Eduardo Abi Jaber

Enzo Miller

Huyên Pham

Integral operator Riccati equations arising in stochastic Volterra control problems

Keywords: MSC Classification: 47G10, 49N10, 34G20 Infinite dimensional Lyapunov equation, integral operator Riccati equation, linear-quadratic control, stochastic Volterra equations

We establish existence and uniqueness for infinite dimensional Riccati equations taking values in the Banach space L 1 (µ ⊗ µ) for certain signed matrix measures µ which are not necessarily finite. Such equations can be seen as the infinite dimensional analogue of matrix Riccati equations and they appear in the Linear-Quadratic control theory of stochastic Volterra equations.

Introduction

Fix d, d ′ , m ∈ N and µ a d × d ′ -matrix signed measure µ. This paper deals with the infinite dimensional Backward Riccati equation

     Γ T (θ, τ ) = 0 Γt (θ, τ ) = (θ + τ )Γ t (θ, τ ) -Q -D ⊤ R 2 + µ(dθ ′ ) ⊤ Γ t (θ ′ , τ ′ )µ(dτ ′ )D -B ⊤ R + µ(dθ ′ ) ⊤ Γ t (θ ′ , τ ) -R + Γ t (θ, τ ′ )µ(dτ ′ )B + S t (θ) ⊤ N -1 t S t (τ ), (1.1)
where

S t (τ ) = C ⊤ R + µ(dθ) ⊤ Γ t (θ, τ ) + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ t (θ ′ , τ ′ )µ(dτ ′ )D, Nt = N + F ⊤ R 2 + µ(dθ) ⊤ Γ t (θ, τ )µ(dτ )F,
and B, D ∈ R d ′ ×d , C, F ∈ R d ′ ×m , Q ∈ R d×d and N ∈ R m×m , and ⊤ is the transpose operation. Here µ is not necessarily finite and satisfies

R + 1 ∧ θ -1/2 |µ|(dθ) < ∞, (1.2)
where |µ| denotes the total variation of µ. We look for solutions Γ : [0, T ] × R 2 + → R d×d with values in L 1 (µ ⊗ µ) (see the precise definition in Section 2) to ensure that equation (1.1) is well-posed.

In particular, if d = d ′ = 1 and µ(dθ) = n i=1 δ θ n i (dθ), (1.1) reduces to a n × n-matrix Riccati equation for Γ n = (Γ(θ n i , θ n j )) 1≤i,j≤n , only written componentwise. Such matrix Riccati equation appears in finite dimensional Linear-Quadratic (LQ) control theory, see e.g. [14, chapter 7]. (One could also recover d × d-matrix Riccati equation by setting d = d ′ and µ = I d δ 0 .)

For more general measures µ, e.g. with infinite support, (1.1) can be seen as the infinitedimensional extension of matrix Riccati equations and one could expect a connection with LQ control in infinite dimension. This is indeed the case, and our motivation for studying the Riccati equation (1.1) comes from infinite dimensional lifts of LQ control theory of non-Markovian stochastic Volterra equations. Setting

K(t) = R +
e -θt µ(dθ), t > 0, one can consider the controlled d-dimensional stochastic linear Volterra equation

X t = t 0 K(t -s) BX s + Cα s ds + t 0 K(t -s) DX s + F α s dW s ,
where W is a one dimensional Brownian motion and α is a suitable control taking its values in R m . Observe that the integrability condition on the measure µ allows singularity of the kernel K at 0, and includes the case of a fractional kernel K H (t) = t H-1/2 with Hurst parameter H ∈ (0, 1/2) with a corresponding measure µ(dθ) = c H θ -H-1/2 dθ, for some normalizing constant c H . The linear-quadratic control problem consisting in the minimization over α of the cost functional

J(α) = E T 0 X ⊤ t QX t + α ⊤ t N α t dt ,
can be explicitly solved using the Riccati equation (1.1), see [START_REF] Jaber | Linear-Quadratic control for a class of stochastic Volterra equations: solvability and approximation[END_REF]. When D = F = 0, the Riccati equation (1.1) also enters in the computation of the Laplace transform of tr T 0 Z ⊤ s QZ s ds , where Z is the d × n-matrix valued Gaussian process

Z t = Z 0 + t 0 K(t -s)BZ s ds + t 0 K(t -s)Cd W s , t ≥ 0,
and W is a m × n matrix Brownian motion, see [START_REF] Jaber | The Laplace transform of the integrated Volterra Wishart process[END_REF].

The Riccati equation (1.1) can be also connected to an operator Riccati equation as follows. Denote by L 1 (µ) the Banach space of µ-a.e. equivalence classes of |µ|-integrable functions ϕ : R + → R d ′ endowed with the norm ϕ L 1 (µ) = R + |µ|(dθ)|ϕ(θ)|, by L ∞ (µ) the space of measurable functions from R + → R d ′ , which are bounded µ-a.e., and introduce the dual pairing:

ϕ, ψ µ := R + ϕ(θ) ⊤ µ(dθ) ⊤ ψ(θ), (ϕ, ψ) ∈ L 1 (µ) × L ∞ (µ ⊤ ).
Given any bounded kernel solution Γ to (1.1), let us consider the corresponding linear integral operator Γ : [0, T ] × L 1 (µ) → L ∞ (µ ⊤ ) defined by

(Γ t ϕ)(θ) = R + Γ t (θ, τ )µ(dτ )ϕ(τ ), t ∈ [0, T ], ϕ ∈ L 1 (µ).
It is then straightforward to see that Γ solves the operator Riccati equation on L 1 (µ):

   Γ T = 0 Γt = -Γ t A mr -(Γ t A mr ) * -Q -D * Γ t D -B * Γ t -(B * Γ t ) * + (C * Γ t + F * Γ t D) * (N + F * Γ t F ) -1 (C * Γ t + F * Γ t D) , t ∈ [0, T ], (1.4)
where A mr is the mean-reverting operator acting on measurable functions ϕ ∈ L 1 (µ) by

(A mr ϕ)(θ) = -θϕ(θ), θ ∈ R + ,
B, D are the integral operators on L 1 (µ) (defined similarly to Γ) induced by the constant matrices B, D, and by misuse of notation, C, F denote the respective constant operators on R m induced by the matrices C, F :

(Ca)(θ) = Ca, (F a)(θ) = F a, θ ∈ R + , a ∈ R m .
Here the symbol * denotes the adjoint operation with respect to the dual pairing. The last equation (1.4) is more in line with the formulation of operator Riccati equations appearing in LQ control theory in Hilbert or Banach spaces, see [START_REF] Curtain | The infinite-dimensional Riccati equation[END_REF], [START_REF] Prato | Direct solution of a Riccati equation arising in stochastic control theory[END_REF], [START_REF] Flandoli | Direct solution of a Riccati equation arising in a stochastic control problem with control and observation on the boundary[END_REF], [START_REF] Guatteri | On the backward stochastic Riccati equation in infinite dimension[END_REF], [START_REF] Lasiecka | Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators[END_REF], [START_REF] Koshkin | Positive semigroup and algebraic Riccati equations in Banach spaces[END_REF], [START_REF] Hu | Stochastic LQ and associated Riccati equation of PDEs driven by state-and control-dependent White noise[END_REF], [START_REF] Artamonov | Solvability of an operator Riccati integral equation in a reflexive banach space[END_REF].

Let us also mention that a related infinite-dimensional Riccati equation appeared in [START_REF] Alfonsi | Capacitary measures for completely monotone kernels via singular control[END_REF] for the minimization problem of an energy functional defined in terms of a non-singular (i.e. K(0) < ∞) completely monotone kernel.

The main contribution of the paper is to establish the existence and uniqueness of a solution to the kernel Riccati equation (1.1). The aforementioned results on the solvability of Riccati equations in infinite dimensional spaces cannot be directly applied in our setting for two reasons. First, they are valid for Hilbert and reflexive Banach spaces, while L 1 (µ) is in general not reflexive, unless µ has finite support, and mostly apply to the cases without multiplicative noise, i.e., D = 0, and without control on the diffusion coefficient, i.e. F = 0, with the noticeable exception in [START_REF] Hu | Stochastic LQ and associated Riccati equation of PDEs driven by state-and control-dependent White noise[END_REF]. Second, they concern the operator Riccati equation (1.4), which is not enough for our purposes, as we still need to argue that Γ is an integral operator induced by some bounded symmetric kernel function Γ satisfying (1.1). We will therefore work directly on the level of the kernel Riccati equation (1.1) (which will also be referred to as integral operator Riccati equation) by adapting the technique used in classical finite-dimensional linear-quadratic control theory [14, Theorem 6.7.2] with the following steps: (i) we first construct a sequence of Lyapunov solutions (Γ i ) i≥0 by successive iterations, (ii) we then show the convergence of (Γ i ) i≥0 in L 1 (µ ⊗ µ), (iii) we next prove that the limiting point is a solution to the Riccati equation (1.1), (iv) we finally prove the continuity and uniqueness for the Riccati solution. We stress that such method has already been applied to prove the existence of operator Riccati equations of the form (1.4) in Hilbert spaces (see [START_REF] Hu | Stochastic LQ and associated Riccati equation of PDEs driven by state-and control-dependent White noise[END_REF]) and in reflexive Banach spaces (see [START_REF] Artamonov | Solvability of an operator Riccati integral equation in a reflexive banach space[END_REF]). However, for the kernel Riccati equation (1.1), the proof is more intricate. The reason is that we need to establish the convergence of the kernels (Γ i ) i≥0 which is a stronger requirement than the usual convergence of the operators (Γ i ) i≥0 . As a consequence, we obtain that the sequence of integral operators (Γ i ) i≥0 converges to some limit which is also an integral operator.

