𝑠 ( 𝜎 ) ∗ 𝜂 for all 𝜎, 𝜏, 𝜂 ∈ ℝ 𝑑 × 𝑑𝑠 , so that thesecond assertion also holds for 𝑅 ′ 𝑠 ( ⋅ ) ∗ . Choosing in particular 𝜏 = 𝑅 𝑠 ( 𝜎 ) ∗ 𝜂 we get | 𝑅 ′ 𝑠 ( 𝜎 ) ∗ 𝜂 | = | 𝑅 ′ 𝑠 ( 𝜎 ) 𝑅 ′ 𝑠 ( 𝜎 ) ∗ 𝜂 ∶ 𝜂 | ≤ 𝐶 | 𝑅 ′ 𝑠 ( 𝜎 ) ∗ 𝜂 | | 𝜂 | , which yields the first assertion for 𝑅 ′ 𝑠 ( ⋅ ) ∗ .D EFINITION
Let 𝜆 𝑠 ∈ (0 , ∞) . We define def:AAs_inertia 𝑠 ∶ 𝑝 → 𝑝 , ℎ = ( ℎ , ℎ , ℎ ) ↦ ⎛⎜⎜⎜⎝ 𝑅 𝑠 ( ℎ ) 𝜆 𝑠 𝑅 𝑠 ( ℎ ) − ℎ 𝜆 𝑠 𝑅 𝑠 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑅 𝑠 ( ℎ ) − ℎ ) + ℎ ) ⎞⎟⎟⎟⎠ and 𝑠 ∶= 𝜆 𝑠 ( 𝐼 − 𝑠 ) (see Assumption Item (ii) for 𝑝 ). According to Proposition and Assumption Item (i) , 𝑠 and 𝑠 are well defined and Lipschitz continuous. As usual,with a slight abuse of notation, we denote operators for different 𝑝 with the same symbol.This manuscript is for review purposes only. S. WALTHER
Let us now consider the smoothed optimization problem {eq:optimization_problem_inertia_s} (4.3) ⎧⎪⎪⎨⎪⎪⎩ min 𝐽 𝑧 ( 𝑢, 𝑣, 𝑞, 𝑓 ) = Ψ 𝑧 ( 𝑢, 𝑣, 𝑞 ) + 𝛼 ‖ 𝑓 ‖ 𝔛 𝑐 , s.t. 𝑄 −1 ( . 𝑢, . 𝑣, . 𝑞 ) + 𝑠 ( 𝑢, 𝑣, 𝑞 ) = 𝑅𝑓 , ( 𝑢, 𝑣, 𝑞 )(0) = ( 𝑢 , 𝑣 , 𝑞 )( 𝑢, 𝑣, 𝑞 ) ∈ 𝐻 ( 𝐻 𝐷 (Ω; ℝ 𝑑 ) × 𝐿 (Ω; ℝ 𝑑 ) × 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 )) ,𝑓 ∈ 𝔛 𝑐 . Analog to Theorem 4.2 one can analogously prove that there exists a global solution of(4.3).As was done in [27, Theorem 4.5 & Corollary 4.6], when 𝑠 and 𝜆 𝑠 are globally “closetogether”, one can prove a result analog to the convergence result in Theorem 4.2 with a se-quence ( 𝑢 𝑠 , 𝑣 𝑠 , 𝑞 𝑠 , 𝑓 𝑠 ) 𝑠> of global solutions to (4.3) when sup ℎ ∈ ‖ 𝜆 𝑠 ( ℎ ) − 𝑠 ( ℎ ) ‖ tendsfast enough to zero relative to 𝜆 𝑠 . The following lemma shows that this is the case when thesame is true for 𝜆 𝑠 sup 𝜏 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ‖ 𝐴 𝜆 𝑠 ( 𝜏 ) − 𝐴 𝑠 ( 𝜏 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) with 𝐴 𝑠 = 𝜆 𝑠 ( 𝐼 − 𝑅 𝑠 ) , whichholds in the case of the von-Mises flow rule investigated in section 5 below for suitable se-quences { 𝜆 𝑠 } 𝜆 𝑠 > and { 𝑠 } 𝑠> , cf. (5.2). Note also that [27, Lemma 3.15] was used in [27, The-orem 4.5 & Corollary 4.6], so that it was in particular required that sup 𝜏 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ‖ 𝐴 𝜆 𝑠 ( 𝜏 ) − 𝐴 𝑠 ( 𝜏 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) tends faster to zero than exp ( 𝜆 𝑠 ) , thus the additional factor 𝜆 𝑠 does not playa big role.L EMMA
The inequality lem:estimate_for_convergence_of_the_regularized_yosida_approximation_inertia ‖ 𝜆 𝑠 ( ℎ ) − 𝑠 ( ℎ ) ‖ ≤ 𝐶 √ 𝜆 𝑠 sup 𝜏 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ‖ 𝐴 𝜆 𝑠 ( 𝜏 ) − 𝐴 𝑠 ( 𝜏 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) holds for all ℎ ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , where 𝐴 𝑠 ∶= 𝜆 𝑠 ( 𝐼 − 𝑅 𝑠 ) and the constant does only dependon ℂ and 𝔹 , 𝐶 = 𝐶 ( ℂ , 𝔹 ) .Proof. Let us abbreviate 𝑀 ∶= sup 𝜏 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ‖ 𝑅 𝜆 𝑠 ( 𝜏 ) − 𝑅 𝑠 ( 𝜏 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) . Due to the definitions of 𝑠 and 𝐴 𝑠 we only have to prove that ‖ 𝜆 𝑠 ( ℎ ) − 𝑠 ( ℎ ) ‖ ≤ 𝐶 √ 𝜆 𝑠 𝑀 {eq:local028102019} (4.4)holds for all ℎ ∈ . To this end let ℎ ∈ be arbitrary and abbreviate 𝑢 ∶= 𝑅 𝜆𝑠 ( ℎ ) , 𝑢 𝑠 ∶= 𝑅 𝑠 ( ℎ ) ∈ 𝐻 𝐷 (Ω; ℝ 𝑑 ) , hence, 𝑢 is the solution of (3.9) with respect to 𝑅 𝜆 𝑠 and 𝑢 𝑠 with respect This manuscript is for review purposes only.
