N A Nguyen

S Olaru

P Rodriguez-Ayerbe

M Hovd

I Necoara
email: ion.necoara@acse.pub.ro

P Rodriguez
email: pedro.rodriguez@centralesupelec.fr

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Keywords: Convex liftings, Parametric convex programming, Inverse optimality, Piecewise affine functions

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.

Introduction

Parametric convex programming (PCP) has attracted significant attention due to its relevance in many related areas, e.g., computational geometry, operational research, control theory, etc. In particular, optimal solution of a linear/quadratic programming problem is known to be a piecewise affine (PWA) function defined over a polyhedral partition of the parameter space. In control theory, this structure of control laws appeared in the last decade as an approximation of the classical nonlinear control laws with respect to a predefined error [START_REF] Johansen | On multi-parametric nonlinear programming and explicit nonlinear model predictive control[END_REF][START_REF] Johansen | Approximate explicit receding horizon control of constrained nonlinear systems[END_REF][START_REF] Grancharova | Explicit nonlinear model predictive control: Theory and Applications[END_REF]. Then, it is shown that this PWA structure is inherited by the exact optimal solution of a linear model predictive control (MPC) problem with respect to a linear/quadratic cost function [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF][START_REF] Tøndel | An algorithm for multi-parametric quadratic programming and explicit mpc solutions[END_REF][START_REF] Seron | Characterisation of receding horizon control for constrained linear systems[END_REF][START_REF] Olaru | A parameterized polyhedra approach for explicit constrained predictive control[END_REF][START_REF] Pistikopoulos | Multi-parametric programming[END_REF][START_REF] Feller | An improved algorithm for combinatorial multi-parametric quadratic programming[END_REF].

Inverse parametric linear/quadratic programming aims to construct a linear constraint set and a linear/quadratic cost function such that the optimal solution of their associated optimization problem is equivalent to a given PWA function, defined over a given polyhedral partition. This inverse optimality problem has been investigated for some years and has resulted in interesting results for the general nonlinear continuous functions [START_REF] Baes | Every continuous nonlinear control system can be obtained by parametric convex programming[END_REF] and recently for continuous PWA functions [START_REF] Hempel | On inverse parametric optimization with an application to hybrid system control[END_REF][START_REF] Nguyen | Inverse parametric convex programming problems via convex liftings[END_REF].

The authors in [START_REF] Baes | Every continuous nonlinear control system can be obtained by parametric convex programming[END_REF] proved that every continuous feedback law can be obtained by PCP. This is an insightful mathematical result; however, it remains purely theoretical; neither a constructive procedure nor a qualitative interpretation of the dimension of the optimization arguments is provided. The present work is motivated by a comment therein: A natural question that can arise from this note would be to particularize our results to piecewise linear controllers: can any continuous piecewise linear feedback law be obtained by parametric linear programming? The answer is positive and one solution to such an inverse optimality problem is recently found in [START_REF] Hempel | On inverse parametric optimization with an application to hybrid system control[END_REF] wherein an indirect solution, built upon a decomposition of a continuous PWA function into the difference of two continuous convex functions, is introduced.

In this paper, we present the results obtained using a different approach based on convex lifting. It will be proved that the proposed method can recover the given PWA function with at most one supplementary variable. The major contributions in this direction are: 1) the introduction of the convex lifting concept for use in the inverse optimality problem; 2) the convex liftability related condition for the existence of a solution of the inverse optimality problem; 3) a constructive procedure based on convex liftings for obtaining a solution of the inverse optimality problem.

The most important result related to Linear Optimal Control can be stated as follows: any continuous piecewise affine control law can be recovered via a linear optimal control problem with a control horizon at most equal to 2 prediction steps.

The key concept used in the developments: the lifting can be defined as an inverse operation of orthogonal projection. As underlined by its definition, this operation allows lifting of a given partition onto a higher dimensional space. In particular, a socalled convex lifting of a given partition in R d amounts to a convex surface in R d+1 such that each pair of neighboring regions are lifted onto two distinct hyperplanes and its image via the orthogonal projection onto R d coincides with the given partition.

It is worth reminding that the lifting notion was introduced for the first time in Maxwell's research publications, e.g., [START_REF] Maxwell | On reciprocal diagrams and diagrams of forces[END_REF] some 150 years ago. Later, a plethora of studies were dedicated for the existence conditions of such a convex lifting for a given partition [START_REF] Crapo | Plane self stresses and projected polyhedra 1: the basic pattern[END_REF][START_REF] Crapo | Spaces of stresses, projections and parallel drawings for spherical polyhedra[END_REF][START_REF] Aurenhammer | Recognising polytopical cell complexes and constructing projection polyhedra[END_REF][START_REF] Aurenhammer | Criterion for the affine equivalence of cell complexes in r d and convex polyhedra in r d+1[END_REF][START_REF] Rybnikov | Polyhedral partitions and stresses[END_REF][START_REF] Schulz | Lifting planar graphs to realize integral 3-polytopes and topics in pseudo-triangulations[END_REF][START_REF] Nguyen | On the lifting problems and their connections with piecewise affine control law design[END_REF]. However, most of these results are difficult to apply in numerical methods such as those usually employed in linear control design. On the other hand, control theory needs a systematic approach for the use of a lifting procedure in the inverse optimality problem. This aspect will be discussed in details in this paper.

2 Notation and Definitions R, R + , N >0 denote the field of real numbers, the non-negative real numbers set and the positive integer set, respectively. The following index set is also defined for ease of presentation, with a given N ∈ N >0 , I N := {1, . . . , N }.

A polyhedron is defined as the intersection of finite number of closed halfspaces. As a sequence, a polyhedron is a closed set. A polytope is defined as a bounded polyhedron. Given a full dimensional polytope S, then V(S) denotes the set of its vertices, int(S) denotes its interior. By dim(S), we denote the dimension of the affine hull of a given set S. Also, conv(S) denotes the convex hull of a given set S. If S is an arbitrary set in R d and S is a subspace of R d , then Proj S S represents the orthogonal projection of S onto the space S. Further, if S ⊆ R d is a full dimensional polyhedron, a face of S is the intersection of S and one of its supporting hyperplanes. A k-face represents a face of dimension k. A 0-face is called a vertex, an 1-face is called an edge, a (d -1)-face is called a facet. F(S) denotes the set of all facets of the polyhedron S.

For a given d ∈ N >0 , 0 d denotes a vector of dimension d whose elements are equal to 0. Similarly, 0 m×n denotes a matrix in R m×n composed of the elements equal to 0. Let us recall also some useful definitions.

Definition 2.1 A collection of N ∈ N >0 full dimensional polyhedra X i ⊂ R d , de- noted by {X i } i∈I N , is called a polyhedral partition of a polyhedron X ⊆ R d iff: 1. X = i∈I N X i , 2. int(X i ) int(X j ) = ∅ with i = j, (i, j) ∈ I 2 N . (X i , X j ) are called neighbours iff (i, j) ∈ I 2 N , i = j and dim(X i ∩ X j ) = d -1. Also, if X is a polytope then {X i } i∈I N is called a polytopic partition.
The definition of a cell complex was presented by Grünbaum in [START_REF] Grünbaum | Convex polytopes[END_REF]. For simplicity, a cell complex, in this paper, should be understood as a polyhedral partition whose face-to-face property is fulfilled, i.e., for any pair of regions, the intersection of faces is either empty or a common face. Accordingly, a polyhedral partition of a polyhedron is a cell complex if any pair of neighboring regions share a common facet. Definition 2.2 For a given polyhedral partition {X i } i∈I N of a polyhedron X ⊆ R d , a piecewise affine lifting is described by the function z : X → R with:

z(x) = A T i x + a i for any x ∈ X i , (1)
and

A i ∈ R d , a i ∈ R, ∀i ∈ I N . Definition 2.3 Given a polyhedral partition {X i } i∈I N of a polyhedron X ⊆ R d , a
piecewise affine lifting: z(x) = A T i x + a i for x ∈ X i , is called a convex piecewise affine lifting iff the following conditions hold true:

z(x) is continuous over X , for each i ∈ I N , z(x) > A T j x + a j for all x ∈ X i \X j and all j = i, j ∈ I N .

The second condition in the above definition implies that z(x) is a convex function defined over X . Moreover, the strict inequalities ensure that any pair of neighboring regions are lifted onto two distinct hyperplanes. For ease of presentation, a slight abuse of notation is herefter used: a convex lifting is understood as a convex piecewise affine lifting. From the above definition, if a polyhedral partition {X i } i∈I N of a polyhedron X admits a convex lifting, then {X i } i∈I N has to be a cell complex. This observation is stated by the following proposition. Proposition 2.1 A polyhedral partition of a polyhedron, which admits a convex lifting, is a cell complex.

Proof: Suppose the given polyhedral partition {X i } i∈I N of a polyhedron X ⊆ R d , which admits a convex lifting, is not a cell complex. By z(x) = A T i x+a i for x ∈ X i , we denote this convex lifting of {X i } i∈I N . Then there exists a pair of neighboring regions, denoted by X i , X j , whose facet-to-facet property is not fulfilled.

According to the definition of convex liftings, the hyperplane H 0 , described by

H 0 := x ∈ R d : A T i x + a i = A T j x + a j , contains X i ∩ X j .
Also, due to the violation of the facet-to-facet property, there exists a point, denoted by x 0 , such that x 0 ∈ H 0 ∩ X i but x 0 / ∈ X j (an illustration can be found in Fig. 1). x 0 ∈ H 0 implies A T i x 0 + a i = A T j x 0 + a j . On the other hand, x 0 ∈ X i , x 0 / ∈ X j lead to A T i x 0 + a i > A T j x 0 + a j . These two last inclusions are clearly contradictory. Therefore, the partition {X i } i∈I N has to be a cell complex.

According to this proposition, a convex lifting is always defined over a cell complex. However, the cell complex characterization of {X i } i∈I N is a necessary condition, but not a sufficient condition for the existence of a convex lifting. Definition 2.4 A given cell complex {X i } i∈I N in R d has an affinely equivalent polyhedron iff there exists a polyhedron X ⊂ R d+1 such that for each i ∈ I N :

1. ∃F i ∈ F( X ) satisfying: Proj R d F i = X i , 2. if z(x) := min z z s.t. x T z T ∈ X , then x T z(x) T ∈ F i for x ∈ X i .
An illustration can be found in Fig. 2 where a cell complex in R consists of the multicolored segments along the horizontal axis. One of its affinely equivalent polyhedra in R 2 is the pink shaded region. Moreover, the lower facets of this polytope are an illustration of the facets F i appearing in Definition 2.4. Note that, given a cell complex {X i } i∈I N of a polyhedron X ⊆ R d , affinely equivalent to a polyhedron X ⊂ R d+1 , if z denotes the last coordinate of X such that x T z T ∈ X , then {X i } i∈I N is nothing other than the cell complex associated with the optimal solution to the following parametric linear programming problem:

z * (x) = min z z subject to x T z T ∈ X .
Also, z * (x) represents a convex lifting for this cell complex.

3 Problem Statement

Parametric Linear/Quadratic Programming Problems

Recall (see [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF][START_REF] Tøndel | An algorithm for multi-parametric quadratic programming and explicit mpc solutions[END_REF][START_REF] Seron | Characterisation of receding horizon control for constrained linear systems[END_REF][START_REF] Olaru | A parameterized polyhedra approach for explicit constrained predictive control[END_REF][START_REF] Pistikopoulos | Multi-parametric programming[END_REF]) that a parametric linear/quadratic programming problem is defined as follows with respect to d x , d U ∈ N >0 :

min U f (U, x) subject to: GU ≤ W + Ex, (2)
where x ∈ R dx represents the parameter vector, U ∈ R d U represents the decision variable and f (U, x) represents a linear/quadratic cost function in U and x. The above problem has a continuous solution denoted as U * (x) (see [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF]), known to be a piecewise affine function defined over a polyhedral partition {X i } i∈I N of the parameter space denoted as X , as a polyhedron:

U * (x) = f pwa (x) = F i x + G i , ∀x ∈ X i . (3)
Note that the optimal solution to a parametric quadratic programming problem is unique if f (U, x) along U is strictly convex; see [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF]. It is already known that this uniqueness may no longer be preserved in the case of a parametric linear programming problem. However, a continuous selection of optimal solution to such a linear problem is shown in [START_REF] Aubin | Differential inclusions: set-valued maps and viability theory[END_REF] to exist. Conversely, given a continuous PWA function defined over a polyhedral partition, the question is whether there exists an optimization problem such that its optimal solution is equivalent to the given PWA function. The answer is shown in [START_REF] Baes | Every continuous nonlinear control system can be obtained by parametric convex programming[END_REF] to be affirmative, although the numerical construction of such an optimization problem is still open. A possible candidate for this optimization problem may be characterized by a linear/quadratic cost function and a set of linear constraints. For the moment, the definition of an inverse parametric linear/quadratic programming problem is introduced.

3.2 Inverse Parametric Linear/Quadratic Programming Problems

In principle, an inverse parametric linear/quadratic programming problem aims to reconstruct an appropriate optimization problem with respect to a given continuous piecewise affine function u(x) = f pwa (x), defined over a given polyhedral partition {X i } i∈I N of the parameter space X ⊆ R dx such that the optimal solution of this reconstructed problem is equivalent 1 to the given PWA function f pwa (x). This problem can be stated as follows:

Problem statement: For a given polyhedral partition {X i } i∈I N of the parameter space X ⊆ R dx , associated with a continuous PWA function f pwa (x) : X → R du , find a linear/quadratic cost function J(x, z, u) and matrices H x , H u , H z , K such that:

f pwa (x) = Proj R du arg min [z u T ] T J(x, z, u) s.t. H x x + H z z + H u u ≤ K. ( 4
)
The convex lifting based solution to such an inverse optimality problem is presented in the sequel. A definition of invertibility needs to be introduced in order to establish the working assumption of the convex lifting based method.