The paper is organized as follows. We formulate precisely our main result in Section 2. Section 3 establishes the existence of a solution to an infinite dimensional Lyapunov equation. Section 4 is devoted to the solvability of the Riccati equation. Finally, we collect in the Appendix some useful results.

Preliminaries and main result

Let us first introduce some notations that will be used in the sequel of the paper. For any d 11 × d 12 -matrix valued measure µ 1 , and d 21 × d 22 -matrix valued measure µ 2 on R + , the Banach space

L 1 (µ 1 ⊗ µ 2 ) consists of µ 1 ⊗ µ 2 -a.e. equivalence classes of |µ 1 | ⊗ |µ 2 |-integrable functions Φ : R 2 + → R d 11 ×d 21 endowed with the norm Φ L 1 (µ 1 ⊗µ 2 ) = R 2 + |µ 1 |(dθ)|Φ(θ, τ )||µ 2 |(dθ) < ∞.
For any such Φ, the integral

R 2 + µ 1 (dθ) ⊤ Φ(θ, τ )µ 2 (dτ )
is well defined by virtue of [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF]Theorem 5.6]. We also denote by L ∞ (µ 1 ⊗ µ 2 ) the set of measurable functions Φ : R 2 + → R d 11 ×d 21 , which are bounded µ 1 ⊗ µ 2 -a.e. We shall prove the existence of a nonnegative symmetric kernel solution to the Riccati equation (1.1) in the following sense.

Definition 2.1. Let Γ : R 2 + → R d×d such that Γ ∈ L ∞ (µ ⊗ µ). We say that Γ is symmetric if Γ(θ, τ ) = Γ(τ, θ) ⊤ , µ ⊗ µ -a.e.

and nonnegative if

R 2 + ϕ(θ) ⊤ µ(dθ) ⊤ Γ(θ, τ )µ(dτ )ϕ(τ ) ≥ 0,
for all ϕ ∈ L 1 (µ).

We denote by S d + (µ ⊗ µ) the set of all symmetric and nonnegative Γ ∈ L ∞ (µ ⊗ µ), and we define on S d + (µ ⊗ µ) the partial order relation Given a kernel Γ, we define the integral operator Γ by

Γ 1 µ Γ 2 whenever (Γ 1 -Γ 2 ) ∈ S d + (µ ⊗ µ).
(Γϕ)(θ) = R + Γ(θ, τ )µ(dτ )ϕ(τ ).
(2.1)

Notice that when Γ ∈ L 1 (µ ⊗ µ), the operator Γ is well-defined on L ∞ (µ), and we have Γϕ ∈ L 1 (µ ⊤ ), for ϕ ∈ L ∞ (µ). In this case Γϕ, ψ µ ⊤ is well defined for all ϕ, ψ ∈ L ∞ (µ). Moreover, when Γ ∈ L ∞ (µ ⊗ µ), the operator Γ is well-defined on L 1 (µ), and we have Γϕ ∈ L ∞ (µ ⊤ ) for ϕ ∈ L 1 (µ). In this case, ϕ, Γψ µ is well defined for all ϕ, ψ ∈ L 1 (µ). Whenever Γ ∈ L ∞ (µ ⊗ µ) is a symmetric kernel, we have

ϕ, Γψ µ , = ψ, Γϕ µ , ϕ, ψ ∈ L 1 (µ),
and Γ is said to be symmetric. For Γ ∈ S d + (µ ⊗ µ), the nonnegativity reads ϕ, Γϕ µ ≥ 0, ∀ϕ ∈ L 1 (µ).

The kernel Riccati equation (1.1) can be compactly written in the form

Γt (θ, τ ) = (θ + τ )Γ t (θ, τ ) -R(Γ t )(θ, τ ), Γ T (θ, τ ) = 0 (2.2)
where we define

R(Γ)(θ, τ ) = Q + D ⊤ R 2 + µ(dθ ′ ) ⊤ Γ(θ ′ , τ ′ )µ(dτ ′ )D + B ⊤ R + µ(dθ ′ ) ⊤ Γ(θ ′ , τ ) + R + Γ(θ, τ ′ )µ(dτ ′ )B -S(Γ)(θ) ⊤ N -1 (Γ)S(Γ)(τ ) (2.3) with S(Γ)(τ ) = C ⊤ R + µ(dθ) ⊤ Γ(θ, τ ) + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ(θ ′ , τ ′ )µ(dτ ′ )D N (Γ) = N + F ⊤ R 2 + µ(dθ) ⊤ Γ(θ, τ )µ(dτ )F.
(2.4)

The following definition specifies the concept of solution to the kernel Riccati equation (2.2). Definition 2.3. By a solution to the kernel Riccati equation (2.2), we mean a function Γ

∈ C([0, T ], L 1 (µ ⊗ µ)) such that Γ t (θ, τ ) = T t e -(θ+τ )(s-t) R(Γ s )(θ, τ )ds, 0 ≤ t ≤ T, µ ⊗ µ -a.e. (2.5)
where R is defined by (2.3). In particular N (Γ t ) given by (2.4) is invertible for all t ≤ T .

Our main result is stated as follows.

Theorem 2.4. Let µ be a d × d ′ -signed matrix measure satisfying (1.2). Assume that

Q ∈ S d + , N -λI m ∈ S m + , (2.6)
for some λ > 0. Then, there exists a unique solution

Γ ∈ C([0, T ], L 1 (µ ⊗ µ)) to the kernel Riccati equation (2.2) such that Γ t ∈ S d + (µ ⊗ µ)
, for all t ≤ T . Furthermore, there exists some positive constant M such that

R + |µ|(dτ )|Γ t (θ, τ )| ≤ M, µ -a.e., 0 ≤ t ≤ T.
(2.7)

The rest of the paper is dedicated to the proof of Theorem 2.4. Lemmas 4.6 and 4.7 provide the existence of a solution in C([0, T ], L 1 (µ ⊗ µ)) such that Γ t ∈ S d + (µ ⊗ µ), for all t ≤ T . The uniqueness statement is established in Lemma 4.8. For each i = 1, 2, we let µ i be a d i1 × d i2 -matrix valued measure on R + , and we define the scalar kernel

Infinite dimensional Lyapunov equation

K i (t) = R + e -θt |µ i |(dθ), t > 0, (3.1) which is in L 2 ([0, T ], R) provided that µ i satisfies R + 1 ∧ θ -1/2 |µ i |(dθ) < ∞ (which we shall assume in this section), see [2, Lemma A.1].
We first establish the existence and uniqueness for the following infinite dimensional Lyapunov equation:

Ψ t (θ, τ ) = T t e -(θ+τ )(s-t) F (s, Ψ s )(θ, τ )ds, t ≤ T, µ 1 ⊗ µ 2 -a.e. (3.2)
where

F (s, Ψ)(θ, τ ) = Qs (θ, τ ) + D1 s (θ) ⊤ R 2 + µ 1 (dθ ′ ) ⊤ Ψ(θ ′ , τ ′ )µ 2 (dτ ′ ) D2 s (τ ) + B1 s (θ) ⊤ R + µ 1 (dθ ′ ) ⊤ Ψ(θ ′ , τ ) + R + Ψ(θ, τ ′ )µ 2 (dτ ′ ) B2 s (τ ), (3.3)
for some coefficients Q, B1 , B2 , D1 , D2 satisfying suitable assumptions made precise in the following theorem.

Theorem 3.1. Let Q : [0, T ] × R 2 + → R d 11
×d 21 be a measurable function and for each i = 1, 2, Bi , Di : [0, T ] × R + → R d i2 ×d i1 be two measurable functions. Assume that there exists κ > 0 such that

| Qs (θ, τ )| + 2 i=1 | Bi s (θ)| + | Di s (θ)| 1/2 ≤ κ, dt ⊗ µ 1 ⊗ µ 2 -a.e. (3.4)
Then, there exists a unique solution

Ψ ∈ C([0, T ], L 1 (µ 1 ⊗ µ 2 )) to (3.2)-(3.3). In particular, sup t≤T Ψ t L 1 (µ 1 ⊗µ 2 ) < ∞. (3.5)
Furthermore, there exists a constant κ ′ > 0 such that

R + |µ 1 |(dθ)|Ψ t (θ, τ )| ≤ κ ′ , µ 2 -a.e., t ≤ T, (3.6)
R + |µ 2 |(dτ )|Ψ t (θ, τ )| ≤ κ ′ , µ 1 -a.e., t ≤ T. (3.7)
Remark 3.2. Since the solution Ψ satisfies (3.5), (3.6) and (3.7), it follows that

|F (Ψ)(θ, τ )| ≤ c, dt ⊗ µ ⊗ µ -a.e.
for some constant c. Combined with (3.2), one gets that Ψ t ∈ L ∞ (µ 1 ⊗ µ 2 ), for all t ≤ T .