PTIMAL CONTROL OF PLASTICITY WITH INERTIA 𝑅 𝑠 , testing both equations with 𝑢 − 𝑢 𝑠 and subtracting the second from the first, we get ‖ ∇ 𝑠 ( 𝑢 − 𝑢 𝑠 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) 𝔻 + ‖‖‖‖ 𝑢 − 𝑢 𝑠 𝜆 𝑠 ‖‖‖‖ 𝐿 (Ω; ℝ 𝑑 ) = − ( 𝔼 ( 𝑅 𝜆 𝑠 ( 𝑤 ) − 𝑅 𝑠 ( 𝑤 𝑠 )) , ∇ 𝑠 ( 𝑢 − 𝑢 𝑠 ) ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) = − ( ( 𝑅 𝜆 𝑠 ( 𝑤 𝑠 ) − 𝑅 𝑠 ( 𝑤 𝑠 )) , 𝑤 − 𝑤 𝑠 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) − ( ( 𝑅 𝜆 𝑠 ( 𝑤 ) − 𝑅 𝜆 𝑠 ( 𝑤 𝑠 )) , 𝑤 − 𝑤 𝑠 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ≤ − ( 𝔻 −1 𝔼 ( 𝑅 𝜆 𝑠 ( 𝑤 𝑠 ) − 𝑅 𝑠 ( 𝑤 𝑠 )) , ∇ 𝑠 ( 𝑢 − 𝑢 𝑠 ) ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) 𝔻 ≤ ‖ 𝔻 −1 𝔼 ( 𝑅 𝜆 𝑠 ( 𝑤 ) − 𝑅 𝑠 ( 𝑤 𝑠 )) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) 𝔻 + 12 ‖ ∇ 𝑠 ( 𝑢 − 𝑢 𝑠 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) 𝔻 ≤ ‖ 𝔼 ⊤ 𝔻 −1 𝔼 ‖ 𝑀 + 12 ‖ ∇ 𝑠 ( 𝑢 − 𝑢 𝑠 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) 𝔻 with 𝑤 ∶= 𝔼 ⊤ ∇ 𝑠 ( 𝑢 − ℎ ) + ℎ ) and 𝑤 𝑠 ∶= 𝔼 ⊤ ∇ 𝑠 ( 𝑢 𝑠 − ℎ ) + ℎ ) , where we used in particularthe monotonicity of 𝑅 𝜆 𝑠 . Thus we obtain ‖ ∇ 𝑠 ( 𝑢 − 𝑢 𝑠 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) 𝔻 + ‖‖‖‖ 𝑢 − 𝑢 𝑠 𝜆 𝑠 ‖‖‖‖ 𝐿 (Ω; ℝ 𝑑 ) ≤ 𝐶𝑀 . {eq:local18112020} (4.5)We get further ‖ 𝑅 𝜆 𝑠 ( 𝑤 ) − 𝑅 𝑠 ( 𝑤 𝑠 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ≤ ‖ 𝑅 𝜆 𝑠 ( 𝑤 ) − 𝑅 𝜆 𝑠 ( 𝑤 𝑠 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ‖ 𝑅 𝜆 𝑠 ( 𝑤 𝑠 ) − 𝑅 𝑠 ( 𝑤 𝑠 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ≤ 𝐶𝜆 𝑠 𝑀 + 𝑀 where we have used (4.5). We arrive at ‖ 𝜆 𝑠 ( ℎ ) − 𝑠 ( ℎ ) ‖ ≤ 𝐶𝑀 + 𝐶𝜆 𝑠 𝑀 which implies (4.4).Let us now turn to optimality conditions. We first need to prove the Fréchet differentiabilityof the smoothed solution operator of the constraint in (4.3). To this end, we need two normgaps in Lemma 4.9 and Proposition 4.10, recall that the corresponding coefficients are fixedin Assumption 4.4 Item (ii).L EMMA 𝑅 𝑠 ). The operator 𝑅 𝑠 is from 𝑝 into lem:T_Frechet_inertia 𝑊 , ̂𝑝𝐷 (Ω; ℝ 𝑑 ) Fréchet differentiable and, for ℎ, 𝑔 ∈ 𝑝 , 𝜂 ∶= ′ 𝑅 𝑠 ( ℎ ) 𝑔 is of class 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 ) and the unique solution of − div( 𝔻 ∇ 𝑠 𝜂 + 𝔼 𝑅 ′ 𝑠 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑢 − ℎ ) + ℎ )( 𝔼 ⊤ ∇ 𝑠 ( 𝜂 − 𝑔 ) + 𝑔 ))) = 𝑔 𝜆 𝑠 + 𝑔 − 𝜂𝜆 𝑠 {eq:TDerivative} (4.6) for all 𝜑 ∈ 𝑊 ,𝑝 ′ 𝐷 (Ω; ℝ 𝑑 ) , where 𝑢 ∶= 𝑅 𝑠 ( ℎ ) .Moreover, there exists a constant 𝐶 such that the extension of ′ 𝑅 𝑠 ( ℎ ) to an element of 𝐿 ( ; 𝐻 𝐷 (Ω; ℝ 𝑑 )) fulfills ‖ ′ 𝑅 𝑠 ( ℎ ) 𝑔 ‖ 𝐻 𝐷 (Ω; ℝ 𝑑 ) ≤ 𝐶 ‖ 𝑔 ‖ for all ℎ ∈ 𝑝 and 𝑔 ∈ .This manuscript is for review purposes only. S. WALTHER
Proof.