Definition 3.1 A continuous PWA function defined over a polyhedral partition is called invertible iff there exists an appropriate constraint set and a cost function such that their associated parametric convex programming problem admits the given continuous PWA function as its optimal solution.

4 Constructive Convex Lifting based Approach for Inverse Parametric Linear/Quadratic Programming

Existing Results on Convex Liftings

Many studies dedicated to the existence of convex liftings for the cell complexes in R 2 , were investigated, e.g., in [START_REF] Maxwell | On reciprocal diagrams and diagrams of forces[END_REF][START_REF] Crapo | Plane self stresses and projected polyhedra 1: the basic pattern[END_REF][START_REF] Crapo | Spaces of stresses, projections and parallel drawings for spherical polyhedra[END_REF][START_REF] Schulz | Lifting planar graphs to realize integral 3-polytopes and topics in pseudo-triangulations[END_REF]. These results were then generalized to the cell complexes in the general dimensional space R d through different studies, e.g., in [START_REF] Rybnikov | Polyhedral partitions and stresses[END_REF]. It is shown therein that there exists a convex lifting for a cell complex in R d if and only if one of the following holds:

it admits a strictly positive d-stress; it is an additively weighted Dirichlet-Voronoi diagram;

it is an additively weighted Delaunay decomposition; it is the section of a (d + 1)-dimensional Dirichlet-Voronoi partition 2 .

The above results cover the general class of cell complexes in R d . Unfortunately, despite the mathematical completeness of the existing results, the verification of these conditions are expensive. Furthermore, they do not provide any hint for the construction of a convex lifting. The next subsection presents such a construction in the general case of cell complexes.

Construction of Convex Liftings

In this subsection, the main objective is to present an algorithm for the construction of a convex lifting for a given cell complex via linear/quadratic programming. This algorithm exploits the continuity and the convexity of a convex lifting for neighboring regions. Note that we restrict our attention in this article to the polytopic partitions. Extensions of the results found here for cell complexes of an unbounded polyhedron have been recently given in [START_REF] Nguyen | Inverse parametric linear/quadratic programming problem for continuous PWA functions defined on polyhedral partitions of polyhedra[END_REF]. Suppose we want to lift a given cell complex {X i } i∈I N of a polytope X ⊂ R d . Let X ⊂ R d+1 be one of its affinely equivalent polyhedra. For each region X i with i ∈ I N , the hyperplane H i , containing the lower facet of X , whose orthogonal projection onto R d coincides with X i , has the following form:

H i = x T z i T ∈ R d+1 : z i = A T i x + a i , (5)
for suitable

A i ∈ R d , a i ∈ R. Let (i, j) ∈ I 2
N be an index pair such that (X i , X j ) are neighbors. The continuity conditions between them are described as follows:

∀x ∈ X i ∩ X j , i = j, z i (x) = z j (x). (6)
Moreover, the convexity conditions between them can be handled as:

∀x ∈ X i \(X i ∩ X j ), z i (x) > z j (x). (7)
Algorithm 1 summarizes the constructive procedure which allows for the computation of the gains (A i , a i ), ∀i ∈ I N of a convex lifting.

The following theorem serves as an explanation of this algorithm.

Theorem 4.1 If the optimization problem (10) is feasible, then the function

z(x) = A T i x + a i for x ∈ X i
represents a convex lifting for the given cell complex {X i } i∈I N .

Algorithm 1 Construction of a convex lifting for a given cell complex

{X i } i∈I N of a polytope X ⊂ R d . Input: {X i } i∈I N and a given constant c > 0. Output: Gains (A i , a i ), ∀i ∈ I N .
1: Register all pairs of neighboring regions in {X i } i∈I N .

2: For each pair of neighboring regions (X i , X j ), (i, j) ∈ I 2 N : -Add continuity conditions ∀v ∈ V(X i ∩ X j ):

A T i v + a i = A T j v + a j . (8)
-Add convexity conditions ∀u ∈ V(X i ), u / ∈ V(X j ):

A T i u + a i ≥ A T j u + a j + c. (9)
3: Solve the following convex optimization problem by minimizing a chosen cost function, e.g., 8), ( 9). ( 10)

min Ai, ai N i=1 (A T i A i + a 2 i ) subject to (
Proof: If the optimization problem ( 10) is feasible, then the continuity conditions on the function z(x) and the convexity conditions of its epigraph are all fulfilled. Accordingly, for two neighboring regions (X i , X j ), it follows that:

A T i x + a i = A T j x + a j for all x ∈ X i ∩ X j , A T i x + a i > A T j x + a j for all x ∈ X i \X j . (11)
The same inclusion holds for the other pairs of neighboring regions. This leads to the continuity of z(x) and for each i ∈ I N :

A T i x + a i > A T j x + a j for all x ∈ X i \X j , ∀j = i, j ∈ I N . (12)
Therefore, function z(x) = A T i x + a i for x ∈ X i is a convex lifting defined over the cell complex {X i } i∈I N , as defined in Definition 2.3.

Note that the cost function chosen in [START_REF] Baes | Every continuous nonlinear control system can be obtained by parametric convex programming[END_REF] aims to avoid the unboundedness of the optimal solution. Other choices of this cost function are possible as long as the boundedness of the optimal solution is guaranteed. Also, as seen in ( 9), the strict convexity condition [START_REF] Olaru | A parameterized polyhedra approach for explicit constrained predictive control[END_REF] can easily be transformed into inequality constraints in an optimization problem by adding a positive constant c on the right-hand side of (9), thus > can be replaced with ≥ . Theoretically, if the given cell complex is convexly liftable, then any choice of this positive constant does not have any effect on the feasibility of the optimization problem [START_REF] Baes | Every continuous nonlinear control system can be obtained by parametric convex programming[END_REF]. Since ( 8) and (9) amount to

(αA i ) T v + (αa i ) = (αA j ) T v + (αa j ) for v ∈ V(X i ∩ X j ) (αA i ) T u + (αa i ) ≥ (αA j ) T u + (αa j ) + αc for u ∈ V(X i ), u / ∈ V(X j ),
for any α > 0. In other words, (x) = (αA i ) T x + (αa i ) for x ∈ X i also represents a convex lifting of cell complex {X i } i∈I N which may be resulted from Algorithm 1 with a given constant αc. Therefore, the optimization problem ( 10) is still feasible with the constant αc > 0. Accordingly, the feasibility of the optimization problem (10) can serve as another necessary and sufficient condition for the existence of a convex lifting of a given cell complex. Furthermore, according to Proposition 2.1, the optimization problem ( 10) is infeasible for the polytopic partitions of polytopes whose facet-to-facet property is not fulfilled. Note also that this construction of convex liftings relies on the vertices of the polytopes in {X i } i∈I N , therefore the computation of its vertices, called vertex enumeration, is required. Although vertex enumeration does not scale exponentially with the dimension of the associated polytopes, see [START_REF] Avis | A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra[END_REF], its computation however becomes demanding as the dimension increases. Fortunately, this computation is performed offline, thus it is reasonable to assume that ample computational resources are available.

To illustrate Algorithm 1, a cell complex in R 2 is presented in Fig. 3. One of its affinely equivalent polyhedra is the shaded polytope with the lower facets multicolored. Fig. 3: A cell complex in R 2 and one of its affinely equivalent polyhedra in R 3 .

Convexly Non-liftable Polyhedral Partitions

It is already known that the parameter partition, associated with the optimal solution to a parametric quadratic programming problem, may not be a cell complex but a polyhedral partition. This case usually takes place in linear model predictive control problems with respect to quadratic cost functions. Therefore, to solve the inverse optimality problem via convex liftings, it is necessary to treat such singular partitions in order that their convex liftability is retrieved. It is shown in [START_REF] Nguyen | On the lifting problems and their connections with piecewise affine control law design[END_REF][START_REF] Nguyen | Explicit robust constrained control for linear systems: analysis, implementation and design based on optimization[END_REF] that any polyhedral partition can be subdivided into a convexly liftable one provided that its internal boundaries are still preserved. This result is recalled here for completeness. Theorem 4.2 ( [20,28]) Given a convexly non-liftable polyhedral partition {X i } i∈I N of a polyhedron X ⊆ R d , there exists at least one subdivision, preserving the internal boundaries of this partition such that the new cell complex is convexly liftable.

Interested readers can find details about the proof in [START_REF] Nguyen | On the lifting problems and their connections with piecewise affine control law design[END_REF][START_REF] Nguyen | Explicit robust constrained control for linear systems: analysis, implementation and design based on optimization[END_REF]. According to its proof, the hyperplane arrangement technique can be of use to carry out this goal. Practically, hyperplane arrangement is only one approach to show the existence of modifications for the given convexly non-liftable polyhedral partition into a convexly liftable one. In control theory, such a modification can increase the complexity of PWA control laws in the implementation. However, such a complete refinement may not be necessary in practice. Many different refinement techniques exist. We refer to [START_REF] Gulan | Implications of inverse parametric optimization in model predictive control[END_REF] for an alternative technique for a class of particular cases in control theory.

Solution to Inverse Parametric Linear/Quadratic Programming Problems

The definition of an inverse parametric linear/quadratic programming problem has been introduced in Subsection 3.2. The solution to such inverse optimality problems is built in this paper upon the convex lifting approach. First, some regularity assumptions need to be stated to make this approach reasonable from a construction point of view. These assumptions are stated with respect to the notation in Subsection 3.2.

Assumption 5.1 The parametric linear/quadratic programming problems are exclusively considered as possible candidates for solutions to the inverse optimality problem. As a consequence, the cost function has the following form:

J(x, z, u) = x T z u T Q x T z u T T + C T x T z u T T , (13)
with positive semidefinite matrix

Q T = Q ≥ 0.
Assumption 5.2 The polytopic partition {X i } i∈I N , associated with a given continuous PWA function, is convexly liftable.

Assumption 5.3 The parameter space X = i∈I N X i is a polytope.

Note that Assumption 5.1 provides a manageable framework for the constructive inverse optimality procedure. Larger classes of objective functions can provide more degrees of freedom, but the linearity of such parametric convex programming problems is lost. Assumption 5.3 restricts the inverse optimality problem to bounded feasible region given by a polytope, since linear constraints are exclusively of interest. Also, the construction presented below can easily be extended to polyhedral partitions of polyhedra, see [START_REF] Nguyen | Inverse parametric linear/quadratic programming problem for continuous PWA functions defined on polyhedral partitions of polyhedra[END_REF]. Assumption 5.2 is not restrictive, since the convex liftability of the given polytopic partition can be enforced by refinement procedures according to Theorem 4.2. Note also that in the scope of this paper, we restrict our attention to the class of continuous PWA functions. It will be shown that due to this continuity property, the optimal solution to the recovered optimization problem is unique.

Inverse optimality for the class of discontinuous PWA functions is studied in [START_REF] Nguyen | Any discontinuous PWA function is optimal solution to a parametric linear programming problem[END_REF].

In this case, it is however shown that the uniqueness of the optimal solution to the recovered optimization problem is lost.

The following intermediate result is necessary for the development of a constructive solution to the inverse optimality problem.

Proposition 5.1 Let Γ s ⊂ R ds be a full dimensional polytope with the set of vertices V(Γ s ) = {s (1) , . . . , s (q) }. For any finite set of points {t (1) , . . . , t (q) } ⊂ R dt defining a full dimensional polytope in R dt , an extension of the family V(Γ s ) can be obtained in higher dimensional space R ds+dt for the concatenated vectors s T t T T defining the set:

V [s T t T ] T := s (1)
t (1) , . . . ,

s (q) t (q) .
The polytope

Γ [s T t T ] T = conv(V [s T t T ] T ) satisfies: V [s T t T ] T = V(Γ [s T t T ] T ).
Proof: Geometrically, this proposition shows that if s (i) is a vertex of Γ s ⊂ R ds , then with any complementary vector t (i) ∈ R dt yielding vector s (i) t (i) ∈ R ds+dt , this vector represents a vertex of the new polytope Γ [s T t T ] T in R ds+dt defined as the convex hull of the extended set of points

V [s T t T ] T . By construction, it can be observed that V(Γ [s T t T ] T ) ⊆ V [s T t T ] T .
Therefore, in order to prove this claim, we will show that

V(Γ [s T t T ] T ) ⊂ V [s T t T ] T leads to a contradiction. In fact, suppose V(Γ [s T t T ] T ) ⊂ V [s T t T ] T .
According to this assumption, there exists a point in V [s T t T ] T which lies in the interior of the polytope Γ [s T t T ] T or can be described by a convex combination of the other points. Without loss of generality, let s (q) t (q) denote this point, then there exists a vector α ∈ R q-1 + such that:

s (q) t (q) = q-1 i=1 α i s (i) t (i) , q-1 i=1 α i = 1. (14)
One can easily see from ( 14) that s (q) , as a vertex of Γ s , is described by a convex combination of the other vertices of Γ s . This inclusion is contradictory to the definition of a vertex of a convex set, see [START_REF] Grünbaum | Convex polytopes[END_REF]. In other words, all elements of V [s T t T ] T are the vertices of Γ [s T t T ] T .

Remark 5.1 Note also that this proposition remains valid for the degenerate case where all points t (1) , . . . , t (q) are placed on a hyperplane in R dt . In this case, the dimension of polytope Γ [s T t T ] T however becomes lower than d s + d t . These particular cases of values t (i) , ∀i ∈ I q , are excluded in the previous result as not relevant for the scope of this paper, even though the mathematical result holds.