The proof of Theorem 3.1 follows from the three following lemmas.

Lemma 3.3. Under the assumptions of Theorem 3.1, there exists a unique

L 1 (µ 1 ⊗ µ 2 )- valued function t ∈ [0, T ] → Ψ t satisfying (3.5), and such that (3.2)-(3.3) hold.
Proof. The proof is an application of the contraction mapping principle. We denote by B T the space of measurable and bounded functions Ψ : [0, T ] → L 1 (µ 1 ⊗ µ 2 ) endowed with the norm

Ψ B T := sup t≤T Ψ t L 1 (µ 1 ⊗µ 2 ) < ∞.
The space (B T , • B T ) is a Banach space. We consider the following family of norms on B T :

Ψ λ := sup t≤T e -λ(T -t)) Ψ t L 1 (µ 1 ⊗µ 2 ) , λ > 0. For every Ψ ∈ B T , define a new function t → (T Ψ) t by (T Ψ) t (θ, τ ) = T t e -(θ+τ )(s-t) F (s, Ψ s )(θ, τ )ds, µ 1 ⊗ µ 2 -a.e.,
where F is given by (3.3). Since the norms • B T and • λ are equivalent, it is enough to find λ > 0 such that T defines a contraction on (B T , • λ ). We thus look for λ > 0 and

M < 1 such that T Ψ -T Φ λ ≤ M Ψ -Φ λ , Ψ, Φ ∈ B T . (3.8)
Step 1: We first prove that T (B T ) ⊂ B T . Fix Ψ ∈ B T and t ≤ T . An application of the triangle inequality combined with the assumption (3.4) leads to

(T Ψ) t L 1 (µ 1 ⊗µ 2 ) ≤ κ R 2 + |µ 1 |(dθ)|µ 2 |(dτ ) T t e -(θ+τ )(s-t) ds + κ R 2 + |µ 1 |(dθ)|µ 2 |(dτ ) T t e -(θ+τ )(s-t) Ψ s L 1 (µ 1 ⊗µ 2 ) ds + κ R 2 + |µ 1 |(dθ)|µ 2 |(dτ ) T t e -(θ+τ )(s-t) R + |µ 1 |(dθ ′ )|Ψ s (θ ′ , τ )| + κ R 2 + |µ 1 |(dθ)|µ 2 |(dτ ) T t e -(θ+τ )(s-t) R + |Ψ s (θ, τ ′ )||µ 2 |(dτ ′ ), = κ(I t + II t + III t + IV t ).
Recalling the definition (3.1), an application of Tonelli's theorem and Cauchy-Schwarz inequality yields

sup t≤T I t = sup t≤T T t K 1 (s -t)K 2 (s -t)ds ≤ K 1 L 2 (0,T ) K 2 L 2 (0,T ) , which is finite due to [2, Lemma A.1]. Similarly, sup t≤T II t ≤ Ψ B T K 1 L 2 (0,T ) K 2 L 2 (0,T ) < ∞.
Now, as e -τ (s-t) ≤ 1, and e -θ(s-t) ≤ 1, for s ≥ t, and θ, τ ∈ R + , another application of Tonelli's theorem leads to

sup t≤T III t ≤ Ψ B T K 1 L 1 (0,T ) < ∞, and sup t≤T IV t ≤ Ψ B T K 2 L 1 (0,T ) < ∞.
Combining the above inequalities proves that T Ψ B T < ∞ and hence T :

B T → B T .
Step 2: We prove that there exists λ > 0 such that (3.8) holds. Fix λ > 0 and Ψ, Φ ∈ S T such that Ψ λ and Φ λ are finite. Similarly to Step 1, the triangle inequality and Tonelli's theorem lead to

sup t≤T e -λ(T -t) (T Ψ) t -(T Φ) t L 1 (µ 1 ⊗µ 2 ) ≤ M (λ) sup t≤T e -λ(T -t) Ψ t -Φ t L 1 (µ 1 ⊗µ 2 ) ,
where

M (λ) = κ T 0 e -λs K 1 (s) + K 2 (s) + K 1 (s)K 2 (s) ds.
By the dominated convergence theorem, M (λ) tends to 0 as λ goes to +∞. We can therefore choose λ 0 > 0 so that (3.8) holds with M (λ 0 ) < 1. An application of the contraction mapping theorem yields the existence and uniqueness statement in (B T , • B T ) such that (3.2) holds, µ 1 ⊗ µ 2 -a.e., for all t ≤ T . The interchange of the quantifiers is possible due to the continuity of t → Ψ t (θ, τ ) µ 1 ⊗ µ 2 -a.e., which ends the proof. Proof. We only prove (3.7), as (3.6) follows by the same argument. Integrating (3.2) over the τ variable leads to

R + Ψ t (θ, τ )µ 2 (dτ ) = R + T t e -(θ+τ )(s-t) F (s, Ψ s )(θ, τ )dsµ 2 (dτ ), t ≤ T, µ 1 -a.e.(3.9)
Let us define the µ 1 -null set N = {θ ∈ R + : (3.9) does not hold}, and fix θ ∈ R + \ N and t ≤ T . The triangle inequality on (3.9) and assumption (3.4) yields

R + |µ 2 |(dτ )|Ψ t (θ, τ )| ≤ κ R + |µ 2 |(dτ ) T t e -(θ+τ )(s-t) ds + κ R + |µ 2 |(dτ ) T t e -(θ+τ )(s-t) Ψ s L 1 (µ 1 ⊗µ 2 ) ds + κ R + |µ 2 |(dτ ) T t e -(θ+τ )(s-t) R + |µ 1 |(dθ ′ )|Ψ s (θ ′ , τ )|ds + κ R + |µ 2 |(dτ ) T t e -(θ+τ )(s-t) R + |µ 2 |(dτ ′ )|Ψ s (θ, τ ′ )|ds.
Using the bound e -θ(s-t) ≤ 1, an application of Tonelli's theorem gives

R + |µ 2 |(dτ )|Ψ t (θ, τ )| ≤κ 1 + sup s≤T Ψ s L 1 (µ 1 ⊗µ 2 ) T 0 1 + K 2 (s) ds + κ T t K 2 (s -t) R + |µ 2 |(dτ ′ )|Ψ s (θ, τ ′ )|ds. (3.10)
After a change of variable, we get that the function f θ defined by

f θ (t) = R + |µ 2 |(dτ )|Ψ T -t (θ, τ )|, t ≤ T, satisfies the convolution inequality f θ (t) ≤ c T + κ t 0 K 2 (t -s)f θ (s)ds, with c T = κ 1 + sup s≤T Ψ s L 1 (µ 1 ⊗µ 2 ) T 0 1 + K 2 (s) ds < ∞. It follows from (3.5) that f θ (t) is finite µ 1 ⊗ dt-a.
Ψ ∈ C([0, T ], L 1 (µ 1 ⊗ µ 2 )).
Proof. We first observe that by virtue of the boundedness of the coefficients (3.4) and the estimates (3.5), (3.6) and (3.7), we have

|F (s, Ψ s )(θ, τ )| ≤ c, dt ⊗ µ 1 ⊗ µ 2 -a.e. (3.11)
for some constant c, where F is given by (3.3). Fix s ≤ t ≤ T . Using (3.2), we write

Ψ s (θ, τ ) -Ψ t (θ, τ ) = t s e -(θ+τ )(u-s) F (u, Ψ u )(θ, τ )ds + T t e -(θ+τ )(u-s) -e -(θ+τ )(u-t) F (u, Ψ u )(θ, τ )ds = I s,t (θ, τ ) + II s,t (θ, τ ) µ 1 ⊗ µ 2 -a.e.
Integrating over the θ and τ variables and successive applications of Tonelli's theorem and Cauchy-Schwarz inequality together with the bound (3.11) lead to

I s,t L 1 (µ 1 ⊗µ 2 ) ≤ c t s K 1 (u -s)K 2 (u -s)du ≤ c K 1 L 2 (0,t-s) K 2 L 2 (0,t-s) .
By virtue of the square integrability of K 1 and K 2 , the right hand side goes to 0 as s ↑ t.