Let ℎ, 𝑔 ∈ 𝑝 . At first we prove that (4.6) has a unique solution 𝜂 ∈ 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 ) with respect to ℎ and 𝑔 . For 𝜎 ∈ ℝ 𝑑 × 𝑑𝑠 we define 𝑏 𝜎 ∶ Ω × ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 by 𝑏 𝜎 ( 𝑥, 𝜏 ) ∶= 𝔻 𝜏 + 𝔼 𝑅 ′ 𝑠 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑢 ( 𝑥 ) − ℎ ( 𝑥 )) + ℎ ( 𝑥 )))( 𝔼 ⊤ 𝜏 + 𝜎 ) for almost all 𝑥 ∈ Ω and all 𝜏 ∈ ℝ 𝑑 × 𝑑𝑠 . The existence of 𝜂 follows now from Corollary 3.9(with 𝜑 ∶= − 𝔼 ⊤ ∇ 𝑠 𝑔 + 𝑔 ), when we have verified the requirements on 𝑏 𝜎 therein. Moreover,Corollary 3.9 also shows that the solution operator of (4.6) is continuous with respect to 𝑔 ∈ 𝑝 (clearly, it is also linear).Clearly, 𝑏 ( 𝑥,
0) = 0 ∈ 𝐿 ∞ (Ω; ℝ 𝑑 × 𝑑𝑠 ) and 𝑏 𝜎 ( ⋅ , 𝜏 ) is measurable as a pointwise limit ofmeasurable functions (see [22, Corollary 3.1.5]), for all 𝜏, 𝜎 ∈ ℝ 𝑑 × 𝑑𝑠 . Moreover, we have ( 𝑏 𝜎 ( 𝑥, 𝜏 ) − 𝑏 𝜎 ( 𝑥, 𝜏 )) ∶ ( 𝜏 − 𝜏 ) ≥ 𝛾 𝔻 | 𝜏 − 𝜏 | + 𝑅 ′ 𝑠 ( 𝑤 ( 𝑥 ))( 𝔼 ⊤ ( 𝜏 − 𝜏 ) + ( 𝜎 − 𝜎 )) ∶ 𝔼 ⊤ ( 𝜏 − 𝜏 ) ≥ 𝛾 𝔻 | 𝜏 − 𝜏 | − 𝐶 | 𝜎 − 𝜎 | | 𝜏 − 𝜏 | , with 𝑤 ∶= 𝔼 ⊤ ∇ 𝑠 ( 𝑢 − ℎ ) + ℎ ) , and | 𝑏 𝜎 ( 𝑥, 𝜏 ) − 𝑏 𝜎 ( 𝑥, 𝜏 ) | ≤ 𝐶 (| 𝜏 − 𝜏 | + | 𝜎 − 𝜎 |) for all 𝜎, 𝜎, 𝜏, 𝜏 ∈ ℝ 𝑑 × 𝑑𝑠 and almost all 𝑥 ∈ Ω , where we have used Lemma 4.6 in bothestimations. Therefore (3.5)–(3.8) are fulfilled.Considering now the equations for 𝑢 𝑔 ∶= 𝑅 𝑠 ( ℎ + 𝑔 ) and 𝑢 ∶= 𝑅 𝑠 ( ℎ ) , we see that − div( 𝔻 ∇ 𝑠 ( 𝑢 𝑔 − 𝑢 − 𝜂 )) + 𝑢 𝑔 − 𝑢 − 𝜂𝜆 𝑠 = div( 𝔼 ( 𝑅 𝑠 ( 𝜇 + 𝜈 𝑔 ) − 𝑅 𝑠 ( 𝜇 ) − 𝑅 ′ 𝑠 ( 𝜇 ) 𝜈 𝑔 ))+ div( 𝔼 𝑅 ′ 𝑠 ( 𝜇 )(( 𝔼 ⊤ ∇ 𝑠 ( 𝑢 𝑔 − 𝑢 − 𝜂 ))) , where 𝜇 ∶= 𝔼 ⊤ ∇ 𝑠 ( 𝑢 − ℎ ) + ℎ ) ,𝜈 𝑔 ∶= 𝔼 ⊤ ∇ 𝑠 ( 𝑢 𝑔 − 𝑢 − 𝑔 ) + 𝑔 ) ∈ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , hence, − div( 𝔻 ∇ 𝑠 ( 𝑢 𝑔 − 𝑢 − 𝜂 ) − 𝔼 𝑅 ′ 𝑠 ( 𝜇 )(( 𝔼 ⊤ ∇ 𝑠 ( 𝑢 𝑔 − 𝑢 − 𝜂 ))) + 𝑢 𝑔 − 𝑢 − 𝜂𝜆 𝑠 = div 𝔼 𝑟 𝜇 ( 𝜈 𝑔 ) , where 𝑟 𝜇 ( 𝜈 𝑔 ) is the remainder term of 𝑅 𝑠 at 𝜇 in direction 𝜈 𝑔 . Applying Corollary 3.9 onceagain with 𝑏 𝜎 ( 𝑥, 𝜏 ) ∶= 𝔻 𝜏 + 𝔼 𝑅 ′ 𝑠 ( 𝜇 ( 𝑥 )) 𝔼 ⊤ 𝜏 (and 𝑝 = ̂𝑝 ) we obtain ‖ 𝑢 𝑔 − 𝑢 − 𝜂 ‖ 𝑊 ,̂𝑝 (Ω; ℝ 𝑑 ) ‖ 𝑔 ‖ 𝑝 ≤ 𝐶 ‖ 𝑟 𝜇 ( 𝜈 𝑔 ) ‖ 𝐿 ̂𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ‖ 𝑔 ‖ 𝑝 ≤ 𝐶 ‖ 𝑟 𝜇 ( 𝜈 𝑔 ) ‖ 𝐿 ̂𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ‖ 𝜈 𝑔 ‖ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) → , as 𝑔 → in 𝑝 , where we also used the Lipschitz continuity of 𝑅 𝑠 and the fact that 𝑅 𝑠 ∶ 𝐿 𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) → 𝐿 ̂𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) is Fréchet differentiable (cf. [9, Theorem 7]).That the extension of ′ 𝑅 𝑠 ( ℎ ) to an element of 𝐿 ( ; 𝐻 𝐷 (Ω; ℝ 𝑑 )) fulfills the asserted in-equality, can be proven as above (one can simply test (4.6) with 𝜂 ∈ 𝐻 𝐷 (Ω; ℝ 𝑑 ) and useLemma 4.6). This manuscript is for review purposes only.