Consider a given cell complex {X i } i∈I N of a polytope X ⊂ R dx satisfying Assumption 5.2 and a continuous PWA function f pwa (x) : X → R du defined over this cell complex. For ease of presentation, let (x) denote a convex lifting for {X i } i∈I N . Define also the following sets:

Π [x T z] T := conv v T (v) T : v ∈ i∈I N V(X i ) , V [x T z u T ] T := v T (v) f T pwa (v) T : v ∈ i∈I N V(X i ) , Π [x T z u T ] T := conv V [x T z u T ] T . ( 15
)
With respect to the above notation, the solution to an inverse parametric linear/quadratic programming problem can be stated as follows.

Theorem 5.4 Given a continuous PWA function f pwa (x) defined over a polytopic partition {X i } i∈I N satisfying Assumptions 5.2, 5.3 and the sets defined in [START_REF] Crapo | Spaces of stresses, projections and parallel drawings for spherical polyhedra[END_REF], the following hold true:

1. V [x T z u T ] T = V(Π [x T z u T ] T ) and Π [x T z] T = Proj [x T z] T Π [x T z u T ] T , 2.
The given PWA function f pwa (x) is the image via the orthogonal projection onto R du of the optimal solution to the optimization problem below:

min [z u T ] T z s.t. x T z u T T ∈ Π [x T z u T ] T . (16)
Proof: 1. The first claim:

V [x T z u T ] T = V(Π [x T z u T ] T )
, is directly deduced from Proposition 5.1. The second claim follows from the construction of Π [x T z u T ] T having all its vertices as extended vectors of the vertices of

Π [x T z] T . 2. It is known that Π [x T z] T ⊂ R dx+1 represents an affinely equivalent polyhedron of the partition {X i } i∈I N . Let F (i) [x T z] T for i ∈ I N denote the lower facet of Π [x T z] T such that: Proj x F (i) [x T z] T = X i and for any x ∈ X i , x T (x) T ∈ F (i) [x T z] T satisfies (x) = min z z s.t. x T z T ∈ Π [x T z] T . Such a facet F (i) [x T z]
T can be represented by:

F (i) [x T z] T = conv v T (v) T : v ∈ V(X i ) .
Also, there exists, in higher dimensional space R dx+du+1 , a d x -face denoted as

F (i) [x T z u T ] T of Π [x T z u T ] T such that: Proj [x T z] T F (i) [x T z u T ] T = F (i) [x T z] T . Thus, a point x T z u T T ∈ Π [x T z u T ] T satisfying x ∈ X i has the minimal value of z if and only if this point locates on F (i) [x T z u T ] T . It is worth stressing that the face F (i) [x T z u T ] T
is defined as follows: Accordingly, if x ∈ X i is represented by x = v∈V(Xi) α(v)v for α(v) ≥ 0 and v∈V(Xi) α(v) = 1, then the optimal solution z * (x) (u * ) T (x)

F (i) [x T z u T ] T := conv v T (v) f T pwa (v) T : v ∈ V(X i ) . X 1 X 2 X 3 z u x F u * 1 (x) u * 2 (x) z * 1 (x) = z * 2 (x)
T to the problem ( 16) is expressed by:

z * (x) u * (x) = v∈V(Xi) α(v) (v) f pwa (v) = (x) f pwa (x)
, ∀x ∈ X i .

Clearly, f pwa (x) is a sub-component of this optimal solution. To complete the proof, the uniqueness of such an optimal solution needs to be clarified. Suppose there exist two different optimal solutions to (16):

z * 1 (x) (u * 1 ) T (x) T = arg min [z u T ] T z, s.t. x T z u T T ∈ Π [x T z u T ] T , z * 2 (x) (u * 2 ) T (x) T = arg min [z u T ] T z, s.t. x T z u T T ∈ Π [x T z u T ] T , then it is clear that z * 1 (x) = z * 2 (x) = (x). Accordingly, if u * 1 (x) = u * 2 (x)
for x ∈ X i , there exists a (d x +1)-face denoted as F of Π [x T z u T ] T (illustrated in Fig. 4) to which two optimal solutions z * 1 (x) (u * 1 ) T (x) T and z * 2 (x) (u * 2 ) T (x) T belong such that F is perpendicular to the space x T z T . This implies that the value of f pwa (v)

is not uniquely defined for vertices v ∈ V(X i ). This consequence contradicts the construction of the constraint set Π [x T z u T ] T presented in [START_REF] Crapo | Spaces of stresses, projections and parallel drawings for spherical polyhedra[END_REF]. Therefore, such two optimal solutions have to be identical, leading to the uniqueness. The constructive procedure towards recovering a continuous PWA function defined over a convexly liftable polytopic partition is summarized through Algorithm 2. Theorem 5.4 proves the existence of an optimization problem with respect to a linear cost function which admits a given PWA function as a sub-component of the optimal solution. The following theorem shows the existence of equivalent optimization problem with respect to a quadratic cost function.

Algorithm 2 Linear equivalent optimization problem

Input: A continuous PWA function f pwa (x) defined over a convexly liftable polytopic partition {X i } i∈I N of a polytope X ⊂ R dx . Output: Π [x T z u T ] T and J(x, z, u).

1: Construct a convex lifting (x) for {X i } i∈I N via Algorithm 1.

2: Compute Π [x T z u T ] T as in [START_REF] Crapo | Spaces of stresses, projections and parallel drawings for spherical polyhedra[END_REF]. 3: Define J(x, z, u) = z. 4: Solve the following parametric linear programming problem:

z * (x) (u * ) T (x) T = arg min [z u T ] T z subject to x T z u T T ∈ Π [x T z u T ] T .
5: Obtain the given PWA function:

Proj u z * (x) u * (x) = f pwa (x).
Theorem 5.5 Consider a continuous PWA function f pwa (x) defined over a polytopic partition {X i } i∈I N satisfying Assumptions 5.2, 5.3 and the sets defined in [START_REF] Crapo | Spaces of stresses, projections and parallel drawings for spherical polyhedra[END_REF]. Function f pwa (x) is the image via the orthogonal projection onto R du of the optimal solution to the following optimization problem:

min [z u T ] T (z -σ(x)) 2 s.t. x T z u T T ∈ Π [x T z u T ] T , (17)
where σ(x) : X → R denotes any function satisfying: σ(x) ≤ (x).

Proof: Consider an affinely equivalent polyhedron Π [x T z] T defined as in [START_REF] Crapo | Spaces of stresses, projections and parallel drawings for spherical polyhedra[END_REF]. According to its definition, we obtain:

(x) = min z z subject to x T z T ∈ Π [x T z] T .
Therefore, for any function σ(x) : X → R satisfying σ(x) ≤ (x), the minimization of (zσ(x)) 2 amounts to the minimization of z subject to the same set of constraints Π [x T z u T ] T . According to Theorem 5.4, the given continuous PWA function f pwa (x) is a sub-component of the optimal solution to (16) as well as [START_REF] Aurenhammer | Criterion for the affine equivalence of cell complexes in r d and convex polyhedra in r d+1[END_REF]. Algorithm 3 summarizes the constructive procedure of an equivalent optimization problem with respect to a quadratic cost function.

Remark 5.2 Theorem 5.5 proposes a cost function of z and x. To return this cost function to the form (13), function σ(x) should be chosen as an affine function of x.

We will present in the sequel the important properties of the solution to inverse parametric linear/quadratic programming problems via convex liftings, i.e., the invertibility and the complexity of the above constructive procedures.

Theorem 5.6 (Invertibility) Given a polytopic partition {X i } i∈I N of a polytope X ⊂ R dx , then any continuous PWA function f pwa (x) : X → R du , defined over {X i } i∈I N , is invertible.

Algorithm 3 Quadratic equivalent optimization problem

Input: A continuous PWA function f pwa (x) defined over a convexly liftable polytopic partition {X i } i∈I N of a polytope X ⊂ R dx . Output: Π [x T z u T ] T and J(x, z, u).

1: Construct a convex lifting (x) for {X i } i∈I N via Algorithm 1.

2: Compute Π [x T z u T ] T as in [START_REF] Crapo | Spaces of stresses, projections and parallel drawings for spherical polyhedra[END_REF].

3: Choose a function 3 σ(x) : X → R such that σ(x) ≤ (x). 4: Define J(x, z, u) = (zσ(x)) 2 . 5: Solve the following parametric quadratic programming problem:

z * (x) (u * ) T (x) T = arg min [z u T ] T (z -σ(x)) 2 s.t. x T z u T T ∈ Π [x T z u T ] T . 6: Project the optimal solution onto R du : Proj u z * (x) u * (x) = f pwa (x).
The proof of Theorem 5.6 is omitted, as it summarizes the above results. The complexity of an inverse parametric linear/quadratic programming problem based on convex liftings is also stated as follows:

Theorem 5.7 (Complexity) Any continuous PWA function defined over a polytopic partition of a polytope can be equivalently obtained by a parametric linear/quadratic programming problem with at most one auxiliary 1-dimensional variable.

Proof: Let {X i } i∈I N denote this given polytopic partition of a polytope X . If {X i } i∈I N is convexly liftable, this 1-dimensional variable describes the convex lifting in the recovered optimization problem. Theorems 5.4, 5.5 show that this PWA function is invertible through the convex lifting based approach. Otherwise, in case the given partition is not convexly liftable, Theorem 4.2 shows that there exists at least one way to subdivide the given convexly non-liftable polytopic partition into a convexly liftable cell complex, denoted by X i i∈I N , meanwhile the internal boundaries are maintained. According to this subdivision, the given PWA function f pwa (x) is also subdivided. This new PWA function, say f pwa (x), is equivalent to f pwa (x) and defined over a convexly liftable cell complex X i i∈I N . Therefore, similar to the first case, a convex lifting of X i i∈I N , represents the 1-dimensional auxiliary variable. Also, as proved in Theorems 5.4, 5.5, f pwa (x), associated with X i i∈I N , is invertible via the convex lifting based method.

Applications to Linear Optimal Control

Consider a linear, discrete time-invariant system:

x k+1 = Ax k + Bu k
where x k , u k denote the state and control variables at time k and A, B represent matrices of suitable dimension. A linear optimal control problem with a prediction horizon N ∈ N >0 for the above system can be written in the following form: U * (x) = argmin U F (x k|k , . . . , x k+N |k , u k|k , . . . , u k+N -1|k ) s.t. G(x k|k , . . . , x k+N |k , u k|k , . . . , u k+N -1|k ) ≤ 0, [START_REF] Rybnikov | Polyhedral partitions and stresses[END_REF] where x k+i|k ∈ R dx , u k+i|k ∈ R du are the state variable, control variable, respectively, at time k + i, predicted at time k; U = u T k|k . . . u T k+N -1|k

T denotes the decision variable composed of the control variables over the prediction horizon; and x k|k = x k , u k|k = u k . Also, G(x k|k , . . . , x k+N |k , u k|k , . . . , u k+N -1|k ) represents a linear function along its variables, F (x k|k , . . . , x k+N |k , u k|k , . . . , u k+N -1|k ) represents a linear/quadratic, real-valued function along its variables. In implementation, the interest of the optimal solution U * (x) is restricted to the first part, i.e., u k . The main message of this paper is stated in the sequel. Proposition 6.1 Any continuous PWA control law is the optimal solution to a parametric linear/quadratic programming problem.

Proof: The proof directly follows from Theorem 5.6 since a continuous PWA control law is a continuous PWA function defined over a polyhedral partition. Theorem 6.1 Any continuous PWA control law can be equivalently obtained through a linear optimal control problem with a linear or quadratic cost function and the control horizon at most equal to 2 prediction steps.

Proof: Let u(x) : R dx → R du denote a given continuous PWA control law defined over a state space partition {X i } i∈I N of a polytope X ⊂ R dx . If {X i } i∈I N is not convexly liftable, it can be subdivided into a convexly liftable cell complex according to Theorem 4.2. Therefore, one can exclusively focus on the case

{X i } i∈I N is convexly liftable. Now, let Π [x T k z u T k ]
T denote the set of constraints in the recovered optimization problem, i.e., min

[z u T k ] T z s.t. x T k z u T k T ∈ Π [x T k z u T k ] T . ( 19
)
For ease of presentation, let

Π [x T k z u T k ]
T be given in the following form:

H x x k + H z z + H u u k ≤ K.
If d u = 1, then it suffices to use z as the second predicted controller, i.e., u k+1|k = z.

Otherwise, the set of constraints H x x k +H u u k +H z z ≤ K amounts to the following constraints:

H x x k + H u u k + [H z 0] z s T T ≤ K, ( 20
)
where 0 denotes a matrix of appropriate dimension, composed of zeros with the number of columns equal to d u -1. Also, s ∈ R du-1 denotes auxiliary variable. Again, apply z s for the next predicted control variable, i.e., u k+1|k = z s . Accordingly, [START_REF] Schulz | Lifting planar graphs to realize integral 3-polytopes and topics in pseudo-triangulations[END_REF] can be written as follows:

min u T k u T k+1|k T 0 T du 1 0 T du-1 u k u k+1|k s.t. H x x k + [H u H z 0] u k u k+1|k ≤ K,
known to be a linear optimal control problem with respect to a linear cost function. On the other hand, according to Theorem 5.5, the recovered optimization problem with a quadratic cost function can also be written in the following form:

min [z u T k ] T (z -σ(x k )) 2 s.t. x T k z u T k T ∈ Π [x T k z u T k ] T , (21)
where σ(x k ) ≤ (x k ), (x k ) denotes the convex lifting for the given cell complex

{X i } i∈I N , used to compute Π [x T k z u T k ] T . Suppose (x k ) = A T i x k + a i for x k ∈ X i , it suffices to choose σ(x k ) = A T i x k + a i over X .
Accordingly, similar to the case of linear optimal control with a linear cost function, ( 21) can be easily written in the form of linear optimal control with respect to a quadratic function of u T k u T k+1|k T . To complete the proof, we will show that (20) can be described as the constraint set in [START_REF] Rybnikov | Polyhedral partitions and stresses[END_REF]. In fact, as G(x k|k , . . . , x k+N |k , u k|k , . . . , u k+N -1|k ) is a linear function of its variables, with a prediction horizon 2, this constraint set can be written as follows:

H 1 x k + H 2 u k + H 3 x k+1|k + H 4 u k+1|k + H 5 x k+2|k ≤ H 6 .
According to the given dynamics and some trivial transformations, (

H 1 + H 3 A + H 5 A 2 = H x , H 4 + H 5 B = [H z 0] , H 2 + H 3 B + H 5 AB = H u , H 6 = K. 20) yields:
The feasibility of the equations in [START_REF] Aubin | Differential inclusions: set-valued maps and viability theory[END_REF] is obvious according to a trivial solution H 3 = 0, H 5 = 0. The choice of matrices H 3 , H 5 of suitable dimension represents a degree of freedom in the recovered optimization problem and its equivalent form of an optimal control problem. The proof is complete.