Similarly, using also that e -(θ+τ )(u-s) ≤ e -(θ+τ )(u-t) , we get

II s,t L 1 (µ 1 ⊗µ 2 ) ≤ c T t K 1 (u -t)K 2 (u -t) -K 1 (u -s)K 2 (u -s) du = c T -t 0 K 1 (u)K 2 (u) -K 1 (t -s + u)K 2 (t -s + u) du = c T -t 0 K 1 (u) -K 1 (t -s + u) K 2 (u)du + c T -t 0 K 1 (t -s + u) K 2 (u) -K 2 (t -s + u) du = c(1 s,t + 2 s,t ).
The right hand side goes also to 0 as s ↑ t. To see this, an application of Cauchy-Schwarz inequality gives

1 s,t ≤ T 0 K 1 (u) -K 1 (t -s + u) 2 du 1/2 K 2 L 2 (0,T ) . Since K 1 is an element of L 2 , it follows from [5, Lemma 4.3] that lim h→0 T 0 |K 1 (u + h) -K 1 (u)| 2 du = 0, (3.12)
showing that 1 s,t converges to 0 as s goes to t. Interchanging the roles of K 1 and K 2 , we also get the convergence 2 s,t → 0 as s ↑ t. Combining the above leads to

Ψ s -Ψ t L 1 (µ 1 ⊗µ 2 ) → 0, as s ↑ t.
Similarly, we get the same conclusion when s ↓ t, and the proof is complete.

Solvability of the Riccati equation

The main goal of this section is to prove Theorem 2.4, i.e., the existence and uniqueness of a function Γ ∈ C([0, T ], L 1 (µ ⊗ µ)) satisfying the kernel Riccati equation (2.5) (recall Definition 2.1), and the estimate (2.7). This is obtained by adapting the technique used in classical linear-quadratic control theory [14, Theorem 6.7.2] to our setting with the following steps:

(i) Construct a sequence of Lyapunov solutions (Γ i ) i≥0 by successive iterations, (ii) Establish the convergence of and under the condition

(Γ i ) i≥0 in L 1 (µ ⊗ µ), (iii)
| Qs (θ, τ )| ≤ κ, dt ⊗ µ ⊗ µ -a.e. | Bs (θ)| + | Ds (θ)| ≤ κ, dt ⊗ µ -a.e. (4.2)
If Qt ∈ S d + (µ ⊗ µ) for all t ≤ T , then (i) t → Ψ t is a non-increasing S d + (µ ⊗ µ)-valued function w.r.t the order relation µ . (ii) t → R 2 + µ(dθ) ⊤ Ψ t (θ, τ )µ(dτ ) is a non-increasing S d + -valued function on [0, T ].
Proof. Note that under (4.1), the Lyapunov equation (3.2) is invariant by transposition and exchange of θ and τ . By uniqueness of the solution, we deduce that Ψ t (θ, τ ) = Ψ t (τ, θ) ⊤ , µ ⊗ µ-a.e., for all t ≤ T . Fix ϕ ∈ L 1 (µ) and t ≤ T , and consider the following equation

d Y s (θ) = -θ Y s (θ) + R + Bs (τ )µ(dτ ) Y s (τ ) ds + R + Ds (τ )µ(dτ ) Y s (τ ) dW s Y t (θ) = ϕ(θ),
which admits a unique L 1 (µ)-valued solution such that

sup t≤s≤T E Y s 4 L 1 (µ) < ∞, (4.3
0 = ϕ, Ψ t ϕ µ - T t Y s , Qs Y s µ ds + T t R + Y s (θ) ⊤ µ(dθ) ⊤ Ds (θ) ⊤ R 2 + µ(dθ ′ ) ⊤ Ψ s (θ ′ , τ ′ )µ(dτ ′ ) Y s (τ ′ )dW s + T t R 2 + Y s (θ ′ ) ⊤ µ(dθ ′ ) ⊤ Ψ s (θ ′ , τ ′ )µ(dτ ′ ) R + Ds (τ )µ(dτ ) Y s (τ )dW s . (4.4)
It is straightforward to check that the local martingales terms are in fact true martingales due to the boundedness conditions (4.2), (3.6) and the moment bound (4.3). Thus, taking the expectation on both sides of (4.4) yields that

ϕ, Ψ t ϕ µ = E T t Y s , Qs Y s µ ds ,
which ensures the positiveness and the non increasingness of t → ϕ, Ψ t ϕ µ for any ϕ ∈ L 1 (µ), since s → Qs is S d + (µ⊗µ)-valued. This proves Assertion (i). Next, by considering the sequence of

L 1 (µ)-valued functions (ϕ n (θ) = z½ [1/n,∞) (θ)) n≥1 for arbitrary z ∈ R d ′ \ {0}, using that Ψ t ∈ L 1 (
µ⊗µ), and taking the limit, we obtain that

t → R 2 + µ(dθ) ⊤ Ψ t (θ, τ )µ(dτ )
is a non increasing S d + -valued function. This proves Assertion (ii) and concludes the proof.

From now on, we work under assumption (2.6). We construct a sequence of Lyapunov solutions (Γ i ) i≥0 by induction as follows.

• Initialization: Let Γ 0 ∈ C([0, T ], L 1 (µ ⊗ µ)) be the unique solution given by Theorem 3.1 to the following Lyapunov equation

       Γ 0 t (θ, τ ) = T t e -(θ+τ )(s-t) F 0 (Γ 0 s )(θ, τ )ds, F 0 (Γ)(θ, τ ) = Q + D ⊤ R 2 + µ(dθ ′ ) ⊤ Γ(θ ′ , τ ′ )µ(dτ ′ )D +B ⊤ R + µ(dθ ′ ) ⊤ Γ(θ ′ , τ ) + R + Γ(θ, τ ′ )µ(dτ ′ )B. Since Q ∈ S d + , an application of Lemma 4.1-(ii) yields that F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ 0 t (θ ′ , τ ′ )µ(dτ ′ )F ∈ S m
+ , for all t ≤ T . Combined with the assumption N -λI m ∈ S m + , we obtain

N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ 0 t (θ ′ , τ ′ )µ(dτ ′ )F -λI m ∈ S m + , t ≤ T.
• Induction:

for i ∈ N, having constructed Γ i ∈ C([0, T ], L 1 (µ ⊗ µ)) such that N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ i t (θ ′ , τ ′ )µ(dτ ′ )F -λI m ∈ S m + , t ≤ T. (4.5)
and

R + |µ|(dτ )|Γ i t (θ, τ )| ≤ κ ′ i , µ -a.e., t ≤ T, (4.6)
for some κ ′ i > 0, we define

Θ i t (τ ) = -N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ i t (θ ′ , τ ′ )µ(dτ ′ )F -1 (4.7) × F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ i t (θ ′ , τ ′ )µ(dτ ′ )D + C ⊤ R + µ(dθ ′ )Γ i t (θ ′ , τ ) , together with the coefficients Qi t (θ, τ ) = Q + Θ i t (θ) ⊤ N Θ i t (τ ), Bi t (τ ) = B + CΘ i t (τ ), Di t (τ ) = D + F Θ i t (τ ). (4.8) Since Γ i ∈ C([0, T ], L 1 (µ ⊗ µ)), we have sup t≤T Γ i t L 1 (µ⊗µ) < ∞.
Combined with the estimate (4.6), this yields the existence of c i > 0 such that

Θ i t (θ) ≤ c i , µ -a.e., t ≤ T.
This implies that the coefficients Qi , Bi , Di satisfy (4.2). Therefore, Theorem 3.1 can be applied to get the existence of a unique solution Γ i+1 ∈ C([0, T ], L 1 (µ ⊗ µ)) to the following Lyapunov equation

       Γ i+1 t (θ, τ ) = T t e -(θ+τ )(s-t) F i (s, Γ i+1 s )(θ, τ )ds, F i (s, Γ)(θ, τ ) = Qi s (θ, τ ) + Di t (θ) ⊤ R 2 + µ(dθ ′ ) ⊤ Γ(θ ′ , τ ′ )µ(dτ ′ ) Di t (τ ) + Bi t (θ) ⊤ R + µ(dθ ′ ) ⊤ Γ(θ ′ , τ ) + R + Γ(θ, τ ′ )µ(dτ ′ ) Bi t (τ ), (4.9)
such that the estimate (4.6) holds also for Γ i+1 . Furthermore, since Qi t clearly lies in S d + (µ ⊗ µ), for all t ≤ T , Lemma 4.1-(ii) yields that (4.5) is satisfied with Γ i t replaced by Γ i+1

t , for all t ≤ T . This ensures that the induction is well-defined.

4.2

Step 2: Convergence of (Γ i t ) i≥0 in L 1 (µ ⊗ µ) Lemma 4.2. For i ∈ N and for a scalar function ξ ∈ L ∞ (|µ|) define the matrix-valued functions

U i = R 2 + µ(dθ) ⊤ Γ i (θ, τ )µ(dτ ), V i (ξ) = R 2 + µ(dθ) ⊤ Γ i (θ, τ )µ(dτ )ξ(τ ).
Then, (i) U i i≥0 is a non-increasing sequence (meaning U i+1 t ≤ U i t , i ∈ N and t ≤ T ) of monotone non-increasing functions on the space C([0, T ], S d + ), converging pointwise to a limit denoted by U ;

(ii) V i (ξ) i≥0 is a uniformly bounded sequence of functions on the space C([0, T ], R d ′ ×d ′ ), converging pointwise to a limit denoted by V(ξ), for any scalar function ξ ∈ L ∞ (|µ|).