PTIMAL CONTROL OF PLASTICITY WITH INERTIA
ROPOSITION 𝑠 ). The mapping 𝑠 is from 𝑝 to prop:AAsFrechet Fréchet differentiable and there exists a constant 𝐶 such that the extension of ′ 𝑠 ( ℎ ) ∈ ( 𝑝 , ) to an element of ( ) fulfills ‖ ′ 𝑠 ( ℎ ) 𝑔 ‖ ≤ 𝐶 ‖ 𝑔 ‖ for all ℎ ∈ 𝑝 and 𝑔 ∈ .For ℎ ∈ 𝑝 and 𝑔 ∈ we have ′ 𝑠 ( ℎ ) 𝑔 = ⎛⎜⎜⎜⎝ ′ 𝑅 𝑠 ( ℎ ) 𝑔 𝜆 𝑠 ′ 𝑅 𝑠 ( ℎ ) 𝑔 − 𝑔 𝜆 𝑠 𝑅 ′ 𝑠 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑅 𝑠 ( ℎ ) − ℎ ) + ℎ )( 𝔼 ⊤ ∇ 𝑠 ( ′ 𝑅 𝑠 ( ℎ ) 𝑔 − 𝑔 ) + 𝑔 ) ⎞⎟⎟⎟⎠ The same is true for 𝑠 = 𝜆 𝑠 ( 𝐼 − 𝑠 ) with ′ 𝑠 ( ℎ ) 𝑔 = 𝜆 𝑠 ( 𝑔 − ′ 𝑠 ( ℎ ) 𝑔 ) for all ℎ ∈ 𝑝 and 𝑔 ∈ .Proof. The assertion follows from Lemma 4.9, Lemma 4.6 for the estimate of ( ′ 𝑠 ( ℎ ) 𝑔 ) ,the fact that 𝑅 𝑠 ∶ 𝐿 ̂𝑝 (Ω; ℝ 𝑑 × 𝑑𝑠 ) → 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) is Fréchet differentiable (cf. [9, Theorem 7])and the chain rule.Now, we can use [27, Theorem 5.5] to derive the differentiability of the solution operatorof the constraint in (4.3) from the differentiability of 𝑠 . To this end, we first introduce thesolution operator inD EFINITION
We denote the solution operator of def:smoothed_solution_operator_inertia 𝑄 −1 ( . 𝑢, . 𝑣, . 𝑞 ) + 𝑠 ( 𝑢, 𝑣, 𝑞 ) = 𝑅𝑓 , ( 𝑢, 𝑣, 𝑞 )(0) = ( 𝑢 , 𝑣 , 𝑞 ) {eq:smoothed_state_equation_inertia} (4.7) by 𝑠 ∶ 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) → 𝐻 ( 𝑝 ) , that is, 𝑠 ( 𝑓 ) = ( 𝑢, 𝑣, 𝑞 ) , which existence follows fromBanachs contraction principle since 𝑠 is Lipschitz continuous according to Definition .Here we use the improved regularity of ( 𝑢 , 𝑣 , 𝑞 ) , see Assumption Item (iii) . P ROPOSITION
The so- prop:frechet_differentiability_of_the_smoothed_solution_operator_inertia lution operator 𝑠 ∶ 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) → 𝐻 ( 𝑝 ) is Lipschitz continuous, 𝑠 ∶ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) → 𝐻 ( ) is Fréchet differentiable and, for 𝑓 , 𝑔 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) , 𝜂 ∶= ′ 𝑠 ( 𝑓 ) 𝑔 ∈ 𝐻 ( ) is the unique solution of 𝑄 −1 . 𝜂 + ′ 𝑠 ( 𝑤 ) 𝜂 = 𝑅𝑔, 𝜂 (0) = 0 , {eq:eta_equation_inertia} (4.8) where 𝑤 ∶= 𝑠 ( 𝑓 ) . Moreover, there exists a constant 𝐶 , such that ‖ ′ 𝑠 ( 𝑓 ) 𝑔 ‖ 𝐻 ( ) ≤ 𝐶 ‖ 𝑔 ‖ 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) holds for all 𝑓 , 𝑔 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) .Proof. Our goal is to use [27, Theorem 5.5], to this end we first consider the transformedequation from subsection 3.3. We again set 𝑝 ∶= − 𝑄 −1 ( 𝑢 , 𝑣 , 𝑞 ) and denote the solutionoperator of . 