Remark 6.1 Note that the second predicted control variable u k+1|k in the recovered optimization problem does not necessarily satisfy the given constraints on the control variable, as it will not be implemented. Therefore, we can use it as an artificial variable. However, making u k+1|k fulfill the constraints on control variable, usually in the form: u min ≤ u k+i|k ≤ u max , can be adapted from the construction of convex lifting by fixing the upper and lower bounds for its value.

Remark 6.2 In the context of generalized predictive control (GPC), it has been shown in [START_REF] Clarke | Generalized predictive controlpart i. the basic algorithm[END_REF] that a large prediction horizon can be beneficial for complex plants. In general, the nature of GPC does not take constraints into account, closed-loop stability guarantee being verified at post-design stage. Therefore, the quality of control algorithm relies mainly on the prediction horizon. It is worth noting that inverse optimality problem for such a GPC design can easily be obtained, as the controller resulted from a GPC problem possesses a linear structure. On the other hand, model predictive control guarantees closed-loop stability due to suitable terminal constraints imposed at the end of prediction horizon and only the first part of its optimal solution is of interest in implementation. Accordingly, the solution of inverse optimality problem presents in a compact manner a set of constraints, considering the input to be implemented and an auxiliary variable as the second control action. That is the main reason for which the inverse optimality problem only needs a prediction horizon of 2 steps.

Remark 6.3 The reduction of prediction horizon to 2 has some consequences in the computation of the associated controller. A remarkable one is that a linear programming problem shares the same first part of its optimal solution with a quadratic programming problem. It is known that a quadratic programming problem is in general more computationally demanding than a linear one. To illustrate this point, we refer the reader to [START_REF] Gulan | Implications of inverse parametric optimization in model predictive control[END_REF][START_REF] Nguyen | On the complexity of the convex liftings-based solution to inverse parametric convex programming problems[END_REF] for more detailed analyses and case studies. Note however that this computational complexity reduction is sometimes obtained at the price of a larger number of constraints as seen in the numerical example in Subsection 7.2. As for the implementation of PWA controllers, the constructed convex lifting turns out to be helpful to avoid storing state-space partition at the hardware level and facilitate the point-location problem [START_REF] Nguyen | Explicit robust constrained control for linear systems: analysis, implementation and design based on optimization[END_REF][START_REF] Nguyen | Convex liftings: Theory and control applications URL[END_REF]. Accordingly, it allows for implementation of PWA controllers onto embedded systems with low computational performance and memory footprint.

Remark 6.4 It is shown in [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF][START_REF] Tøndel | An algorithm for multi-parametric quadratic programming and explicit mpc solutions[END_REF][START_REF] Seron | Characterisation of receding horizon control for constrained linear systems[END_REF] that in the case of non-degeneracies, the optimal cost function of a parametric quadratic programming problem represents a strictly convex, piecewise quadratic function (this can also be proven using dynamic programming).

In the context of inverse optimality, the idea of using this optimal cost function as an auxiliary variable is proposed in [START_REF] Baes | Every continuous nonlinear control system can be obtained by parametric convex programming[END_REF] and is recalled here. For ease of presentation, let µ(x) denote the controller defined over a convex set Ω to be recovered and g 0 (x) represent a strictly convex function over Ω. Accordingly, the set of constraints denoted therein by E for inverse optimality problem can be computed as follows:

S := (x, µ(x), t) : x ∈ Ω, g 0 (x) ≤ t , E := conv(S).

It can be seen that taking the present controller into account, t as an auxiliary variable is also used to represent the second control action. In the case of non-degenerate parametric quadratic programming, its optimal cost function, being strictly convex and piecewise quadratic, can be used as g 0 (x). Note also that the controller µ(x) in this case is a continuous PWA function. Unfortunately, it becomes very difficult to be able to obtain an explicit representation of the constraint set E as in [START_REF] Aurenhammer | Voronoi diagrams: A survey of a fundamental data structure[END_REF]. First, it is not clear whether E could be described by a set of quadratic constraints. Second, to our best knowledge, solvers for parametric programming problems subject to quadratic constraints are still unavailable; instead, only approximate optimal solution can be obtained. These obstacles prevent this optimal cost function from the application in the inverse optimality problem. It is worth stressing that a convex lifting used in this paper is not strictly convex, but convex, as there does not exist a strictly convex, piecewise affine function. Moreover, the recovered optimization problem is shown to have the unique optimal solution which was not treated in [START_REF] Baes | Every continuous nonlinear control system can be obtained by parametric convex programming[END_REF].

Illustrative Examples

This section considers numerical examples to illustrate the above results.

Double Integrator System

To illustrate the above results, consider a double integrator system:

x k+1 = 1 0.5 0 1 x k + 0.125 0.5 u k , y k = 1 0 x k (24)
An MPC problem, constructed with the minimization of a quadratic cost function over a prediction horizon N = 5, is presented as follows, with respect to weighting matrices Q = 10 0 0 10 , R = 0.5:

J = 4 i=0 (x T k+i|k Qx T k+i|k + u T k+i|k Ru k+i|k ) + x T k+5|k P x k+5|k ,
where P is computed via the Riccati equation. Constraints on the present control variable and output signal are given by:

u k ∈ [-2, 2] , y k ∈ [-5, 5] (25)
The terminal constraint is chosen as the maximal output admissible set as shown in [START_REF] Gilbert | Linear systems with state and control constraints: The theory and application of maximal output admissible sets[END_REF]. The feedback control law is depicted in Fig. 7. Its inverse optimization problem in form (4) can be composed of the components presented in [START_REF] Avis | A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra[END_REF]. Accordingly, the new linear optimal control problem with the control horizon of 2 is described as follows:

min [uk u k+1|k ] T [0 1] u k u k+1|k s.t. H x x k + H u u k + H z u k+1|k ≤ K, (26)
where H x , H u , H z , K are presented in [START_REF] Avis | A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra[END_REF]. Consider the same MPC problem but with a prediction horizon 3, we obtain the inverse optimization problem in form (4) associated with the ingredients in [START_REF] Nguyen | Explicit robust constrained control for linear systems: analysis, implementation and design based on optimization[END_REF]. Finally, an equivalent linear optimal control problem with the prediction horizon 2 is represented as in [START_REF] Nguyen | Inverse parametric linear/quadratic programming problem for continuous PWA functions defined on polyhedral partitions of polyhedra[END_REF] with the information in [START_REF] Nguyen | Explicit robust constrained control for linear systems: analysis, implementation and design based on optimization[END_REF].

J(x, z, u) = z Hx =                                            0.
                                           , Hz =                                           
-0.1565 0 -0.1546 -0.0266 -0.0881 -0.0308 0 -0.0887 -0.0295 -0.0256 -0.0291 -0.0253 -0.0288 -0.0251

0 0 0 0 0 0 0 0 0 0                                            , Hu =                                            0.
0 0 0 0 0 0 0 0 0                                            , K =                                            -3
                                           . ( 27
) J(x, z, u) = z Hx =                            -0.
                           , Hz =                            -0.5271 0 -0.0658 -0.0230 -0.4955 -0.0244 0 -0.5108 -0.0235 -0.0222 0 0 0 0 0 0                            , Hu =                            -0.0132 1 -0.3961 0.3567 0.0279 -0.3788 -1 0.0128 -0.3796 0.3582 0 0 0 0 0 0                            , K =                            -6.5119 2
                           . ( 28
)

Non-Minimum Phase System

To illustrate the comparison between GPC design [START_REF] Boucher | La commande prédictive[END_REF][START_REF] Maciejowski | Predictive control: with constraints[END_REF] and inverse optimality in terms of prediction horizon, we consider the following non-minimum phase system in the discrete-time domain:

H(z) = -0.375z + 0.625 z 2 -2z + 1 . (29)
The parameters in GPC design are chosen below: the minimum output horizon N 1 = 1; the maximum output horizon N 2 = 5; the control horizon N u = 5 and weighting value λ = 1. The reference signal (the green line) is represented by a square wave as shown in Fig. 5. Clearly, the controller designed with the above parameters is not stabilizing, as the output signal (the blue one) diverges from the reference signal. As a consequence, if N 2 ≤ 5, the obtained controller cannot stabilize the system. For now, we increase the maximum output horizon N 2 = 6, whereas the other parameters are still the same. The result presented in Fig. 6 verifies that the controller designed with sufficiently large horizon stabilizes the given system. Note also that the green and blue signals in Fig. 6 represent the reference and output, respectively. To clarify the advantage of inverse optimality, we consider model predictive control problem for the same system with the same prediction horizon, i.e., N = 5. Note that a state-space representation of the discrete system (29) is described by:

x k+1 = 1 0 0.5 1 x k + 1 0.25 u k , y k = -0.5 0.5 x k .
We choose the weighting matrices Q = 0.25 -0.25 -0.25 0.25 , R = 1. The cost function is as in Subsection 7.1; constraints on present output and control variable are as in [START_REF] Nguyen | Recognition of additively weighted Voronoi diagrams and weighted Delaunay decompositions[END_REF]. Fig. 8 depicts the controller obtained by the above MPC problem. Accordingly, a linear optimal control problem in form (26) with prediction horizon 2, which admits this controller as its optimal solution, is composed of the matrices defined in [START_REF] Nguyen | Any discontinuous PWA function is optimal solution to a parametric linear programming problem[END_REF].

Conclusions

This article presents a method to solve inverse parametric linear/quadratic programming problems. This method relies on convex lifting. It is shown that for any continuous PWA function defined over a polytopic partition, an appropriate equivalence of this function can be obtained by another parametric linear/quadratic programming problem with a supplementary variable of dimension equal to 1. In view of linear optimal control, it has been shown that any continuous PWA control law can be obtained via a linear optimal control problem with the prediction horizon equal to 2 prediction steps. Several numerical examples prove the effectiveness of this method.

Hx =                                                                                            -0.
                                                                                           , Hz =                                                                                            -0.
                                                                                           , , Hu =                                                                                            -0
                                                                                           , K =                                                                                            -7
                                                                                           . ( 30
)

Fig. 1 :

 1 Fig. 1: An illustration for Proposition 2.1.

Fig. 2 :

 2 Fig. 2: An illustration of affinely equivalent polyhedron.

Fig. 4 :

 4 Fig. 4: An illustration of two different optimal solutions.

Fig. 5 :

 5 Fig. 5: Reference tracking designed by GPC with N 2 = 5.

Fig. 6 :Fig. 7 :

 67 Fig. 6: Reference tracking designed by GPC with N 2 = 6.

Fig. 8 :

 8 Fig. 8: Controller obtained from the MPC problem in Subsection 7.2.

The equivalence hereafter means that the boundary between two regions of the parameter space partition, corresponding to two different affine functions, must be preserved and a subdivision or refinement of the regions corresponding to the same affine function is possible.

Other related results can be found in Konstantin Rybnikov's thesis[START_REF] Rybnikov | Polyhedral partitions and stresses[END_REF], equally in[START_REF] Aurenhammer | Criterion for the affine equivalence of cell complexes in r d and convex polyhedra in r d+1[END_REF][START_REF] Aurenhammer | Voronoi diagrams: A survey of a fundamental data structure[END_REF][START_REF] Aurenhammer | Power diagrams: properties, algorithms and applications[END_REF][START_REF] Nguyen | Recognition of additively weighted Voronoi diagrams and weighted Delaunay decompositions[END_REF].

One can choose σ(x) to be an affine function composing (x), i.e., σ(x) = A T i x + a i .

Acknowledgements The authors would like to thank the Associate Editor and reviewers for constructive and helpful comments. The research leading to these results has benefited from the financial support of the European Unions 7th Framework Programme under EC-GA No. 607957 TEMPO -Training in Embedded Model Predictive Control and Optimization. The first author would like to thank Dr. Martin Gulan and Prof. Luigi Del Re for their support and fruitful discussions. The last author acknowledges support from UEFISCDI Romania, PNII-RU-TE, project MoCOBiDS, no. 176/01.10.2015.


Hantaek Bae
email: hantaek@unist.ac.kr

Rafael Granero-Belinch Ón

Omar Lazar
email: omar.lazar@icmat.es

GLOBAL EXISTENCE OF WEAK SOLUTIONS TO DISSIPATIVE TRANSPORT EQUATIONS WITH NONLOCAL VELOCITY

Keywords: Mathematics Subject Classification. Primary 35A01, 35D30, 35Q35, 35Q86 Fluid equations, 1D models, Global weak solution

We consider 1D dissipative transport equations with nonlocal velocity field:

where N is a nonlocal operator given by a Fourier multiplier. Especially we consider two types of nonlocal operators:

(1) N = H, the Hilbert transform, (2) N = (1 -∂xx) -α . In this paper, we show several global existence of weak solutions depending on the range of γ and δ. When 0 < γ < 1, we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variables), which have infinite energy, when γ = 1.