Proof. Throughout the proof we consider the intermediate scalar sequences

U i t (ϕ) = Γ i t ϕ, ϕ µ ⊤ and V i t (ϕ, ψ) = Γ i t ϕ, ψ µ ⊤ , t ≤ T, i ∈ N, ϕ, ψ ∈ L ∞ (µ),
which are well-defined since Γ i t ∈ L 1 (µ ⊗ µ) for all t ≤ T . We also Set Θ -1 = 0, and for i ≥ 0, and define

∆ i = Γ i -Γ i+1 , ρ i = Θ i-1 -Θ i ,
where we recall that Θ i is given by (4.7). Straightforward computations, detailed in Lemma A.2 and Remark A.3, yield that ∆ i solves the Lyapunov equation:

       ∆ i t (θ, τ ) = T t e -(θ+τ )(s-t) F δ i (s, ∆ i s )(θ, τ )ds, F δ i (t, ∆)(θ, τ ) = Qi,δ t (θ, τ ) + Di t (θ) ⊤ R 2 + µ(dθ ′ ) ⊤ ∆(θ ′ , τ ′ )µ(dτ ′ ) Di t (τ ) + Bi t (θ) ⊤ R + µ(dθ ′ ) ⊤ ∆(θ ′ , τ ) + R + ∆(θ, τ ′ )µ(dτ ′ ) Bi t (τ ), (4.10)
where

Qi,δ t (θ, τ ) = ρ i t (θ) ⊤ N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ i t (θ ′ , τ ′ )µ(dτ ′ )F ρ i t (τ ).
• Fix i ∈ N. Since Qi,δ t ∈ S d + (µ ⊗ µ), an application of Lemma 4.1-(i) on (4.10) shows that t → ∆ i t is a non-increasing S d + (µ ⊗ µ)-valued function. Thus, for any ϕ ∈ L 1 (µ),

ϕ, Γ 0 0 ϕ µ ≥ ϕ, Γ 0 t ϕ µ ≥ ϕ, Γ i t ϕ ≥ ϕ, Γ i+1 t ϕ µ ≥ 0, t ≤ T, i ∈ N,
Since for all t ≤ T , Γ i t is also an element of L 1 (µ ⊗ µ), the density of simple functions in L ∞ (µ) with respect to the uniform norm, implies that

0 ≤ U i+1 t (ϕ) ≤ U i t (ϕ) ≤ U 0 t (ϕ) ≤ U 0 0 (ϕ).
for all ϕ ∈ L ∞ (µ). This implies that the sequence of functions (U i (ϕ)) i≥0 is non-increasing, nonnegative and converging pointwise to a limit that we denote by U t (ϕ) for any t ∈ [0, T ]. Furthermore, t → U i t (ϕ) is continuous, for all i ∈ N and ϕ ∈ L ∞ (µ), thanks to the continuity of t → Γ t in L 1 (µ ⊗ µ), see Lemma 3.5. The claimed statement (i) for U now follows by evaluating with ϕ(θ) ≡ z, where z ranges through

R d ′ . • Since Γ i t -Γ j t ∈ S d + (µ ⊗ µ) for any i ≤ j, an application of the Cauchy-Schwartz inequality (see Lemma A.1) yields ϕ, Γ i t -Γ j t ψ 2 µ ≤ ϕ, Γ i t -Γ j t ϕ µ ψ, Γ i t -Γ j t ψ µ , ϕ, ψ ∈ L 1 (µ). (4.11)
Invoking once again the density of simple functions in L ∞ (µ) with respect to the uniform norm and the fact that for all t ≤ T , Γ t ∈ L 1 (µ ⊗ µ), (4.11) gives

V i t (ϕ, ψ) -V j t (ϕ, ψ) 2 ≤ U i t (ϕ) -U j t (ϕ) U i t (ψ) -U j t (ψ) , ϕ, ψ ∈ L ∞ (µ).
Whence, the sequence of real valued functions t → V i t (ϕ, ψ) i≥0 is uniformly bounded. Furthermore, this also shows that the sequence t → V i t (ϕ, ψ) i≥0 is a real-valued Cauchy sequence that converges pointwise to a limit that we denote by V t (ϕ, ψ), for any ϕ, ψ ∈ L ∞ (µ). To obtain the continuity of V i (ϕ, ψ), note that Γ i t -Γ i s ∈ S d + (µ ⊗ µ) for any t ≤ s and Γ i s -Γ i t ∈ S d + for any s ≤ t, which allows us once again to apply the Cauchy Schwartz inequality (see Lemma A.1) coupled with the density argument to obtain for any s, t ∈ [0, T ]:

V i t (ϕ, ψ) -V i s (ϕ, ψ) 2 ≤ U i t (ϕ) -U i s (ϕ) U i t (ψ) -U i s (ψ) .
Consequently, the continuity of U i (ϕ) for any ϕ ∈ L ∞ (µ) implies that of V i (ϕ, ψ) for any ϕ, ψ ∈ L ∞ (µ). Fix ξ ∈ L ∞ (|µ|), the claimed statement (ii) for V(ξ) now follows by evaluating with ϕ(θ) ≡ z and ψ(θ) = ξ(θ)z ′ , where z, z ′ range through R d ′ .

Lemma 4.3. There exists a constant κ > 0 such that for every i ∈ N and t ∈ [0, T ]

R + Γ i t (θ, τ )µ(dτ ) ≤ κ µ -a.e. ( 4

.12)

Proof. Lemma 4.2 ensures that there exists a constant M > 0 such that for every i ∈ N

sup t≤T R 2 + µ(dθ) ⊤ Γ i t (θ, τ )µ(dτ ) ∨ sup t≤T R 2 + e -θt µ(dθ) ⊤ Γ i t (θ, τ )µ(dτ ) ≤ M. (4.13)
Fix i ∈ N. We proceed as in the proof of Lemma 3.4 to bound the quantity

g i t (θ) = R + Γ i+1 t (θ, τ )µ(dτ )
. By construction Γ i+1 solves (4.9), so that an integration over the τ -variable combined with (4.13) and the triangle inequality yield

g i+1 t (θ) ≤4rT + 4r T t (1 + |K(t -s)|) g i s (θ) + g i+1 s (θ) ds, (4.14)
where r is a constant only depending on B, D, Q N and M . Let us now show the desired inequality (4.12). For n ≥ 0, let us define

G n t (θ) = sup i=0,...,n R + Γ i t (θ, τ )µ(dτ ) .
The inequality (4.14) yields for every i ≥ 0

G i+1 t (θ) ≤ 4rT + 4r T t (1 + 2|K(t -s)|) G i+1 s (θ)ds.
Consequently, the generalized Gronwall inequality implies that there exists a constant c only depending on B, C, D, F, N, T, K and M such that for every n ∈ N, t ∈ [0, T ] we have |G n t (θ)| ≤ c for µ-almost every θ and t ∈ [0, T ].

Lemma 4.4. The sequence of functions (U i ) i≥0 converges uniformly on C([0, T ], S d + ) to its simple limit U introduced in Lemma 4.2.

Proof. From Lemma 4.2, we have that (U i ) i≥0 is a non increasing sequence of continuous functions converging pointwise to U . To obtain the uniform convergence it suffices to show that U is continuous and apply Dini's theorem. To do so our strategy is to show that t → U t solves an equation whose solutions are continuous.