𝑝 = 𝑠 ( 𝑅𝐹 − 𝑄𝑝 ) , 𝑝 (0) = 𝑝 {eq:local009042020} (4.9)by ̃ 𝑠 ∶ 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) → 𝐻 ( 𝑝 ) , that is, ̃ 𝑠 ( 𝐹 ) = 𝑝 . Thus we have 𝑠 ( 𝑓 ) = 𝑅 𝜌 𝑓 − 𝑄 ̃ 𝑠 ( 𝜌 𝑓 ) for all 𝑓 ∈ 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) .We can now apply [27, Lemma 5.3 & Theorem 5.5] (with = 𝐿 (Ω; ℝ 𝑑 ) , = 𝑝 , = , 𝑧 = 𝑝 and 𝑧 = 𝑝 ), note that the assumptions in [27, Assumption 5.1 (ii)] are satisfiedthanks to Proposition 4.10. Thus the solution operator ̃ 𝑠 ∶ 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) → 𝐻 ( 𝑝 ) isLipschitz continuous and ̃ 𝑠 ∶ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) → 𝐻 ( ) is Fréchet differentiable, hence,the desired Lipschitz continuity and Fréchet differentiability also hold for 𝑠 . Furthermore, This manuscript is for review purposes only. S. WALTHER the asserted inequality holds and we have 𝜂 = 𝑅 𝜌 𝑔 − 𝑄 ̃𝜂 , where 𝜂 ∶= ′ 𝑠 ( 𝑓 ) 𝑔 and ̃𝜂 ∶= ̃ ′ 𝑠 ( 𝜌 𝑓 ) 𝜌 𝑔 . [27, Theorem 5.5] also shows that ̃𝜂 is the unique solution of 𝜕 𝑡 ̃𝜂 = ′ 𝑠 ( 𝑅 𝜌 𝑓 − 𝑄𝑝 )( 𝑅 𝜌 𝑔 − 𝑄 ̃𝜂 ) , ̃𝜂 (0) = 0 , where 𝑝 ∶= ̃ 𝑠 ( 𝜌 𝑓 ) . Taking into account that ̃𝜂 = 𝑅 𝑔 − 𝑄 −1 𝜂 and 𝜕 𝑡 𝑔 = 𝑔 , we see that 𝜂 is the solution of (4.8). Remark 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) is Fréchet differentiable. The norm gaps, which arisefrom the exponents in Assumption 4.4 Item (ii), are only needed for the differentiability of 𝑅 𝑠 but not in the control space. Unfortunately, we still require the compactness propertyimposed on 𝔛 𝑐 in Theorem 4.2 to use the convergence results in subsection 3.3. However, wecan avoid taking a subspace of 𝐻 ( 𝐿 ̃𝑝 (Ω; ℝ 𝑑 )) , for a certain ̃𝑝 > , as the control space.Let us now consider the following reduced optimization problem min 𝑓 ∈ 𝔛 𝑐 𝐹 𝑧 ( 𝑓 ) , {eq:optimization_problem_r_inertia} (4.10)where the reduced objective function 𝐹 𝑧 ∶ 𝔛 𝑐 → ℝ is defined by 𝐹 𝑧 ( 𝑓 ) ∶= 𝐽 𝑧 ( 𝑠 ( 𝑓 ) , 𝑓 ) .Clearly, (4.10) and (4.3) are equivalent.We can finally present our main result.T HEOREM
Let 𝑓 ∈ 𝔛 𝑐 and abbreviate thm:KKT_conditions_inertia ( 𝑢, 𝑣, 𝑞 ) ∶= 𝑠 ( 𝑓 ) ∈ 𝐻 ( 𝑝 ) and 𝑤 ∶= 𝑅 𝑠 ( 𝑢, 𝑣, 𝑞 ) ∈ 𝐻 ( 𝑊 ,𝑝𝐷 (Ω; ℝ 𝑑 × 𝑑𝑠 )) . Then the varia-tional equation 𝐹 ′ 𝑧 ( 𝑓 ) 𝑔 = Ψ ′ 𝑧 ( 𝑠 ( 𝑓 )) ′ 𝑠 ( 𝑓 ) 𝑔 + 𝛼 ( 𝑓 , 𝑔 ) 𝔛 𝑐 = 0 {eq:Fz_derivative_inertia} (4.11) holds for all 𝑔 ∈ 𝔛 𝑐 if and only if there exists an unique adjoint state ( 𝜑, 𝜂 ∗ ) =( 𝜑 , 𝜑 , 𝜑 , 𝜂 ∗ ) ∈ 𝐻 ( × 𝐻 𝐷 (Ω; ℝ 𝑑 )) such that the following optimality system is satis-fied: eq:KKT_conditions_inertia State equation: ⎛⎜⎜⎜⎝ . 𝑢 . 𝑣 . 𝑞 ⎞⎟⎟⎟⎠ = 1 𝜆 𝑠 ⎛⎜⎜⎝ 𝑤 − 𝑢 ( 𝑤 − 𝑢 ) ∕ ( 𝜌𝜆 𝑠 ) − 𝑣 ∕ 𝜌 ( ℂ + 𝔹 )( 𝑝 − 𝑞 ) ⎞⎟⎟⎠ + ⎛⎜⎜⎝ 𝑓 ∕ 𝜌 ⎞⎟⎟⎠ (4.12a) − div( 𝔻 ∇ 𝑠 𝑤 + 𝔼 𝑝 ) = 𝑣 ∕ 𝜆 𝑠 + ( 𝑤 − 𝑢 ) ∕ 𝜆 𝑠 (4.12b) 𝑝 = 𝑅 𝑠 ( 𝔼 ∇ 𝑠 ( 𝑤 − 𝑢 ) + 𝑞 ) (4.12c) ( 𝑢, 𝑣, 𝑞 )(0) = ( 𝑢 , 𝑣 , 𝑞 ) (4.12d) Adjoint equation : ⎛⎜⎜⎝ . 𝜑 . 𝜑 . 𝜑 ⎞⎟⎟⎠ = 1 𝜆 𝑠 ⎛⎜⎜⎝ 𝜂 ∗ − 𝜑 𝜂 ∗ − 𝜑 ) ∕ ( 𝜌𝜆 𝑠 ) − 𝜑 ∕ 𝜌 ( ℂ + 𝔹 )( 𝑟 ∗ − 𝜑 ) ⎞⎟⎟⎠ − 𝑄 Ψ ′ 𝑧 ( 𝑢, 𝑣, 𝑞 ) {eq:KKT_conditions_adjoint_evolution_inertia}{eq:KKT_conditions_adjoint_evolution_inertia} (4.12e) − div( 𝔻 ∇ 𝑠 𝜂 ∗ + 𝔼 𝑟 ∗ ) = 𝜑 ∕ 𝜆 𝑠 + ( 𝜂 ∗ − 𝜑 ) ∕ 𝜆 𝑠 (4.12f) 𝑟 ∗ = 𝑅 ′ 𝑠 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑤 − 𝑢 ) + 𝑞 ) ∗ ( 𝔼 ⊤ ∇ 𝑠 ( 𝜂 ∗ − 𝜑 ) + 𝜑 ) (4.12g) ( 𝜑 , 𝜑 , 𝜑 )( 𝑇 ) = 0 (4.12h) This manuscript is for review purposes only.