Introduction

In this paper, we consider transport equations with nonlocal velocity. Here, the non-locality means that the velocity field is defined through a nonlocal operator that is represented in terms of a Fourier multiplier. For example, in the two dimensional Euler equation in vorticity form, ω t + u • ∇ω = 0, the velocity is recovered from the vorticity ω through u = ∇ ⊥ (-∆) -1 ω or equivalently u(ξ) = iξ ⊥ |ξ| 2 ω(ξ). Other nonlocal and quadratically nonlinear equations appear in many applications. Prototypical examples are the surface quasi-geostrophic equation, the incompressible porous medium equation, Stokes equations, magneto-geostrophic equation in multi-dimensions. For more details on nonlocal operators in these equations, see [START_REF] Bae | Global existence for some transport equations with nonlocal velocity[END_REF].

We here study 1D models of physically important equations. The 1D reduction idea were initiated by Constatin-Lax-Majda [START_REF] Constantin | A simple one-dimensional model for the three dimensional vorticity[END_REF]: they proposed the following 1D model θ t = θHθ for the 3D Euler equation in the vorticity form and proved that Hθ blows up in finite time under certain conditions. Motivated by this work, other similar models were proposed and analyzed in the literature [START_REF] Bae | Global existence for some transport equations with nonlocal velocity[END_REF][START_REF] Baker | Analytic structure of 1D transport equations with nonlocal fluxes[END_REF][START_REF] Carrillo | A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity[END_REF][START_REF] Castro | Global existence, singularities and ill-posedness for a nonlocal flux[END_REF][START_REF] Castro | Self-similar solutions for a transport equation with non-local flux[END_REF][START_REF] Chae | Finite time singularities in a 1D model of the quasi-geostrophic equation[END_REF][START_REF] Córdoba | Formation of singularities for a transport equation with nonlocal velocity[END_REF][START_REF] Gregorio | On a one-dimensional model for the 3D vorticity equation[END_REF][START_REF] Kiselev | Regularity and blow up for active scalars[END_REF][START_REF] Li | Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation[END_REF][START_REF] Li | On a one-dimensional nonlocal flux with fractional dissipation[END_REF][START_REF] Morlet | Further properties of a continuum of model equations with globally defined flux[END_REF]. In this paper, we consider the following 1D equation: θ t + uθ x + δu x θ + νΛ γ θ = 0, u = N (θ).

(1.1)

Depending on a nonlocal operator N , (1.1) has structural similarity of several important fluid equations as described below. The goal of this paper is to show the existence of weak solutions with rough initial data. To this end, we will choose functionals carefully to extract more information from the structure of the nonlinearity to construct weak solutions.

1.1. The case N = H. We first take the case N = H, the Hilbert transform. Then, (1.1) becomes θ t + (Hθ) θ x + δθΛθ + νΛ γ θ = 0, (1.2) where the range of γ and δ will be specified below. We note that (1.2) is considered as an 1D model of the dissipative surface quasi-geostrophic equation. The surface quasi-geostrophic equation describes the dynamics of the mixture of cold and hot air and the fronts between them in 2 dimensions [START_REF] Constantin | Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar[END_REF][START_REF] Pedlosky | Geophysical Fluid Dynamics[END_REF].

The equation is of the form

θ t + u • ∇θ + νΛ γ θ = 0, u = (-R 2 θ, R 1 θ) , (1.3)
where the scalar function θ is the potential temperature and R j is the Riesz transform

R j f (x) = 1 2π p.v. R 2
(x jy j )f (y) |x -y| 3 dy, j = 1, 2.

As Constatin-Lax-Majda did for the Euler equation, the equation (1.2) is derived by replacing the Riesz transforms with the Hilbert transform. The case δ = 0 and δ = 1 correspond to (1.3) in non-divergence and divergence form, respectively. We take a parameter δ ∈ [0, 1] to cover more general nonlinear terms in (1.2). We note that there are several singularity formation results when ν = 0: 0 < δ < 1 3 and δ = 1 [START_REF] Morlet | Further properties of a continuum of model equations with globally defined flux[END_REF], 0 < δ ≤ 1 [START_REF] Chae | Finite time singularities in a 1D model of the quasi-geostrophic equation[END_REF], and δ = 0 [START_REF] Córdoba | Formation of singularities for a transport equation with nonlocal velocity[END_REF][START_REF] Li | Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation[END_REF][START_REF] Silvestre | On a transport equation with nonlocal drift[END_REF]. By contrast, we look for weak solutions of (1.2) globally in time (see e.g. [START_REF] Dong | Well-posedness for a transport equation with nonlocal velocity[END_REF]). From now on, we set ν = 1 for notational simplicity.

We assume that θ 0 satisfies the conditions

θ 0 (x) > 0, θ 0 ∈ L 1 ∩ H 1 2 . (1.4)
Since (1.2) satisfies the minimum principle (see Section 2) when δ ≥ 0, θ(t, x) ≥ 0 for all time. Moreover, the structure of the nonlinearity enables us to use the following function space

A T = L ∞ 0, T ; L p ∩ H 1 2 ∩ L 2 0, T ; H γ+1 2
for all p ∈ (1, ∞).

Definition 1.1. We say θ is a weak solution of (1.2) if θ ∈ A T and (1.2) holds in the following sense: for any test function

ψ ∈ C ∞ c ([0, T ) × R), T 0 R [θψ t + (Hθ) θψ x + (1 -δ)Λθθψ -θΛ γ ψ] dxdt = R θ 0 (x)ψ(0, x)dx
holds for any 0 < T < ∞.

Theorem 1.1. Let γ ∈ (0, 1) and δ ≥ 1 2 . Then, for any θ 0 satisfying (1.4), there exists a weak solution of (1.2) in A T for all T > 0. Moreover, a weak solution is unique when γ = 1.

When γ = 1, we consider infinite energy solutions of (1.2). More precisely, we take a family of

weights w β = 1 + |x| 2 -β 2 , 0 < β < 1
, and take initial data satisfying

θ 0 (x) ≥ 0, θ 0 ∈ H 1 2 (w β dx) ∩ L ∞ (1.5)
where weighted Sobolev spaces are defined in Section 2. We note that θ 0 can decay (slowly) at infinity. For example, as long as

β + 2η > 1, |θ 0 (x)| ≃ |x| -η , η ≥ 1/2 is allowed to stay in L 2 (w β dx) |x|≥1 |x| 2η (1 + |x| 2 ) β 2 dx < ∞.
But, we can still use the energy method to obtain a weak solution of (1.2). Let

B T = L ∞ 0, T ; H 1 2 (w β dx) ∩ L 2 0, T ; H 1 (w β dx) .
Theorem 1.2. Let γ = 1 and δ ≥ 1 2 . Then, for any θ 0 satisfying (1.5) with θ 0 L ∞ being sufficiently small, there exists a unique weak solution of (1.2) in B T for all T > 0.

In Theorem 1.1 and Theorem 1.2, we have restrictions on the sign of initial data and the range of δ. We can remove these conditions by looking for a solution of (1.2) in function spaces defined by the Fourier transform. Let

A α = f ∈ L 1 loc : f A α = R (1 + |ξ| α )| f (ξ)|dξ < ∞ .
We also define

W T = L ∞ 0, T ; W 1,∞ ∩ W 1,∞ (0, T ; L ∞ ) ∩ L 1 (0, T ; W 2,∞ ).
Theorem 1.3. Let γ = 1 and δ ∈ R. Then, for any θ 0 ∈ A 1 with

θ 0 A 0 < √ π √ 2(1 + |δ|) , (1.6)
there exists a unique weak solution of (1.2) verifying the following inequality for all T > 0

θ ∈ W T , sup t∈[0,T ] θ(t) A 1 + 1 - √ 2(1 + |δ|) θ 0 A 0 √ π T 0 θ x (t) A 1 dt ≤ θ 0 A 1 .
We note that θ 0 ∈ A 1 can have infinite energy. For example, we take

θ 0 (ξ) = e -|ξ| √ |ξ| for ξ = 0. Then, θ 0 ∈ A 1 but θ 0 / ∈ L 2 .
1.2. The case N = (1 -∂ xx ) -α and δ = 0. In this case, (1.1) is changed to the equation

θ t + uθ x + Λ γ θ = 0, u = (1 -∂ xx ) -α θ. (1.7)
This equation is closely related to a generalized Proudman-Johnson equation [START_REF] Okamoto | Some similarity solutions of the Navier-Stokes equations and related topics[END_REF][START_REF] Proudman | Boundary-layer growth near a rear stagnation point[END_REF][START_REF] Wunsch | The Generalized Proudman-Johnson Equation Revisited[END_REF]:

f txx + f f xxx + δf x f xx = νf xxxx
which is derived from the 2D incompressible Navier-Stokes equations via the separation of space variables when δ = 1. By taking w = f xx ,

w t + f w x + δf x w = νw xx , f = (∂ xx ) -1 w.
The inviscid case with δ = 2 is equivalent to the Hunter-Saxton equation arising in the study of nematic liquid crystals [START_REF] Hunter | Dynamics of director fields[END_REF]. The equation (1.7) is also considered as a model equation of the Lagrangian averaged Navier-Stokes equations [START_REF] Marsden | The anisotropic Lagrangian averaged Euler and Navier-Stokes equations[END_REF] which are given by

∂ t 1 -σ 2 ∆ u + u • ∇ 1 -σ 2 ∆ u + (∇u) T • 1 -σ 2 ∆ u = -∇p + ν∆ 1 -σ 2 ∆ u, ∇ • u = 0
We first deal with (1.7) with initial data in

L 2 ∩ L ∞ . Let C T = L ∞ (0, T ; L p ) ∩ L 2 0, T ; H γ 2 for all p ∈ [2, ∞].
Definition 1.2. We say θ is a weak solution of (1.7) if θ ∈ C T and (1.7) holds in the following sense: for any test function

ψ ∈ C ∞ c ([0, T ) × R), T 0 R [θψ t + u x θψ + uθψ x -θΛ γ ψ] dxdt = R θ 0 (x)ψ(0, x)dx
holds for any 0 < T < ∞.

Theorem 1.4. Let γ ∈ (0, 2) and α = 1 2 -γ 4 . Then, for any θ 0 ∈ L 2 ∩ L ∞ , there exists a weak solution of (1.7) in C T for all T > 0.

We note that θ 0 ∈ L 2 ∩ L ∞ is enough to construct a weak solution in Theorem 1.4, but we do not know whether it is unique or not.

When γ = 1, we consider weights w β = 1 + |x| 2 -β 2 with 0 < β < 1, and take initial data in

H 1 (w β dx) ∩ L ∞ . Let α = 1 4 and D T = L ∞ 0, T ; H 1 (w β dx) ∩ L 2 0, T ; H 3 2 (w β dx) .
Theorem 1.5. Let γ = 1 and α = 1 4 . Then, for any θ 0 ∈ H 1 (w β dx) ∩ L ∞ , there exists a unique global weak solution of (1.7) in D T for all T > 0.

Compared to Theorem 1.2, we do not assume that θ 0 L ∞ is small to prove Theorem 1.5.

In Theorem 1.4 and Theorem 1.5, we have restrictions on the range of α. Again, we can remove these conditions by looking for a solution of (1.7) in function spaces defined by the Fourier variables.

Theorem 1.6. Let γ = 1 and α ≥ 0. Then, for any θ 0 ∈ A 1 satisfying

θ 0 A 0 < √ π √ 2 , (1.8)
there exists a unique weak solution of (1.7) verifying the following inequality for all T > 0

θ ∈ W T , sup t∈[0,T ] θ(t) A 1 + 1 - √ 2 θ 0 A 0 √ π T 0 θ x (t) A 1 dt ≤ θ 0 A 1 .
Remark 1. We note that Theorem 1.6 remains valid with straightforward changes in the spirit of Theorem 1.3 when δ = 0.

Preliminaries

All constants will be denoted by C that is a generic constant. In a series of inequalities, the value of C can vary with each inequality. For s ∈ R, H s is a Hilbert space with

f 2 H s = R (1 + |ξ| 2 ) s f (ξ) 2 dξ.
2.1. Hilbert transform and fractional Laplacian. The Hilbert transform is defined as

Hf (x) = p.v. R f (y)
xy dy.

The differential operator Λ γ = ( √ -∆) γ is defined by the action of the following kernels [START_REF] Córdoba | A maximum principle applied to quasi-geostrophic equations[END_REF]:

Λ γ f (x) = c γ p.v. R f (x) -f (y) |x -y| 1+γ dy, (2.1)
where

c γ > 0 is a normalized constant. When γ = 1, Λf (x) = Hf x (x).
Moreover, we have the following identity:

H (θ x (Hθ x )) = 1 2 (Λθ) 2 -(θ x ) 2 . (2.2)
We also recall the following pointwise property of Λ α .