Step 1. Equation satisfied by U . By definition Γ i+1 is solution to (4.9), thus by integrating over τ, θ and applying Fubini's theorem we get

U i+1 t = T t Fi (t, r) U i+1 dr, Fi (t, r)(U i+1 ) = I i (t, r) + II i (t, r) + III i (t, r) + III i (t, r) ⊤ , (4.15)
where

I i (t, r) = R 2 + e -θ(r-t) µ(dθ) ⊤ Qi r (θ, τ )µ(dτ )e -τ (r-t) , II i (t, r) = R + e -θ(r-t) µ(dθ) ⊤ Di r (θ) ⊤ U i+1 r R + Di r (τ )µ(dτ )e -τ (r-t) , III i (t, r) = R + e -θ(r-t) µ(dθ) ⊤ Bi r (θ) ⊤ V i r (e •(t-r) ),
with Bi , Di and Qi defined as in (4.8). The pointwise convergences and the uniform bounds stated in Lemma 4.2 allow us to apply the dominated convergence theorem to (4.15) to get

U t = T t F (t, r) (U ) dr, F (t, r) (U ) = I(t, r) + II(t, r) + III(t, r) + III(t, r) ⊤ , (4.16)
where

I(t, r) = K(r -t) ⊤ QK(r -t) + Θ(t, r) ⊤ N Θ(t, r), II(t, r) = DK(r -t) + F ⊤ Θ(t, r) ⊤ U r DK(r -t) + F ⊤ Θ(t, r) , III(t, r) = V r e •(t-r) BK(t -r) + C ⊤ Θ(t, r) , Θ(t, r) = -N + F ⊤ U r F -1 F ⊤ U r DK(r -t) + C ⊤ V r e •(t-r) .
Step 2. Continuity of t → U t . We first observe that by virtue of Lemma 4.2 t → U t and t → V t are bounded on [0, T ] so that there exists c > 0 such that

F (t, r)(U ) ≤ c 1 + |K(r -t)| 2 , t ≤ r ≤ T.
(4.17) Fixing t ≤ s ≤ T , it follows from (4.16) that

U t -U s = s t F (t, r) (U ) dr + T s F (t, r) (U ) -F (s, r) (U ) dr = 1 t,s + 2 t,s .
By (4.17),

|1 t,s | ≤ c s -t + K 2 L 2 (0,s-t) .
By virtue of the square integrability of K, the right hand side goes to 0 as t ↑ s. Similarly,

using u ⊤ Qu -v ⊤ Qv = (u + v) ⊤ Q(u -v), we get |2 t,s | ≤c T s |K(r -t) -K(r -s)| |K(r -t) + K(r -s)| dr + c T s |V r (e •(t-r) ) -V r (e •(s-r) )|dr ≤ A t,s + B t,s ,
where c is a constant. The first term can be easily handled with Cauchy-Schwarz inequality

A t,s ≤2c K 2 L 2 (0,T ) T 0 |K(r + s -t) -K(r)| 2 dr,
which shows that A t,s converges to zero as t goes to s, recall (3.12). For the second term note that for all i ∈ N, t ≤ s ≤ r ≤ T ,

R 2 + µ(dθ) ⊤ Γ i r (θ, τ )µ(dτ )e -τ (r-t) - R 2 + µ(dθ) ⊤ Γ i r (θ, τ )µ(dτ )e -τ (r-s) ≤ ess sup τ ′ ∈R + R + µ(dθ) ⊤ Γ i r (θ, τ ′ ) R + e -τ (r-t) -e -τ (r-s) |µ|(dτ ) ≤ κ K(r -s) -K(r -t) , (4.18)
where κ is the uniform bound from Lemma 4.3 and

K(t) = R + e -θt |µ|(dθ), t > 0. (4.19)
Taking the limit i → ∞ in (4.18) and invoking Lemma 4.2, we obtain

V r (e •(t-r) ) -V r (e •(s-r) ) ≤ κ K(r -s) -K(r -t) .
Thus, similarly as for A t,s we get that B t,s converges to 0 as t goes to s. As a result U is continuous.

Lemma 4.5. For any t ≤ T , (Γ i t ) i≥0 is a Cauchy sequence in L 1 (µ ⊗ µ). Proof. Let t ≤ T and i ≤ j. Let Θ j , Bj , Dj be defined as in (4.8) for any j ∈ N. Then cumbersome but straightforward computations, detailed in Lemma A.2, yield that ∆ i,j t = Γ i t -Γ j t solves the Lyapunov equation

           ∆ ij t (θ, τ ) = T t e -(θ+τ )(s-t) F δ ij (s, ∆ ij s )(θ, τ )ds, F δ ij (t, ∆)(θ, τ ) = Qij,δ t (θ, τ ) + Dj-1 t (θ) ⊤ R 2 + µ(dθ ′ ) ⊤ ∆(θ ′ , τ ′ )µ(dτ ′ ) Dj-1 t (τ ) + Bj-1 t (θ) ⊤ R + µ(dθ ′ ) ⊤ ∆(θ ′ , τ ) + R + ∆(θ, τ ′ )µ(dτ ′ ) Bj-1 t (τ ) +S ij t (θ) ⊤ ρ ij t (τ ) + ρ ij t (θ) ⊤ S ij t (τ ), (4.20)
where

ρ ij = Θ i-1 -Θ j-1 , Qij,δ t (θ, τ ) = ρ ij t (θ) ⊤ N + F ⊤ U i t F ρ ij t (τ ), S ij t (τ ) = C ⊤ R + µ(dθ ′ ) ⊤ Γ i t (θ ′ , τ ) + F ⊤ U i t D + N + F ⊤ U i t F Θ j-1 s (τ ),
and U is defined as in Lemma 4.2. We will show that ∆ ij t L 1 (µ⊗µ) → 0 as i, j → ∞ by successive applications of Gronwall inequality and by showing that ρ ij is small enough. For this, we fix ǫ > 0 and we denote by c > 0 a scalar independent of i, j, t, τ and θ that may vary from line to line throughout the proof.

Step 1. We bound the terms |ρ ij t (τ )| and R + ∆ ij t (θ, τ )µ(dτ ) . We first write

ρ ij t (τ ) = N + F ⊤ U j t F -1 -N + F ⊤ U i t F -1 F ⊤ U j t D + C ⊤ R + µ(dθ ′ ) ⊤ Γ j t (θ ′ , τ ) -N + F ⊤ U i t F -1 F ⊤ U i t -U j t D + C ⊤ R + µ(dθ ′ ) ⊤ ∆ ij t (θ ′ , τ ) . (4.21)
By the uniform convergence of the sequence of functions U i i≥0 , obtained in Lemma 4.4, one can find n ′ ∈ N such that

U i t -U j t + N + F ⊤ U j t F -1 -N + F ⊤ U i t F -1 ≤ ǫ, t ≤ T, i, j ≥ n ′ , (4.22)
where the bound for the second term comes from the matrix identity A -1 -B -1 = B -1 (B -A)A -1 . Furthermore, it follows from Lemmas 4.2 and 4.3 that

U i t ∨ R + Γ i t (θ, τ )µ(dτ ) ≤ c,
|ρ ij t (τ )| ≤ c ǫ + R + ∆ ij s (θ, τ )µ(dτ ) , µ -a.e., t ≤ T, i, j ≥ n ′ .
In addition, (4.23) yields that

|Θ i t (θ)| ≤ c, µ -a.e., t ≤ T, i ≥ 0, (4.24)
which in turn implies

R + e -θ(s-t) Θ i t (θ) |µ|(dθ) ≤ κ ′ K(s -t), s ≤ t ≤ T, i ≥ 0,
where K is given by (4.19). Fix i, j ≥ n ′ and t ≤ T . Combining all the above and integrating equation (4.20) over the τ variable leads to

R + ∆ ij t (θ, τ )µ(dτ ) ≤ c T t (1 + K 2 (s -t)) ǫ + R + ∆ ij s (θ, τ )µ(dτ ) ds, µ -a.e.
An application of the generalized Gronwall inequality for convolution equation with R the resolvent of c(1 + K 2 ), see [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF]Theorem 9.8.2], yields

R + ∆ ij t (θ, τ )µ(dτ ) ≤ ǫc T + K 2 L 2 (0,T ) 1 + R L 1 (0,T ) , µ -a.e. ( 4
∆ ij t L 1 (µ⊗µ) ≤c T t 1 + K 2 (t -s) ǫ + ∆ ij s L 1 (µ⊗µ) ds.
Another application of the generalized Gronwall inequality for convolution equations yields that

∆ ij t L 1 (µ⊗µ) ≤ ǫc T + K 2 2 L 2 (0,T ) 1 + R L 1 (0,T ) .
This proves that (Γ i t ) i≥0 is a Cauchy sequence in L 1 (µ ⊗ µ) for every t ∈ [0, T ].

4.3

Step 3: The limiting point of (Γ i t ) i≥0 solves the Riccati equation ensures that there exists a constant c > 0 such that

|Γ i t (θ, τ )| ≤ c T t e -(θ+τ )(s-t) ds ≤ (1 ∨ T ) 1 ∧ (θ + τ ) -1 since 1 -e -θt ≤ (1 ∨ t) 1 ∧ θ -1 .
Hence the dominated convergence theorem yields

R 2 + µ(dθ) ⊤ Γ i t (θ, τ )µ(dτ ) → R 2 + µ(dθ) ⊤ Γ t (θ, τ )µ(dτ ), R 2 + µ(dθ) ⊤ Γ i t (θ, τ ) → R 2 + µ(dθ) ⊤ Γ t (θ, τ ) and R 2 + Γ i t (θ, τ )µ(dτ ) → R 2 + Γ t (θ, τ )µ(dτ ), µ -a.e.
Thus, as i → ∞ we have

Θ i t (θ) → Θ t (θ) = N + F ⊤ R 2 + µ(dθ) ⊤ Γ t (θ, τ )µ(dτ )F -1 × F ⊤ R 2 + µ(dθ) ⊤ Γ t (θ, τ )µ(dτ )D + C ⊤ R + Γ t (θ, τ )µ(dτ ) B i t (θ) → B + C ⊤ Θ t (θ) D i t (θ) → D + F ⊤ Θ t (θ) (4.28)
By plugging these convergences into (4.9) we obtain that the limit Γ solves

Γ t (θ, τ ) = T t e -(θ+τ )(s-t) R s (θ, τ )ds, µ ⊗ µ -a.e. (4.29)
with

R t (θ, τ ) =Q + Θ t (θ)N Θ t (τ ) + R + Γ t (θ, τ )µ(dτ ) B + C ⊤ Θ t (τ ) + B + C ⊤ Θ t (θ) ⊤ R + µ(dθ) ⊤ Γ s (θ, τ ) + D + F ⊤ Θ t (θ) ⊤ R 2 + µ(dθ) ⊤ Γ t (θ, τ )µ(dτ ) D + F ⊤ Θ t (τ ) .
By using the expression of Θ exhibited in (4.28), we get that R t (θ, τ ) = R(Γ t )(θ, τ ), where R is given by (2.3), so that (4.29) is the desired Riccati equation. Finally, the uniform bounds obtained in Lemmas 4.3 and 4.4 and plugged into (4.29) imply (4.27).