PTIMAL CONTROL OF PLASTICITY WITH INERTIA Gradient equation: ( 𝜑 , 𝑔 ) 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) = 𝛼 ( 𝑓 , 𝑔 ) 𝔛 𝑐 ∀ 𝑔 ∈ 𝔛 𝑐 . (4.12i) In particular, if 𝑓 is locally optimal for (4.10) , then there exists a unique adjoint state ( 𝜑, 𝜂 ∗ ) ∈ 𝐻 ( × 𝐻 𝐷 (Ω; ℝ 𝑑 )) such that (4.12) is fulfilled.Proof. At first we proof that the assertion holds when we exchange (4.12) with 𝑄 −1 ( . 𝑢, . 𝑣, . 𝑞 ) + 𝑠 ( 𝑢, 𝑣, 𝑞 ) = 𝑅𝑓 , ( 𝑢, 𝑣, 𝑞 )(0) = ( 𝑢 , 𝑣 , 𝑞 ) ,𝑄 −1 . 𝜑 + ′ 𝑠 ( 𝑢, 𝑣, 𝑞 ) ∗ 𝜑 = −Ψ ′ 𝑧 ( 𝑢, 𝑣, 𝑞 ) , 𝜑 ( 𝑇 ) = 0 , ( 𝜑 , 𝑔 ) 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) = 𝛼 ( 𝑓 , 𝑔 ) 𝔛 𝑐 ∀ 𝑔 ∈ 𝔛 𝑐 . {eq:abstract_KKT_conditions_inertia} (4.13)To this end, let 𝜑 be the solution of the second equation in (4.13) (which unique existencefollows as in [27, Lemma 5.11]) and 𝜂 ∶= ′ 𝑠 ( 𝑓 ) 𝑔 ∈ 𝐻 ( ) for an arbitrary 𝑔 ∈ 𝔛 𝑐 , then ( 𝜑 , 𝑔 ) 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) = ( 𝜑, 𝑅𝑔 ) 𝐿 ( ) = ( 𝜑, 𝑄 −1 . 𝜂 ) 𝐿 ( ) + ( 𝜑, ′ 𝑠 ( 𝑢, 𝑣, 𝑞 ) 𝜂 ) 𝐿 ( ) = ( 𝑄 −1 . 𝜑, 𝜂 ) 𝐿 ( ) + ( ′ 𝑠 ( 𝑢, 𝑣, 𝑞 ) ∗ 𝜑, 𝜂 ) 𝐿 ( ) = − ( Ψ ′ 𝑧 ( 𝑠 ( 𝑓 )) , 𝜂 ) 𝐿 ( ) holds for all 𝑔 ∈ 𝔛 𝑐 , which implies the equivalence between (4.11) and the last equation in(4.13). Moreover, it is well known that if 𝑓 is locally optimal for (4.3), then (4.11) must hold.Let us now prove the equivalence between (4.13) and (4.12). We choose ℎ, 𝜉 ∈ anddenote by 𝜂 ∗ ∈ 𝐻 𝐷 (Ω; ℝ 𝑑 × 𝑑𝑠 ) the solution of − div( 𝔻 ∇ 𝑠 𝜂 ∗ + 𝔼 𝑅 ′ 𝑠 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑅 𝑠 ( ℎ ) − ℎ ) + ℎ ) ∗ ( 𝔼 ⊤ ∇ 𝑠 ( 𝜂 ∗ − 𝜉 ) + 𝜉 ))) = 𝜉 𝜆 𝑠 + 𝜉 − 𝜂 ∗ 𝜆 𝑠 {eq:local1029102019} (4.14)for all 𝜙 ∈ 𝐻 𝐷 (Ω; ℝ 𝑑 ) (the existence of 𝜂 ∗ follows as in Lemma 4.9, note that the inequalitiesin Lemma 4.6 hold also for the adjoint operator). Then ′ 𝑠 ( ℎ ) ∗ 𝜉 = ⎛⎜⎜⎜⎝ 𝜂 ∗1 𝜆 𝑠 𝜂 ∗ − 𝜉 𝜆 𝑠 𝑅 ′ 𝑠 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑅 𝑠 ( ℎ ) − ℎ ) + ℎ ) ∗ ( 𝔼 ⊤ ∇ 𝑠 ( 𝜂 ∗ − 𝜉 ) + 𝜉 ) ⎞⎟⎟⎟⎠ holds, which can be seen as follows: Let 𝑔 ∈ and abbreviate 𝜂 ∶= ′ 𝑅 𝑠 ( ℎ ) 𝑔, 𝜂 𝑣 ∶= 𝜂 − 𝑔 𝜆 𝑠 , 𝜂 𝑞 ∶= 𝑅 ′ 𝑠 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑅 𝑠 ( ℎ ) − ℎ ) + ℎ )( 𝔼 ⊤ ∇ 𝑠 ( 𝜂 − 𝑔 ) + 𝑔 ) ,𝜂 ∗ 𝑣 ∶= 𝜂 ∗ − 𝜉 𝜆 𝑠 , 𝜂 ∗ 𝑞 ∶= 𝑅 ′ 𝑠 ( 𝔼 ⊤ ∇ 𝑠 ( 𝑅 𝑠 ( ℎ ) − ℎ ) + ℎ ) ∗ ( 𝔼 ⊤ ∇ 𝑠 ( 𝜂 ∗ − 𝜉 ) + 𝜉 ) . Testing (4.6) with 𝜙 = 𝜉 − 𝜂 ∗ gives ( 𝔻 ∇ 𝑠 𝜂, ∇ 𝑠 ( 𝜉 − 𝜂 ∗ ) ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ( 𝜂 𝑣 − 𝑔 , 𝜂 ∗ 𝑣 ) 𝐿 (Ω; ℝ 𝑑 ) = ( 𝔼 𝜂 𝑞 , ∇ 𝑠 ( 𝜂 ∗ − 𝜉 ) ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) = ( 𝔼 ⊤ ∇ 𝑠 ( 𝜂 − 𝑔 ) + 𝑔 , 𝜂 ∗ 𝑞 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) − ( 𝜂 𝑞 , 𝜉 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , This manuscript is for review purposes only. S. WALTHER and testing (4.14) with 𝜙 = 𝜂 − 𝑔 yields ( 𝔻 ∇ 𝑠 𝜂 ∗ , ∇ 𝑠 ( 𝜂 − 𝑔 ) ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ( 𝜉 − 𝜂 ∗ 𝑣 , 𝜂 𝑣 ) 𝐿 (Ω; ℝ 𝑑 ) = ( 𝔼 𝜂 ∗ 𝑞 , ∇ 𝑠 ( 𝑔 − 𝜂 ) ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , thus, adding both equations together, we arrive at ( 𝔻 ∇ 𝑠 𝜂, ∇ 𝑠 𝜉 ) 𝐻 (Ω; ℝ 𝑑 ) + ( 𝜂 𝑣 , 𝜉 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ( 𝜂 𝑞 , 𝜉 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) = ( 𝔻 ∇ 𝑠 𝜂 ∗ , ∇ 𝑠 𝑔 ) 𝐻 (Ω; ℝ 𝑑 ) + ( 𝜂 ∗ 𝑣 , 𝑔 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) + ( 𝜂 ∗ 𝑞 , 𝑔 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) , which is equivalent to ( ′ 𝑠 ( ℎ ) 𝑔, 𝜉 ) = ( 𝑔, ′ 𝑠 ( ℎ ) ∗ 𝜉 ) . Now one only has to use the definitions of 𝑠 and 𝑅 to obtain the equivalence between(4.12) and (4.13). sec:examples
5. Examples.
Let us conclude with examples about a concrete objective function, thegradient equation in Theorem 4.14 regarding a concrete control space and finally a realizationof the maximal monotone operator 𝐴 (which will be the von-Mises flow rule). Objective Function.