Lemma 2.1. [START_REF] Córdoba | A maximum principle applied to quasi-geostrophic equations[END_REF] Let 0 ≤ α ≤ 2 and f ∈ S. Then,

f (x)Λ α f (x) ≥ 1 2 Λ α f 2 (x) , f 2 (x)Λf (x) ≥ 1 3 Λ f 3 (x) when f ≥ 0 2.2.
Minimum and Maximum Principles. In Theorem 1.1 and 1.2, we assume θ 0 > 0. To obtain global-in-time solutions, we need θ(t, x) ≥ 0 for all time. We first assume that θ(t, x) ∈ C 1 ([0, T ] × R) and x t be a point such that m(t) = θ(t, x t ). If m(t) > 0 for all time, nothing is left to prove. So, we check a point (t, x t ) where m(t) = 0. Since m(t) is a continuous Lipschitz function, it is differentiable at almost every t by Rademacher's theorem. From the definition of Λ γ ,

d dt m(t) = -δθ(t, x t )p.v. R θ(t, x t ) -θ(t, y) |x t -y| 1+γ dy -p.v. R θ(t, x t ) -θ(t, y) |x t -y| 1+γ dy ≥ -δp.v. R θ(t, x t ) -θ(t, y) |x t -y| 1+γ dy m(t).
Since the quantity in the bracket is nonnegative when δ ≥ 0, we have that m(t) is non-decreasing in time if θ 0 > 0 and thus θ(t, x) ≥ 0 for all time. Similarly, maximum values of θ(t, x) are nonincreasing in time when θ 0 > 0 with θ 0 ∈ L ∞ . For general initial data satisfying (1.4) and (1.5), we can use regularization method. For such a regularized problem with smooth solution θ ǫ , the same argument works. Then, we construct θ as the limit of θ ǫ . As θ will be also the pointwise limit of θ ǫ almost everywhere, we conclude that θ(t, x) ≥ 0. Since (1.7) is purely a dissipative transport equation, we immediately have that

θ(t) L ∞ ≤ θ 0 L ∞ .
2.3. The Wiener spaces A α . The Wiener space is defined as

A 0 = f ∈ L 1 loc : f (ξ) ∈ L 1 ,
where f denotes the Fourier transform of f

f (ξ) = 1 √ 2π R f (x)e -ix•ξ dx.
A 0 is a Banach space endowed with the norm

f A 0 = f L 1 .
Furthermore, using Fubini's Theorem, A 0 is a Banach algebra, i.e.

f g A 0 ≤ f A 0 g A 0 .
Once we have defined A 0 , we can define the full scale of homogeneous, Ȧα , and inhomogeneous, A α , Wiener spaces as

Ȧα = f ∈ L 1 loc : f Ȧα = R |ξ| α | f (ξ)|dξ < ∞ , A α = f ∈ L 1 loc : f A α = R (1 + |ξ| α )| f (ξ)|dξ . (2.3)
For these spaces, the following inequalities hold

f C(R) ≤ f A 0 (R d ) ∀ f ∈ A 0 (R) (2.4) f Ȧα (R) ≤ f 1-θ A 0 (R d ) f θ Ȧ α θ (R d ) ∀ 0 < θ < 1, α ≥ 0, f ∈ A 0 (R) ∩ A α θ (R). (2.5)
As a consequence of (2.4), we obtain that if u ∈ A 0 has infinite energy then lim sup

|x|→∞ |u(x)| + lim inf |x|→∞ |u(x)| < ∞.
2.4. Commutator estimate. In the proof of Theorem 1.1, we need to estimate a commutator term involving Λ 1 2 . To do this, we first recall Hardy-Littlewood-Sobolev inequality in 1D. Let

K α (x) = 1 |x| λ and T λ f = K λ * f . Then, T λ f L q ≤ C f L p , 1 q + 1 = 1 p + λ. Lemma 2.2. For f ∈ L 3 2 , g ∈ L 3 2 and ψ ∈ W 1,∞ , Λ 1 2 , ψ f -Λ 1 2 , ψ g L 6 ≤ C ψ W 1,∞ f -g L 3 2 .
Proof. By the definition of Λ

1 2 , we have Λ 1 2 , ψ f -Λ 1 2 , ψ g (x) = c 1 p.v. (ψ(y) -ψ(x))(f (y) -g(y)) |x -y| 3 2
dy and thus

Λ 1 2 , ψ f -Λ 1 2 , ψ g (x) ≤ C ∇ψ L ∞ |f (y) -g(y)| |x -y| 1 2
dy.

(2.6)

Using Hardy-Littlewood-Sobolev inequality, we obtain that

Λ 1 2 , ψ f -Λ 1 2 , ψ g L 6 ≤ C ∇ψ L ∞ f -g L 3 2
(2.7) which completes the proof.

2.5. Muckenhoupt weights. We briefly introduce weighted spaces. A weight w is a positive and locally integrable function. A measurable function θ on R belongs to the weighted Lebesgue spaces L p (wdx) with 1 ≤ p < ∞ if and only if

θ p L p (wdx) = R |θ(x)| p w(x)dx < ∞.
An important class of weights is the Muckenhoupt class A p for 1 < p < ∞ [START_REF] Coifman | Wavelets: Calderón-Zygmund and Multilinear Operators[END_REF][START_REF] Muckenhoupt | Weighted norm inequalities for the Hardy maximal function[END_REF]. Let 1 < p < ∞, we say that w ∈ A p if and only if there exists a constant C p,w > 0 such that

sup r>0,x 0 ∈R 1 2r [x 0 -r,x 0 +r] wdx 1 2r [x 0 -r,x 0 +r] w 1 1-p dx p-1 ≤ C p,w .
This class satisfies the following properties. (1) Calderón-Zygmund type operators are bound on L p (wdx) when w ∈ A p and 1 < p < ∞ [START_REF] Stein | Harmonic Analysis: Real variable methods, Orthogonality and Oscillatory Integrals[END_REF].

(2) Let w ∈ A p . We define weighted Sobolev spaces as follows

f ∈ H 1 (wdx) ⇐⇒ f ∈ L 2 (wdx) and f x ∈ L 2 (wdx), f ∈ H 1 (wdx) ⇐⇒ (1 -∂ xx ) 1 2 f ∈ L 2 (wdx) ⇐⇒ f ∈ L 2 (wdx) and Λf ∈ L 2 (wdx), f ∈ H 1 2 (wdx) ⇐⇒ (1 -∂ xx ) 1 4 f ∈ L 2 (wdx) ⇐⇒ f ∈ L 2 (wdx) and Λ 1 2 f ∈ L 2 (wdx).
(2.8)

(3) Gagliardo-Nirenberg type inequalities (see e.g [START_REF] Maz | Sobolev spaces with applications to elliptic partial differential equations[END_REF])

Λ 1 2 f L 2 (wdx) ≤ C f 1 2 L 2 (wdx) Λf 1 2 L 2 (wdx) , θ L 4 (wdx) ≤ C θ 1 2 L 2 (wdx) Λ 1 2 θ 1 2 L 2 (wdx)
.

(2.9) This latter inequality can be proved for instance by using the weighted Sobolev embedding H 1 4 (wdx) ֒→ L 4 (wdx), and then by weighted interpolation one recover the second inequality in (2.9).

In this paper, we take weights w β = (1 + |x| 2 ) -β 2 , 0 < β < 1, which belongs to the A p class of Muckenhoupt for all 1 < p < ∞. These weights also satisfy the following properties. For the proofs, see [START_REF] Lazar | Infinite energy solutions for a 1D transport equation with nonlocal velocity[END_REF].

Lemma 2.3. Let w β = (1 + |x| 2 ) -β 2 , 0 < β < 1. (1) For 2 ≤ p < ∞ such that β(1 -p -1 ) < 1/2, the commutator 1 w β Λ 1 2 , w β is bounded from L p (w β dx) to L p (w β dx). (2) |∂ x w β (x)| ≤ Cw β (x) and |Λw β (x)| ≤ Cw β (x)
, where C > 0 depends only on β.

Note that one can also derive commutator estimates for generalized Muckenhoupt weights of the type

w β = (1 + |x| k ) -β k ∈ A ∞
where k is an even integer (see [START_REF] Lazar | On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion[END_REF] for instance). However, our aim here is just to show the existence of global infinite energy solutions not to be optimal in the family of weights.

2.6. Compactness. Since we look for weak solutions, we use compactness arguments when we pass to the limit in weak formulations.

Lemma 2.4. [START_REF] Temam | Navier-Stokes Equations: theory and numerical analysis[END_REF] Let X 0 , X, X 1 be reflexive Banach spaces such that

X 0 ⊂⊂ X ⊂ X 1 ,
where X 0 is compactly embedded in X. Let T > 0 be a finite number and let α 0 and α 1 be two finite numbers such that α

i > 1. Then, Y = {u ∈ L α 0 (0, T ; X 0 ) , ∂ t u ∈ L α 1 (0, T ; X 1 )} is compactly embedded in L α 0 (0, T ; X). Lemma 2.5 ([9]). Consider a sequence (θ ǫ ) ∈ C([0, T ] × B R (0)) that is uniformly bounded in L ∞ ([0, T ], W 1,∞ (B R (0))). Assume further that the weak derivative dθ ǫ dt is in L ∞ ([0, T ], L ∞ (B R (0))) (not necessarily uniform) and is uniformly bounded in L ∞ ([0, T ], W -2,∞ * (B R (0))). Finally suppose that θ ǫ x ∈ C([0, T ] × B R (0)).
Then there exists a subsequence of (θ ǫ ) that converges strongly in

L ∞ ([0, T ] × B R (0)).
3. Proof of Theorem 1.1 3.1. A priori estimates. We first obtain a priori bounds of the equation

θ t + (Hθ) θ x + δθΛθ + Λ γ θ = 0, (3.1)
We note that by the minimum principle applied to (3.1), we have θ(t, x) ≥ 0 for all t ≥ 0.

To obtain H 1 2 bound of θ, we begin with the L 2 bound. We multiply (3.1) by θ and integrate over R. Then,

1 2 d dt θ 2 L 2 + Λ γ 2 θ 2 L 2 = -[(Hθ) θ x θ] dx -δ θ 2 Λθ dx = 1 2 -δ θ 2 Λθ dx.
Since θ ≥ 0, we have

θ 2 Λθ dx = (θ(x) -θ(y)) 2 |x -y| 2 • θ(x) + θ(y) 2 dxdy ≥ 0
and thus

θ(t) 2 L 2 + 2 t 0 Λ γ 2 θ(s) 2 L 2 ds ≤ θ 0 2 L 2 . (3.2)
We next estimate θ in Ḣ 1 2 . We multiply (3.1) by Λθ and integrate over R:

1 2 d dt Λ 1 2 θ 2 L 2 + Λ 1+γ 2 θ 2 L 2 = -[(Hθ) θ x Λθ] dx -δ θ (Λθ) 2 dx.
By (2.2), we have

-[(Hθ) θ x Λθ] dx = [θH (θ x (Hθ x ))] dx = 1 2 θ (Λθ) 2 -(θ x ) 2 dx,
and hence 1 2

d dt Λ 1 2 θ 2 L 2 + Λ 1+γ 2 θ 2 L 2 = 1 2 -δ θ (Λθ) 2 dx - 1 2 θ (θ x ) 2 dx ≤ 0,
where we use the sign conditions θ ≥ 0 and δ ≥ 1 2 . This leads to the inequality

Λ 1 2 θ(t) 2 L 2 + 2 t 0 Λ 1+γ 2 θ(s) 2 L 2 ds ≤ Λ 1 2 θ 0 2 L 2 . (3.3)
By (3.2) and (3.3), we obtain that

θ(t) 2 H 1 2 + 2 t 0 Λ γ 2 θ(s) 2 H 1 2 ds ≤ θ 0 2 H 1 2 . ( 3.4)
We finally estimate θ in L 1 . Since θ ≥ 0,

d dt θ L 1 = d dt θdx = (1 -δ) θΛθdx ≤ C Λ 1 2 θ 2 L 2
and thus we conclude that

θ(t) L 1 ≤ θ 0 L 1 + C t 0 θ(s) 2 Ḣ 1 2 ds ≤ θ 0 L 1 + Ct θ 0 2 H 1 2
.

(3.5)

3.2. Approximation and passing to limit. We first regularize initial data as θ ǫ 0 = ρ ǫ * θ 0 where ρ ǫ is a standard mollifier. We then regularize the equation by putting the Laplacian with the coefficient ǫ:

θ ǫ t + (Hθ ǫ ) θ ǫ x + δθ ǫ Λθ ǫ + Λ γ θ ǫ = ǫθ ǫ xx . (3.6)
For the proof of the existence of a global-in-time smooth solution, see [START_REF] Lazar | Infinite energy solutions for a 1D transport equation with nonlocal velocity[END_REF] (Section 6). Moreover, (θ ǫ ) satisfies that

θ ǫ (t) L 1 + θ ǫ (t) 2 H 1 2 + 2 t 0 Λ γ 2 θ ǫ (s) 2 H 1 2 ds + ǫ ∇θ ǫ 2 H 1 2 ≤ θ 0 L 1 + C(1 + t) θ 0 2 H 1 2
.