Step 4: Continuity and uniqueness

We now establish the estimate (2.7) for the solutions of the Riccati equation, which in turn implies continuity.

Lemma 4.7. Assume that there exists a L 1 (µ ⊗ µ)-valued function t → Γ t such that (2.5) holds with (4.27). Then, the estimate (2.7) holds and

Γ ∈ C([0, T ], L 1 (µ⊗µ)). If in addition Q ∈ S d + , then Γ t ∈ S d + (µ ⊗ µ)
, for all t ≤ T . Proof. The proof of the estimate follows the same lines as that of Lemma 3.4, with constant coefficients. The only difference is the nonlinear term

V t (θ, τ ) = T t e -(θ+τ )(s-t) S(Γ s )(θ) ⊤ N -1 (Γ s )S(Γ s )(τ )ds, which we can bound as follows. Let Ŝ(Γ)(s)(θ) = |F ||D| Γ s L 1 (µ⊗µ) +|C| R + |µ|(dτ ′ )|Γ(θ, τ ′ )|.
Integration over the τ -variable, using the bound e -θ(s-t) ≤ 1 and Tonelli's theorem give for a constant c that may vary from line to line

R + |µ|(dτ )|V t (θ, τ )| ≤ R + |µ|(dτ ) T t e -(θ+τ )(s-t) Ŝ(Γ)(s)(θ)|N -1 | Ŝ(Γ)(s)(τ )ds ≤ c sup s≤T Γ s L 1 (µ⊗µ) T 0 (1 + K(s))ds + c sup s≤T Γ s L 1 (µ⊗µ) T t (1 + K(s -t)) R + |µ|(dτ ′ )|Γ s (θ, τ ′ )|ds,
where K is defined as in (4.19). The first four terms appearing in R + |µ|(dτ )|Γ t (θ, τ )| lead to inequality (3.10), with (Γ, K, µ, c) instead of (Ψ, K 2 , µ 2 , κ). Adding the previous bound for the fifth nonlinear term yields

R + |µ|(dτ )|Γ t (θ, τ )| ≤c 1 + sup s∈[0,T ] Γ s L 1 (µ 1 ⊗µ) T 0 1 + K(s) ds + c T t (1 + K(s -t)) R + |µ|(dτ ′ )|Γ s (θ, τ ′ )|ds.
The claimed estimate now follows from the generalized Gronwall inequality for convolution equations, see [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF]Theorem 9.8.2].

To argue continuity, we recall that the Riccati equation (2.5) can be recast as a Lyapunov equation as in (4.29). The claimed continuity is therefore a consequence of Lemma 3.5 provided that the coefficients of (4.29) are bounded, which amounts to showing that the functions t

→ R 2 + µ(dθ) ⊤ Γ t (θ, τ )µ(dτ ) and (t, θ) → R + Γ t (θ, τ )µ(dτ ) are bounded.
The boundedness of the former is ensured by (4.27) and that of the latter follows from the estimate (2.7). If in addition Q ∈ S d + , then Lemma 4.1 applied for (4.29) yields that Γ t ∈ S d + (µ ⊗ µ) for any t ≤ T .

Finally, exploiting once more the correspondence with the Lyapunov equation, uniqueness for the Riccati equation is obtained as a consequence of Theorem 3.1 and Lemma 4.1.

Lemma 4.8. There exists at most one solution to (2.5) such that (2.7) and (4.27) hold.

Proof. Let Γ a and Γ b be two solutions of (2.5) such that (2.7) and (4.27) hold. For i ∈ {a, b}, observe that Γ i can be recast as a solution to a Lyapunov equation with bounded coefficients in the form (4.9). As a result, ∆ = Γ a -Γ b can be written as a solution to the following Lyapunov equation with bounded coefficients (see Lemma A.2 for details): where

           ∆ t (θ, τ ) = T t e -(θ+τ )(s-t) F ab (s, ∆ s )(θ, τ )ds, F ab (t, ∆)(θ, τ ) = Q ab t (θ, τ ) + D b t (θ) ⊤ R 2 + µ(dθ ′ ) ⊤ ∆(θ ′ , τ ′ )µ(dτ ′ )D b t (τ ) +B b t (θ) ⊤ R + µ(dθ ′ ) ⊤ ∆(θ ′ , τ ) + R + ∆(θ, τ ′ )µ(dτ ′ )B
ρ ab = Θ a -Θ b , Q ab t (θ, τ ) = ρ ab t (θ) ⊤ N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ a t (θ ′ , τ ′ )µ(dτ ′ )F ρ ab t (τ ), S ab t (τ ) = C ⊤ R + µ(dθ ′ ) ⊤ Γ a (θ ′ , τ ) + F ⊤ R 2 + µ(dθ ′ )Γ a s (θ ′ , τ ′ )µ(dτ ′ )D+ + N + F ⊤ R 2 + µ(dθ ′ )Γ a s (θ ′ , τ ′ )µ(dτ ′ )F Θ a s (τ ).
The fact that the coefficients are bounded comes from (2.7) and (4.27) on Γ a and Γ b . Now, one can note similarly as in (4.21) that ρ ab can be re-written as

ρ ab s (τ ) = -N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ a t (θ ′ , τ ′ )µ(dτ ′ )F -1 × F ⊤ R 2 + µ(dθ ′ ) ⊤ ∆ t (θ ′ , τ ′ )µ(dτ ′ )D + C ⊤ R + µ(dθ ′ ) ⊤ ∆ t (θ ′ , τ ) +   N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ b t (θ ′ , τ ′ )µ(dτ ′ )F -1 -N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ a t (θ ′ , τ ′ )µ(dτ ′ )F -1   × F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ b t (θ ′ , τ ′ )µ(dτ ′ )D + C ⊤ R + µ(dθ ′ ) ⊤ Γ b t (θ ′ , τ ) = A(τ ) + B(τ ),
which is linear in ∆ since B(τ ) can be rewritten as

N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ b t (θ ′ , τ ′ )µ(dτ ′ )F -1 -N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ a t (θ ′ , τ ′ )µ(dτ ′ )F -1 = N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ b t (θ ′ , τ ′ )µ(dτ ′ )F -1 F ⊤ R 2 + µ(dθ ′ ) ⊤ ∆ t (θ ′ , τ ′ ) µ(dτ ′ )F × N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ a t (θ ′ , τ ′ )µ(dτ ′ )F -1
.

Consequently, ∆ = Γ a -Γ b is solution to a homogeneous linear Lyapunov equation with bounded coefficients, and no affine term. Thus, the generalized Gronwall inequality for convolution equations, see [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF]Theorem 9.8.2] ensures that ∆ t L 1 (µ⊗µ) = 0 for every t ∈ [0, T ], which proves uniqueness.

A Some elementary results By expanding the square we obtain a non negative second order polynomial in λ whose discriminant must be non positive. This combined with Ψ(θ, τ ) = Ψ(τ, θ) ⊤ yields the claimed inequality.