Let us consider a tracking type objective function, that is, Ψ( 𝑢, 𝑣, 𝑧 ) = 12 ‖ ( 𝑢, 𝑣, 𝑧 ) − ( 𝑢 𝑑 , 𝑣 𝑑 , 𝑧 𝑑 ) ‖ 𝐿 ( ) with a desired state ( 𝑢 𝑑 , 𝑣 𝑑 , 𝑧 𝑑 ) ∈ 𝐿 ( ) . Then Ψ 𝑧 ( 𝑢, 𝑣, 𝑞 ) = 12 ‖ ( 𝑢, 𝑣, ( ℂ + 𝔹 ) −1 ( ℂ ∇ 𝑠 𝑢 − 𝑞 )) − ( 𝑢 𝑑 , 𝑣 𝑑 , 𝑧 𝑑 ) ‖ 𝐿 ( ) and Ψ ′ 𝑧 ( 𝑢, 𝑣, 𝑞 ) = ⎛⎜⎜⎝ ̂𝑢𝑣 − 𝑣 𝑑 ( ℂ + 𝔹 ) −1 ( ℂ ∇ 𝑠 𝑢 − 𝑞 )) − 𝑧 𝑑 , ⎞⎟⎟⎠ where ̂𝑢 is such that − div( 𝔻 ∇ 𝑠 ( ̂𝑢 − 𝑢 + 𝑢 𝑑 ) − (( ℂ + 𝔹 ) −1 ( ℂ ∇ 𝑠 𝑢 − 𝑞 ) − 𝑧 𝑑 )) = 0 , hence, inthis example the adjoint equation in (4.12) has to be completed by this equation. Note thatwhen one uses a finite element approach to solve (4.12) numerically, then one can eliminatethis additional equation after multiplying (4.12e) with a test function, that is, taking the -scalar product. When the -scalar product of 𝑄 Ψ ′ 𝑧 ( 𝑢, 𝑣, 𝑞 ) and a test function ( 𝜂 , 𝜂 , 𝜂 ) isevaluated, the term ( 𝔻 ∇ 𝑠 ̂𝑢, 𝜂 ) 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) arises, then one can use the additional equation toeliminate ̂𝑢 (respectively the equation). Control space.
Let us consider the space 𝔛 𝑐 ∶= { 𝑓 ∈ 𝐻 ( 𝐿 (Ω; ℝ 𝑑 )) ∩ 𝐿 ( 𝐻 (Ω; ℝ 𝑑 )) ∶ 𝑓 (0) = 𝑓 ( 𝑇 ) = 0} with the scalar product ( 𝑓 , 𝑔 ) 𝔛 𝑐 = ( . 𝑓 , . 𝑔 ) 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) + (∇ 𝑓 , ∇ 𝑔 ) 𝐿 𝐿 (Ω; ℝ 𝑑 × 𝑑 ) , see Example 4.3. The Gradient equation in (4.12) then becomes 𝛼 ( . 𝑓 , . 𝑔 ) 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) + 𝛼 (∇ 𝑓 , ∇ 𝑔 ) 𝐿 𝐿 (Ω; ℝ 𝑑 × 𝑑 ) = ( 𝜑 , 𝑔 ) 𝐿 ( 𝐿 (Ω; ℝ 𝑑 )) This manuscript is for review purposes only.
PTIMAL CONTROL OF PLASTICITY WITH INERTIA 𝑔 ∈ 𝔛 𝑐 , which is the weak formulation of .. 𝑓 + Δ 𝑓 = − 𝜑 𝛼 . Maximal monotone operator 𝐴 . We consider the case of linear kinematic hardeningwith the von Mises yield condition, cf. [12] for a detailed description of this model. In thiscase, 𝐴 is the convex subdifferential of the indicator functional 𝐼 (Ω) of the following set ofadmissible stresses (Ω) ∶= { 𝜏 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ∶ | 𝜏 𝐷 ( 𝑥 ) | ≤ 𝛾 f.a.a. 𝑥 ∈ Ω} , where 𝜏 𝐷 ∶= 𝜏 − 𝑑 tr( 𝜏 ) 𝐼 is the deviator of 𝜏 ∈ ℝ 𝑑 × 𝑑𝑠 and 𝛾 denotes the initial uni-axial yieldstress, a given material parameter. The domain of 𝐴 = 𝜕𝐼 (Ω) is trivially (Ω) , which isclosed and convex. For the Yosida approximation of 𝜕𝐼 (Ω) , one obtains by a straightforwardcalculation {eq:proj} (5.1) 𝐴 𝜆 ( 𝜏 ) = 1 𝜆 ( 𝜏 − 𝜋 ( 𝜏 )) = 1 𝜆 max { , 𝛾 | 𝜏 𝐷 | } 𝜏 𝐷 , with the resolvent 𝑅 𝜆 = 𝜋 (Ω) , where 𝜋 (Ω) denotes the projection on (Ω) in 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ,i.e., 𝜋 ( 𝜎 ) ∶= arg min 𝜏 ∈ (Ω) ‖ 𝜏 − 𝜎 ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) . We get in particular 𝑅 𝜆 ( 𝜏 ) = 𝜋 (Ω) ( 𝜏 ) = 𝜏 − max { , 𝛾 | 𝜏 𝐷 | } 𝜏 𝐷 , so that (2.2) is fulfilled.To smoothen this function let 𝑠 > and max 𝑠 ∶ ℝ → ℝ 𝑟 ↦ { max{0 , 𝑟 } , | 𝑟 | ≥ 𝑠, 𝑠 ( 𝑟 + 𝑠 ) , | 𝑟 | < 𝑠, we then set 𝑅 𝑠 ∶ ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 , 𝜏 ↦ 𝜏 − max 𝑠 ( 𝛾 | 𝜏 𝐷 | ) 𝜏 𝐷 . One easily checks that max 𝑠 ∈ 𝐶 ( ℝ ) and we obtain ‖ 𝜕𝐼 𝜆,𝑠 ( 𝜏 ) − 𝜕𝐼 𝜆 ( 𝜏 ) ‖ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) ≤ 𝜆 ( ˆ Ω ||| max 𝑠 ( 𝛾 | 𝜏 ( 𝑥 ) 𝐷 | ) − max ( , 𝛾 | 𝜏 ( 𝑥 ) 𝐷 | )||| | 𝜏 ( 𝑥 ) 𝐷 | ) ∕ ≤ | Ω | 𝛾𝑠 𝜆 (1 − 𝑠 ) {eq:smoothYosidaEstimate} (5.2)for all 𝜏 ∈ 𝐿 (Ω; ℝ 𝑑 × 𝑑𝑠 ) . Moreover, it is also easy to verify that 𝑅 𝑠 ∶ ℝ 𝑑 × 𝑑𝑠 → ℝ 𝑑 × 𝑑𝑠 is Fréchetdifferentiable with 𝑅 ′ 𝑠 ( 𝜏 ) ℎ = ℎ − max ′ 𝑠 ( 𝛾 | 𝜏 𝐷 | ) 𝛾 | 𝜏 𝐷 | ( 𝜏 𝐷 ∶ ℎ 𝐷 ) 𝜏 𝐷 − max 𝑠 ( 𝛾 | 𝜏 𝐷 | ) ℎ 𝐷 and the following lemma shows that 𝑅 𝑠 is monotone and Lipschitz continuous, thus Assump-tion 4.4 Item (i) is satisfied. Note also that 𝑅 ′ 𝑠 ( 𝜏 ) is self adjoint for every 𝜏 ∈ ℝ 𝑑 × 𝑑𝑠 , hence, ′ 𝑠 ( ℎ ) is also self adjoint for all ℎ ∈ (cf. the proof of Theorem 4.14). This manuscript is for review purposes only. S. WALTHER L EMMA
For every 𝑠 ∈ (0 , , the mapping 𝑅 𝑠 is monotone and Lipschitz continuouswith constant 1.Proof. It is well known that, since max 𝑠 is continuously differentiable and convex, max 𝑠 ( 𝑥 ) − max 𝑠 ( 𝑦 ) ≥ max 𝑠 ( 𝑦 ) ′ ( 𝑥 − 𝑦 ) {eq:local011122019} (5.3)holds for all 𝑥, 𝑦 ∈ ℝ .Let 𝜏, 𝜎 ∈ ℝ 𝑑 × 𝑑𝑠 , w.l.o.g. we can assume that | 𝜎 𝐷 | ≥ | 𝜏 𝐷 | > and max 𝑠 ( 𝛾 | 𝜎 𝐷 | ) ≥ max 𝑠 ( 𝛾 | 𝜏 𝐷 | ) , then, using (5.3) with 𝑥 = 1 − 𝛾 | 𝜏 𝐷 | and 𝑦 = 1 − 𝛾 | 𝜎 𝐷 | , we get | max 𝑠 ( 𝛾 | 𝜏 𝐷 | ) 𝜏 𝐷 − max 𝑠 ( 𝛾 | 𝜎 𝐷 | ) 𝜎 𝐷 | ≤ (max 𝑠 ( 𝛾 | 𝜎 𝐷 | ) − max 𝑠 ( 𝛾 | 𝜏 𝐷 | ) ) | 𝜏 𝐷 | + max 𝑠 ( 𝛾 | 𝜎 𝐷 | )| 𝜏 𝐷 − 𝜎 𝐷 | ≤ 𝛾 max ′ 𝑠 ( 𝛾 | 𝜎 𝐷 | )| 𝜏 𝐷 | | | 𝜏 𝐷 | − 1 | 𝜎 𝐷 | | + max 𝑠 ( 𝛾 | 𝜎 𝐷 | )| 𝜏 𝐷 − 𝜎 𝐷 | , taking into account that 𝛾 | 𝜏 𝐷 | | | 𝜏 𝐷 | − 1 | 𝜎 𝐷 | | = 𝛾 ||| | 𝜏 𝐷 | − | 𝜎 𝐷 || 𝜎 𝐷 | ||| ≤ 𝛾 | 𝜎 𝐷 | | 𝜏 𝐷 − 𝜎 𝐷 | we obtain | max 𝑠 ( 𝛾 | 𝜏 𝐷 | ) 𝜏 𝐷 − max 𝑠 ( 𝛾 | 𝜎 𝐷 | ) 𝜎 𝐷 | ≤ ( max ′ 𝑠 ( 𝛾 | 𝜎 𝐷 | ) 𝛾 | 𝜎 𝐷 | + max 𝑠 ( 𝛾 | 𝜎 𝐷 | ))| 𝜏 𝐷 − 𝜎 𝐷 | ≤ max 𝑠 (1) | 𝜏 − 𝜎 | = | 𝜏 − 𝜎 | , where we used (5.3) again with 𝑥 = 1 and 𝑦 = 1 − 𝛾 | 𝜎 𝐷 | , and the fact that | 𝜏 𝐷 − 𝜎 𝐷 | ≤ | 𝜏 − 𝜎 | .This proves the Lipschitz continuity of 𝑅 𝑠 . We also get ( 𝑅 𝑠 ( 𝜏 ) − 𝑅 𝑠 ( 𝜎 )) ∶ ( 𝜏 − 𝜎 )= | 𝜏 − 𝜎 | − ( max 𝑠 ( 𝛾 | 𝜏 𝐷 | ) 𝜏 𝐷 − max 𝑠 ( 𝛾 | 𝜎 𝐷 | ) 𝜎 𝐷 ) ∶ ( 𝜏 − 𝜎 ) ≥ | 𝜏 − 𝜎 | − | 𝜏 − 𝜎 | = 0 which shows the monotonicity of 𝑅 𝑠 . Acknowledgements.
I would like to thank Christian Meyer (Technische UniversitätDortmund) for fruitful discussions regarding Corollary 3.9.
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