Therefore, (θ ǫ ) is bounded in A T uniformly in ǫ > 0. From this, we have uniform bounds

Hθ ǫ ∈ L 4 0, T ; L 4 , θ ǫ ∈ L 2 0, T ; L 4
and hence

((Hθ ǫ ) θ ǫ ) x ∈ L 4 3 0, T ; H -1 .
Moreover,

Λ γ θ ǫ + ǫθ ǫ xx ∈ L 2 0, T ; H -1 .
To estimate θ ǫ Λθ ǫ , we use the duality argument. For any χ ∈ L 2 0, T ;

H 2 , | θ ǫ Λθ ǫ , χ | ≤ θ ǫ Λθ ǫ (ξ) χ(ξ) dξ ≤ θ ǫ (ξ -η) |η| θ ǫ (η) | χ(ξ)| dηdξ ≤ θ ǫ (ξ -η) |η| 1 2 |ξ -η| 1 2 + |ξ| 1 2 θ ǫ (η) | χ(ξ)| dηdξ = |ξ -η| 1 2 θ ǫ (ξ -η) |η| 1 2 θ ǫ (η) | χ(ξ)| dηdξ + θ ǫ (ξ -η) |η| 1 2 θ ǫ (η) |ξ| 1 2 | χ(ξ)| dηdξ ≤ Λ 1 2 θ ǫ 2 L 2 χ L 1 + Λ 1 2 θ ǫ L 2 θ L 1 Λ 1 2 χ L 2 ≤ C (1 + Λ) 1 2 + γ 4 θ ǫ 2 L 2 χ H 2 .
Since (1 + Λ)

1 2 + γ 4 θ ǫ ∈ L 4 0, T ; L 2 uniformly in ǫ > 0, we have | θ ǫ Λθ ǫ , χ | dt ≤ C (1 + Λ) 1 2 + γ 4 θ ǫ 2 L 4 T L 2 χ L 2 T H 2 .
This implies that θ ǫ Λθ ǫ ∈ L 2 0, T ; H -2 . So, we conclude that from the equation of

θ ǫ t θ ǫ t ∈ L 4 3 0, T ; H -2 .
We now extract a subsequence of (θ ǫ ), using the same index ǫ for simplicity, and a function θ ∈ A T such that

θ ǫ ⋆ ⇀ θ in L ∞ 0, T ; L p ∩ H 1 2
for all p ∈ (1, ∞),

θ ǫ ⇀ θ in L 2 0, T ; H γ+1 2 , θ ǫ → θ in L 2 0, T ; H 1 2 , θ ǫ → θ in L 2 0, T ; L p loc for all p ∈ (1, ∞) (3.7)
where we use Lemma 2.4 for the strong convergence. We now multiply (3.6) by a test function ψ ∈ C ∞ c ([0, T ) × R) and integrate over R. Then,

T 0 [θ ǫ ψ t + (Hθ ǫ ) θ ǫ ψ x + (1 -δ)Λθ ǫ θ ǫ ψ -θ ǫ Λψ + ǫθ ǫ ψ xx ] dxdt = θ ǫ 0 (x)ψ(0, x)dx
which can be rewritten as

T 0 θ ǫ ψ t + (Hθ ǫ ) θ ǫ ψ x I -θ ǫ Λψ + ǫθ ǫ ψ xx dxdt -θ ǫ 0 (x)ψ(0, x)dx = -(1 -δ) T 0 Λ 1 2 θ ǫ Λ 1 2 , ψ θ ǫ II dxdt -(1 -δ) T 0 Λ 1 2 θ ǫ 2 ψ III dxdt. (3.8)
By Lemma 2.4 with

X 0 = L 2 0, T ; H 1 2 , X = L 2 0, T ; L 2 loc , X 1 = L 2 0, T ; H -2 ,
we can pass to the limit to I. Moreover, since

Λ 1 2 , ψ θ ǫ → Λ 1 2 , ψ θ
strongly in L 2 0, T ; L 6 by Lemma 2.2 and Λ 1 2 θ ǫ converges weakly in L 2 0, T ; L 2 by (3.7), we can pass to the limit to II. Lastly, Lemma 2.4 with

X 0 = L 2 0, T ; H 1+γ 2 , X = L 2 0, T ; H 1 2 loc , X 1 = L 2 0, T ; H -2 ,
allows to pass to the limit to III. Combining all the limits together, we obtain that

T 0 [θψ t + (Hθ) θψ x + (1 -δ)Λθθ ǫ ψ] dxdt = θ 0 (x)ψ(0, x)dx.
(3.9)

3.3. Uniqueness when γ = 1. To show the uniqueness of a weak solution, let θ = θ 1θ 2 . Then, θ satisfies the following equation:

θ t + Λθ = -(Hθ) θ 1x -(Hθ 2 ) θ x -δθΛθ 1 -δθ 2 Λθ, θ(0, x) = 0. (3.10)
We multiply θ to (3.10) and integrate over R. Then,

1 2 d dt θ 2 L 2 + Λ 1 2 θ 2 L 2 = [-(Hθ) θ 1x -(Hθ 2 ) θ x -δθΛθ 1 -δθ 2 Λθ] θdx.
The first three terms in the right-hand side are easily bounded by

C θ 1x L 2 θ 2 L 4 + C θ 2x L 2 θ 2 L 4 .
Moreover, the last term is bounded by using Lemma 2.1

-δ θ 2 θΛθdx ≤ - δ 2 θ 2 Λθ 2 dx = - δ 2 θ 2 Λθ 2 dx ≤ C θ 2x L 2 θ 2 L 4 .
Hence we derive that

d dt θ 2 L 2 + Λ 1 2 θ 2 L 2 ≤ C θ 1x L 2 θ 2 L 4 + C θ 2x L 2 θ 2 L 4 ≤ C θ 1x 2 L 2 + θ 2x 2 L 2 θ 2 L 2 + 1 2 Λ 1 2 θ 2 L 2 . Since θ 1x ∈ L 2 0, T : L 2 , θ 2x ∈ L 2 0, T : L 2
when γ = 1, we conclude that θ = 0 in L 2 and thus a weak solution is unique. This completes the proof of Theorem 1.1.

4.

Proof of Theorem 1.2

4.1.

A priori estimate. We consider the equation

θ t + (Hθ) θ x + δθΛθ + Λθ = 0. (4.1)
Since (4.1) satisfies the minimum and maximum principles, we have

θ(t, x) ≥ 0, θ(t) L ∞ ≤ θ 0 L ∞ .
We begin with the L 2 (w β dx) bound. For notational simplicity, we suppress the dependence of β.

We multiply (4.1) by θw and integrate in x. Then, Since 1 2δ ≤ 0 and θ ≥ 0, we use Lemma 2.1 to have

1 2 -δ θ 2 (Λθ) wdx ≤ - 1 3 1 2 -δ Λ θ 3 wdx = - 1 3 1 2 -δ θ 3 Λwdx ≤ C θ L ∞ θ 2 wdx ≤ C θ 0 L ∞ θ 2 L 2 (wdx) ,
where we use Lemma 2.3 to bound Λw by w. Moreover, by the L 2 (wdx) boundedness of the Hilbert transform, we also have 1 2 (Hθ)

θ 2 w x dx ≤ C θ L ∞ |θ| |Hθ| wdx ≤ C θ 0 L ∞ θ 2 L 2 (wdx) .
We finally estimate the commutator term. By Lemma 2.3,

Λ 1 2 θ Λ 1 2 , w θdx ≤ Λ 1 2 θ w 1 w Λ 1 2 , w θ dx ≤ C Λ 1 2 θ L 2 (wdx) 1 w Λ 1 2 , w θ L 2 (wdx) ≤ C Λ 1 2 θ L 2 (wdx) θ L 2 (wdx) ≤ 1 2 Λ 1 2 θ 2 L 2 (wdx) + C θ 2 L 2 (wdx) .
Collecting all terms together, we obtain that

d dt θ 2 L 2 (wdx) + Λ 1 2 θ 2 L 2 (wdx) ≤ C(1 + θ 0 L ∞ ) θ 2 L 2 (wdx) . (4.2)
We next multiply (4.1) by Λ

d dt Λ 1 2 θ L 2 (wdx) + Λθ L 2 (wdx) = -(Hθ) θ x Λ 1 2 wΛ 1 2 θ dx -δ θ (Λθ) Λ 1 2 wΛ 1 2 θ dx -Λθ Λ 1 2 , w Λ 1 2 θdx = I+II+III. 2
We note that since δ > 0 and θ ≥ 0

II = -δ θ |Λθ| 2 wdx -δ θΛθ Λ 1 2 , w Λ 1 2 θdx ≤ -δ θΛθ Λ 1 2 , w Λ 1 2 θdx (4.3)
and thus we only need to estimate I and III and the right-hand side of (4.3). (There is an extra term θ in the right-hand side of (4.3) but we can take the L ∞ norm to θ and it does not affect the proof.) These bounds are obtained in [START_REF] Lazar | Infinite energy solutions for a 1D transport equation with nonlocal velocity[END_REF]:

I+III -δ Λθ Λ 1 2 , w Λ 1 2 θdx ≤ C θ 0 L ∞ + θ 0 4 L ∞ θ 2 L 2 (wdx) + C Λ 1 2 θ 2 L 2 (wdx) + C θ 0 L ∞ Λθ 2 L 2 (wdx) + 1 2 Λθ 2 L 2 (wdx) . Hence we have that d dt Λ 1 2 θ 2 L 2 (wdx) + Λθ 2 L 2 (wdx) ≤ C θ 0 L ∞ + θ 0 4 L ∞ θ 2 L 2 (wdx) + C Λ 1 2 θ 2 L 2 (wdx) + C θ 0 L ∞ Λθ 2 L 2 (wdx) (4.4)
By (4.2) and (4.4),

d dt θ 2 H 1 2 (wdx) + Λ 1 2 θ 2 L 2 (wdx) + Λθ 2 L 2 (wdx) ≤ C θ 0 L ∞ + θ 0 4 L ∞ θ 2 H 1 2 (wdx) + C θ 0 L ∞ Λθ 2 L 2 (wdx) . If θ 0 L ∞ is sufficiently small, d dt θ 2 H 1 2 (wdx) + Λ 1 2 θ 2 H 1 2 (wdx) ≤ C θ 0 L ∞ + θ 0 4 L ∞ θ 2 H 1 2 (wdx)
and hence we derive the following inequality

θ(t) 2 H 1 2 (wdx) + t 0 Λ 1 2 θ(s) 2 H 1 2 (wdx) ds ≤ θ 0 2 H 1 2 (wdx) exp C θ 0 L ∞ + θ 0 4 L ∞ t . (4.5)
4.2. Approximation and passing to limit. To show the existence of a weak solution in D T , we first approximate the initial data θ 0 . Let χ be a smooth positive function such that χ(x) = 1 for |x| ≤ 1 and χ(x) = 0 for |x| ≥ 2. Let χ R (x) = χ(x/N ), N ∈ N, and consider truncated initial data θ N 0 (x) = θ 0 (x)χ N (x). Then, a direct computation shows that lim

N →∞ θ N 0 -θ 0 H 1 2 (wdx) = 0.
Moreover, this truncation does not alter the non-negativity and does not increase the L ∞ norm. So, if θ 0 L ∞ is sufficiently small, there is a global-in-time solution of

∂ t θ N + Hθ N ∂ x θ N + δθ N Λθ N + Λθ N = 0, θ N (0, x) = θ N 0 (x). (4.6)
¿From the a priori estimates, the sequence (θ N ) is bounded in

L ∞ ([0, T ], H 1 2 (wdx)) ∩ L 2 ([0, T ], H 1 (wdx)) uniformly with respect to N . We now take a test function ψ ∈ C ∞ c ([0, T ) × R). Then, ψθ N is bounded in L 2 ([0, T ], H 1 ). Moreover, since θ N ∈ L ∞ ([0, T ] × R) ψ Hθ N ∂ x θ N + δθ N Λθ N + Λθ N = (ψHθ N θ N ) x -ψ x Hθ N θ N + (δ -1)ψθ N Λθ N + ψΛθ N ∈ L 2 ([0, T ], H -1 ).
By Lemma 2.4, we can pass to the limit to the weak formulation,

T 0 θ N ψ t + Hθ N θ N ψ x + (1 -δ)Λθ N θ N ψ -θ N Λψ dxdt = θ N 0 (x)ψ(0, x)dx,
to obtain a weak solution θ which is also in

L ∞ ([0, T ], H 1 2 (wdx)) ∩ L 2 ([0, T ], H 1 (wdx)).
4.3. Uniqueness. To show the uniqueness of a weak solution, we consider the equation of θ = θ 1θ 2 given by

θ t + Λθ = -(Hθ) θ 1x -(Hθ 2 ) θ x -δθΛθ 1 -δθ 2 Λθ, θ(0, x) = 0. ( 4.7)
We multiply wθ to (4.7) and integrate over R. Then,

1 2 d dt θ 2 L 2 (wdx) + Λ 1 2 θ 2 L 2 (wdx) = [-(Hθ) θ 1x -(Hθ 2 ) θ x -δθΛθ 1 -δθ 2 Λθ] θwdx -Λ 1 2 θ Λ 1 2 , w θdx.
As before, the last term is bounded by

Λ 1 2 θ Λ 1 2 , w θdx ≤ 1 2 Λ 1 2 θ 2 L 2 (wdx) + C θ 2 L 2 (wdx) .
The first three terms in the right-hand side are easily bounded by

C θ 1x L 2 (wdx) + θ 2x L 2 (wdx) + θ 2 L 2 (wdx) θ 2 L 4 (wdx) .
Moreover, since δ > 0, θ 2 ≥ 0 and w ≥ 0, the fourth term is bounded by using Lemma 2.1

-δ θ 2 θΛθwdx ≤ - δ 2 θ 2 wΛθ 2 dx = - δ 2 θ 2 Λ(θ 2 w)dx = δ 2 H(θ 2 )(θ 2 w) x dx ≤ C θ 2x L 2 (wdx) + θ 2 L 2 (wdx) θ 2 L 4 (wdx) . Hence we obtain that d dt θ 2 L 2 (wdx) + Λ 1 2 θ 2 L 2 (wdx) ≤ C θ 1x L 2 (wdx) + θ 2x L 2 (wdx) + θ 2 L 2 (wdx) θ 2 L 4 (wdx) + C θ 2 L 2 (wdx) ≤ C 1 + θ 1x 2 L 2 (wdx) + θ 2x 2 L 2 (wdx) + θ 2 2 L 2 (wdx) θ 2 L 2 (wdx) + 1 2 Λ 1 2 θ 2 L 2 (wdx)
, where we use (2.9) to obtain the last inequality. Since

θ 1x ∈ L 2 0, T : L 2 (wdx) , θ 2x ∈ L 2 0, T : H 1 (wdx) ,
we conclude that θ = 0 in L 2 (wdx) and thus a weak solution is unique. This completes the proof of Theorem 1.2.

5.

Proof of Theorem 1.3

5.1.

A priori estimates. Taking the Fourier transform of (1.2), we have that

∂ t | θ(ξ)| = θ(ξ)∂ t θ(ξ) + θ(ξ)∂ t θ(ξ) 2| θ(ξ)| = Re θ(ξ)∂ t θ(ξ) | θ(ξ)| = -Re R -iζ |ζ| θ(ζ)i(ξ -ζ)) θ(ξ -ζ) -δ θ(ζ)|ξ -ζ| θ(ξ -ζ)dζ θ(ξ) | θ(ξ)| 1 √ 2π -|ξ|| θ|. Consequently, d dt θ A 0 ≤ (1 + |δ|) R R | θ(ζ)||ξ -ζ|| θ(ξ -ζ)|dζdξ 1 √ 2π -θ Ȧ1 ≤ (1 + |δ|) R R | θ(ζ)||ξ -ζ|| θ(ξ -ζ)|dξdζ 1 √ 2π -θ Ȧ1 ≤ (1 + |δ|) θ A 0 √ 2π -1 θ Ȧ1 .
Thus, if θ 0 satisfies the condition (1.6), we have

θ(t) A 0 + 1 - (1 + |δ|) θ 0 A 0 √ 2π ∞ 0 θ(s) Ȧ1 ds ≤ θ 0 A 0 . (5.1)
Similarly,

∂ t |ξ θ| = -Re R -iζ |ζ| θ(ζ)(i(ξ -ζ)) 2 θ(ξ -ζ) + |ζ| θ(ζ)i(ξ -ζ)) θ(ξ -ζ)dζ -δ R iζ θ(ζ)|ξ -ζ| θ(ξ -ζ) + θ(ζ)i(ξ -ζ)|ξ -ζ| θ(ξ -ζ)dζ iξ θ(ξ) |ξ|| θ(ξ)| 1 √ 2π -|ξ| 2 | θ|.
Thus, using (2.5), we have that

d dt θ Ȧ1 ≤ θ A 0 θ Ȧ2 + θ 2 A 1 1 + |δ| √ 2π -θ Ȧ2 ≤ 2(1 + |δ|) θ A 0 √ 2π -1 θ Ȧ2 .
As a consequence, we obtain that, if θ 0 satisfies the condition (1.6), we also have

θ(t) Ȧ1 + 1 - 2(1 + |δ|) θ 0 A 0 √ 2π ∞ 0 θ(s) Ȧ2 ds ≤ θ 0 Ȧ1 . (5.2)
By (5.1) and (5.2), we conclude that

θ(t) A 1 + 1 - 2(1 + |δ|) θ 0 A 0 √ 2π t 0 θ x (s) A 1 ds ≤ θ 0 A 1 (5.3)
for all t ≥ 0.

5.2. Approximation and passing to the limit. Define e ǫ∂ 2 x the heat semigroup, i.e.

e ǫ∂ 2 x f (x) = e -ǫξ 2 f (ξ), and g ǫ (x) = e -ǫx 2 . Note that

ĝǫ (ξ) = 1 √ 2ǫ e -ξ 2 4ǫ , ĝǫ L 1 = √ 2π.
Given θ 0 (x) ∈ A 1 , we consider θ ǫ 0 (x) = g ǫ (x)e ǫ∂ 2 x θ 0 (x). As θ 0 (x) is a bounded function, we have that θ ǫ 0 is infinitely smooth and has finite total mass:

θ ǫ 0 L 1 ≤ g ǫ L 1 e ǫ∂ 2 x θ 0 L ∞ ≤ π ǫ θ 0 L ∞ .
Furthermore, using Young's inequality and the definition of g ǫ ,

θ ǫ 0 A 0 = 1 √ 2π ĝǫ * e -ǫξ 2 θ0 L 1 ≤ e -ǫξ 2 θ0 L 1 ≤ θ 0 A 0 . Similarly, θ ǫ 0 Ȧ1 = ∂ x θ ǫ 0 A 0 ≤ θ 0 Ȧ1 + ∂ x g ǫ e ǫ∂ 2 x θ 0 A 0 = θ 0 Ȧ1 + cǫ θ 0 A 0 . Now we define the approximated problems θ ǫ t + (Hθ ǫ ) θ ǫ x + δθ ǫ Λθ ǫ + Λθ ǫ = ǫ∂ 2 x θ ǫ , (5.4)
with finite energy approximated initial data θ ǫ 0 . These problems have unique smooth solutions denoted by θ ǫ . Moreover, (θ ǫ ) satisfies a uniform bound

θ ǫ (t) A 1 + 1 - 2(1 + |δ|) θ 0 A 0 √ 2π t 0 θ ǫ x (s) A 1 ds + ǫ t 0 θ ǫ x (s) A 2 ds ≤ θ 0 A 1 .
uniformly in ǫ. Thus, the a priori estimates lead to the following uniform-in-ǫ bounds

sup t∈[0,∞) θ ǫ (t) C 0 (R) ≤ sup t∈[0,∞) θ ǫ (t) A 0 ≤ θ 0 A 0 < √ 2π 1 + |δ| , sup t∈[0,∞) θ ǫ (t) Ċ1 (R) ≤ sup t∈[0,∞) θ ǫ (t) Ȧ1 < θ 0 Ȧ1 + cǫ θ 0 A 0 , sup t∈[0,∞) Λθ ǫ (t) C 0 (R) < θ 0 Ȧ1 + cǫ θ 0 A 0 . (5.5)
Moreover, from the equation (5.4) we also obtain uniform bounds

sup t∈[0,∞) ∂ t θ ǫ (t) C 0 (R) + sup t∈[0,∞) ∂ t Hθ ǫ (t) C 0 (R) ≤ 2 sup t∈[0,∞) ∂ t θ ǫ (t) A 0 (R) ≤ F 1 ( θ 0 A 1 , δ), Hθ ǫ C 1 ([0,∞)×R) + θ ǫ C 1 ([0,∞)×R) ≤ F 2 ( θ 0 A 1 , δ) (5.6)
where F 1 and F 2 only depend on the quantities in the right-hand side of (5.5). Due to Banach-Alaoglu Theorem, there exists a subsequence (denoted by ǫ) and a limit function

θ ∈ W 1,∞ ([0, ∞)× R) such that θ ǫ ⋆ ⇀ θ in L ∞ 0, T ; W 1,∞ , Λθ ǫ ⋆ ⇀ Λθ in L ∞ (0, T ; L ∞
) for all T > 0. Using Lemma 2.5, we have the following strong convergence

lim ǫ→0 θ ǫ -θ L ∞ (K) + Hθ ǫ -Hθ L ∞ (K) = 0 for K = [0, T ] × [-R, R] for any R > 0. Since L ∞ (K) ⊂ L 2 (K)
, we can pass to the limit to the weak formulation

T 0 [θ ǫ ψ t + (Hθ ǫ ) θ ǫ ψ x + (1 -δ)Λθ ǫ θ ǫ ψ -θ ǫ Λψ + ǫθ ǫ ψ xx ] dxdt = θ ǫ 0 (x)ψ(0, x)dx
to obtain a weak solution θ satisfying

θ ∈ W T , sup t∈[0,T ] θ(t) A 1 + 1 - √ 2 θ 0 A 0 √ π T 0 θ x (t) A 1 dt ≤ θ 0 A 1
for all T > 0.

5.3. Uniqueness. To show the uniqueness of a weak solution, we consider the equation of θ = θ 1θ 2 given by

θ t + Λθ = -(Hθ) θ 1x -(Hθ 2 ) θ x -δθΛθ 1 -δθ 2 Λθ, θ(0, x) = 0. (5.7)
Taking the Fourier transform of (5.7) and multiply by

θ | θ| , we have d dt θ A 0 + θ Ȧ1 ≤ 1 + |δ| √ 2π θ 1 Ȧ1 θ A 0 + 1 + |δ| √ 2π θ 2 A 0 θ Ȧ1 . (5.8) Since (1 + |δ|) θ 2 A 0 √ 2π < 1, θ 1 (t) Ȧ1 ∈ L 1 ([0, ∞)),
we have θ(t, x) = 0 in A 0 for all time. This implies that a weak solution is unique.

6. Proof of Theorem 1.4

6.1. A priori estimate. We consider the equation

θ t + uθ x + Λ γ θ = 0, u = (1 -∂ xx ) -α θ (6.1)
Since (6.1) satisfies the maximum principle, we have

θ(t) L ∞ ≤ θ 0 L ∞ .
We also obtain that 1 2

d dt θ 2 L 2 + Λ γ 2 θ 2 L 2 = -uθ x θdx ≤ θ L ∞ u x L 2 θ L 2 ≤ C θ 0 L ∞ + θ 0 2 L ∞ θ 2 L 2 + 1 2 Λ γ 2 θ 2 L 2
where we use the condition α = L 2 e C(1+ θ 0 2 L ∞ )t .

6.2. Approximation and passing to limit. We consider the following equation with regularized initial data:

θ ǫ t + u ǫ θ ǫ x + Λ γ θ ǫ = 0, θ ǫ 0 = ρ ǫ * θ 0 . (6.2)
Then, there exists a global-in-time smooth solution θ ǫ . Moreover, θ ǫ satisfies that

θ ǫ (t) L ∞ + θ ǫ (t) 2 L 2 + t 0 Λ γ 2 θ ǫ (s) 2 L 2 ds + ǫ ∇θ ǫ 2 L 2 ≤ θ 0 L ∞ + θ 0 2
L 2 e C(1+ θ 0 2 L ∞ )t . (6.3) Therefore, (θ ǫ ) is bounded in C T uniformly in ǫ > 0. This implies the uniform bounds u ǫ ∈ L 4 0, T ; L 4 , θ ǫ ∈ L 4 0, T ; L 4 , u ǫ x ∈ L 2 0, T ; L 2 . Moreover, these bounds with the equation

θ ǫ t = -u ǫ θ ǫ x -Λ γ θ ǫ + ǫθ ǫ xx = -(u ǫ θ ǫ ) x + u ǫ x θ ǫ -Λ γ θ ǫ + ǫθ ǫ xx
we also have that θ ǫ t ∈ L 4 3 0, T ; H -2 . We now extract a subsequence of (θ ǫ ), using the same index ǫ for simplicity, and a function θ ∈ C T and u = (1 -∂ xx ) -α θ such that θ ǫ ⋆ ⇀ θ in L ∞ (0, T ; L p ) for all p ∈ (1, ∞),

θ ǫ ⇀ θ in L 2 0, T ; H γ 2 , θ ǫ → θ in L 2 0, T ; L p loc for all p ∈ (1, ∞), u ǫ x ⇀ u x in L 2 0, T ; L 2 , (6.4)
where we use Lemma 2.4 to obtain the strong convergence. We now multiply (6.2) by a test function ψ ∈ C ∞ c ([0, T ) × R) and integrate over R. Then, T 0 [θ ǫ ψ t + u ǫ θ ǫ ψ x + u ǫ x θ ǫ ψθ ǫ Λ γ ψ + ǫθ ǫ ψ xx ] dxdt = θ ǫ 0 (x)ψ(0, x)dx.

By Lemma 2.4 with

X 0 = L 2 0, T ; H γ 2
, X = L 2 0, T ; L 2 loc , X 1 = L 2 0, T ; H -2 and using (6.4), we can pass to the limit to obtain that T 0 [θψ t + uθψ x + u x θψ -θΛ γ ψ] dxdt = θ 0 (x)ψ(0, x)dx (6.5) This completes the proof. we conclude that θ = 0 in L 2 (wdx) and thus a weak solution is unique. This completes the proof of Theorem 1.5.

Proof of Theorem 1.6

Taking the Fourier transform of (1.7), we have that

∂ t | θ(ξ)| = -Re R 1 (1 + |ζ| 2 ) α θ(ζ)i(ξ -ζ)) θ(ξ -ζ)dζ θ(ξ) | θ(ξ)| 1 √ 2π -|ξ|| θ|.
Consequently, ignoring the factor 1 (1+|ζ| 2 ) α , we follow the proof of Theorem 1.3 with δ = 0 and the smallness condition (1.8) to obtain that

θ(t) A 1 + 1 - √ 2 θ 0 A 0 √ π t 0 θ x (s) A 1 ds ≤ θ 0 A 1 (8.1)
for all t ≥ 0. We also follow the proof of Theorem 1.3 to obtain a unique weak solution via the approximation procedure. This completes the proof.

1 2

 1 θ and integrate in x. Then, 1

7 . 5 7. 1 . 2 L 2 1 4 1 2

 7512211 Proof of Theorem 1.A priori estimate. We consider the equationθ t + uθ x + Λθ = 0, u = (1 -∂ xx ) -(wdx) + ΛTo estimate terms involving u, we use θ = (1 -∂ xx ) u and (2.8) to obtain that Since θ 2 ∈ L 2 0, T : H 1 (wdx) , θ 1 ∈ L 2 0, T : H (wdx) ,

  1 2 -γ 4 to bound u x asu x L 2 ≤ C θ L 2 + Λ

					γ 2 θ	L 2 .
	Therefore, we obtain that
	θ(t) 2 L 2 +	0	t	Λ	γ 2 θ(s)

2

L 2 ds ≤ θ 0 2

Acknowledgments H.B. was supported by NRF-2015R1D1A1A01058892. R.G.B. is partially supported by the Labex MILYON. Both O. L. and R.G.B. were partially supported by the Grant MTM2014-59488-P from the former Ministerio de Economía y Competitividad (MINECO, Spain). O.L. was partially supported by the Marie-Curie Grant, acronym: TRANSIC, from the FP7-IEF program.