Lemma A.2. Let (Γ i ) i≥0 be the sequence defined in (4.9). Then for any 1 ≤ i < j,

∆ ij = Γ i -Γ j is solution to            ∆ ij t (θ, τ ) = T t e -(θ+τ )(s-t) F δ ij (s, ∆ ij s )(θ, τ )ds, F δ ij (t, ∆)(θ, τ ) = Q ij,δ t (θ, τ ) + D j-1 t (θ) ⊤ R 2 + µ(dθ ′ ) ⊤ ∆(θ ′ , τ ′ )µ(dτ ′ )D j-1 t (τ ) +B j-1 t (θ) ⊤ R + µ(dθ ′ ) ⊤ ∆(θ ′ , τ ) + R + ∆(θ, τ ′ )µ(dτ ′ )B j-1 t (τ ) +S ij t (θ) ⊤ ρ ij t (τ ) + ρ ij t (θ) ⊤ S ij t (τ ), (A.1)
where

ρ ij = Θ i-1 -Θ j-1 , Q ij,δ t (θ, τ ) = ρ ij t (θ) ⊤ N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ i t (θ ′ , τ ′ )µ(dτ ′ )F ρ ij t (τ ), S ij t (τ ) = C ⊤ R + µ(dθ ′ ) ⊤ Γ i (θ ′ , τ ) + F ⊤ R 2 + µ(dθ ′ )Γ i s (θ ′ , τ ′ )µ(dτ ′ )D+ + N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ i s (θ ′ , τ ′ )µ(dτ ′ )F Θ j-1 s (τ ).
Remark A.3. Note that when j = i + 1, then S i(i+1) = 0. Indeed, in such case we have

S i(i+1) s (θ) =C ⊤ R + µ(dθ ′ )Γ i s (θ ′ , τ ) + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ i s (θ ′ , τ ′ )D j-1 s (τ ) + N Θ i s (τ ) =C ⊤ R + µ(dθ ′ )Γ i s (θ ′ , τ ) + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ i s (θ ′ , τ ′ )D + N + F ⊤ R 2 + µ(dθ ′ ) ⊤ Γ i s (θ ′ , τ ′ )µ(dτ ′ )F Θ i s (τ ) = 0.
As a consequence, in the particular case where j = i + 1, ∆ i = ∆ i(i+1) is solution to (4.10).

Proof.

Remark 2 . 2 .

 22 (On utilise cette notation dans le lemme 4.1(ii)) S d + (δ 0 ⊗ δ 0 ) reduces to S d + , the cone of symmetric semidefinite d × d-matrices.

Fix d 11

 11 , d 12 , d 21 , d 22 ∈ N.

Lemma 3 . 4 .

 34 The function Ψ constructed in Lemma 3.3 satisfies the estimates (3.6)-(3.7).

  e., so that an application of the generalized Gronwall inequality for convolution equations, see [9, Theorem 9.8.2], yields the estimate (3.7). Lemma 3.5. Under the assumptions of Theorem 3.1, let t ∈ [0, T ] → Ψ t be such that (3.2) holds, with (3.5), (3.6) and (3.7). Then,

  Prove that the limiting point is a solution to the Riccati equation (2.5), (iv) Derive the estimate (2.7), the continuity and the uniqueness for the Riccati solution.

4. 1 Lemma 4 . 1 .

 141 Step 1: Construction of a sequence of Lyapunov solutions(Γ i t ) i≥0 Let Ψ ∈ C([0, T ], L 1 (µ ⊗ µ))denote the solution to the Lyapunov equation (3.2) produced by Theorem 3.1 for the configuration d 11 = d 21 = d, d 12 = d 22 = d ′ , µ 1 = µ 2 = µ, B1 = B2 = B, D1 = D2 = D. (4.1)

Lemma A. 1 . 2 +

 12 Let Ψ ∈ S d + (µ ⊗ µ), and Ψ its corresponding linear integral operator. Then for any ϕ, ψ ∈ L 1 (µ) ϕ, Ψψ 2 µ ≤ ϕ, Ψϕ µ . ψ, Ψψ µ Proof. Since Ψ ∈ S d + (µ ⊗ µ), then for any ϕ, ψ ∈ L 1 (µ) and λ ∈ R we have R (ϕ(θ) + λψ(θ)) ⊤ µ(dθ) ⊤ Ψ(θ, τ )µ(dτ ) (ϕ(τ ) + λψ(τ )) ≥ 0.

  ) see[START_REF] Jaber | Linear-Quadratic control for a class of stochastic Volterra equations: solvability and approximation[END_REF] Theorem 4.1]. Recall the bold notation G in (2.1) for the integral operator generated by a kernel G. Note that, by virtue of Remark 3.2, Ψ s ∈ L ∞ (µ ⊗ µ), so that Ψ s : L 1 (µ) → L ∞ (µ ⊤), for all s ≤ T . This shows that s → Y s , Ψ s Y s µ is well-defined P-a.s. An application of Itô's formula (see[START_REF] Jaber | Linear-Quadratic control for a class of stochastic Volterra equations: solvability and approximation[END_REF] Lemma 5.1]) to the process s → Y s , Ψ s Y s µ yields, due to the vanishing terminal condition for Ψ, and after successive applications of Fubini's theorem:

  Lemma 4.6. Assume that (2.6) holds. For each t ≤ T , denote by Γ t the limiting point in L 1 (µ ⊗ µ) of the sequence Γ i t i≥0 obtained from Lemma 4.5. Then, t → Γ t solves the Riccati equation (2.5) with Proof. Fix t ≤ T . By virtue of the L 1 (µ ⊗ µ) convergence,Γ i t (θ, τ ) → Γ t (θ, τ ) µ ⊗ µa.e.

	sup t≤T	Γ t L 1 (µ⊗µ) < +∞.	(4.27)
	Furthermore the boundedness of (i, t) → R 2 +	µ(dθ) ⊤ Γ i t (θ, τ )µ(dτ ) , (i, t, θ) → R + Γ i t (θ, τ )µ(dτ )
	and (i, t, τ ) → R + µ(dθ) ⊤ Γ i t (θ, τ ) , see Lemmas 4.2 and 4.3, combined with equation (4.9)

The work of this author is supported by FiME (Finance for Energy Market Research Centre) and the "Finance et Développement Durable -Approches Quantitatives" EDF -CACIB Chair.


Hoa Phuoc

Truong

Damien Flieller
email: damien.flieller@insa-strasbourg.fr

Ngac Ky Nguyen
email: ngacky.nguyen@ensam.eu

Jean Mercklé
email: jean.merckle]@uha.fr

Guy Sturtzer
email: guy.sturtzer@insa-strasbourg.fr

Torque Ripple Minimization in Non-Sinusoidal Synchronous Reluctance Motors Based on Artificial Neural Networks

Keywords: Non-sinusoidal Synchronous Reluctance Motor, Torque Ripple, Optimal Currents, Lagrange Optimization, Adaline, Artificial Neural Networks

This paper proposes a new method based on Artificial Neural Networks for reducing the torque ripple in a non-sinusoidal Synchronous Reluctance Motor. The Lagrange optimization method is used to solve the problem of calculating optimal currents in the d-q frame. A neural control scheme is then proposed as an adaptive solution to derive the optimal stator currents giving a constant electromagnetic torque and minimizing the ohmic losses.

Thanks to the online learning capacity of neural networks, the optimal currents can be obtained online in real time. With this neural control, each machine's parameters estimation errors and current controller errors can be compensated. Simulation and experimental results are presented which confirm the validity of the proposed method.

Introduction

ynchronous reluctance motor (SynRM) has received much attention for many applications in recent years due to its structural simplicity, low manufacturing cost and rugged construction [START_REF] Taghavi | A Sizing Methodology of the Synchronous Reluctance Motor for Traction Applications[END_REF]- [START_REF] Moghaddam | Novel High-Performance SynRM Design Method: An Easy Approach for A Complicated Rotor Topology[END_REF]. However, a high level of torque ripple which produces mechanical vibrations and acoustic noise is one of the major drawbacks of this motor.

Torque smoothness is an essential requirement in many applications. Therefore, many authors have proposed different methods to minimize the torque ripple with this kind of machine. In [START_REF] Jahns | Pulsating torque minimization techniques for permanent magnet AC motor drives-a review[END_REF], the authors pointed out that there are two approaches that minimize the torque ripple of synchronous motors. The first one consists of techniques to adjust the machine's stator and rotor design in order to cancel the undesirable torque ripple. The authors in [START_REF] Vagati | Design of low-torque-ripple synchronous reluctance motors[END_REF]- [START_REF] Bianchi | Rotor Flux-Barrier Design for Torque Ripple Reduction in Synchronous Reluctance and PM-Assisted Synchronous Reluctance Motors[END_REF] proposed the methods for reducing the torque ripple by adjusting the flux barrier in rotor structure. The effect of rotor skewing to minimize the torque ripple has been studied in detail [START_REF] Bomela | Effect of stator chording and rotor skewing on performance of reluctance synchronous machine[END_REF]- [START_REF] Islam | Permanent-Magnet Synchronous Motor Magnet Designs With Skewing for Torque Ripple and Cogging Torque Reduction[END_REF]. The authors in [START_REF] Bomela | Effect of stator chording and rotor skewing on performance of reluctance synchronous machine[END_REF] and [START_REF] Islam | Permanent-Magnet Synchronous Motor Magnet Designs With Skewing for Torque Ripple and Cogging Torque Reduction[END_REF] show that the torque ripple is minimized when the rotor is skewed with an angle which is equal to a stator slot pitch.

The second approach is based on the active control schemes which modify the stator currents and propose the best currents for cancelling the undesired torque ripple. The authors in [START_REF] Hamiti | Modelling of a synchronous reluctance machine accounting for space harmonics in view of torque ripple minimization[END_REF] worked in the d-q frame in order to calculate the optimal currents. The copper losses in this method are not minimized because the direct current is forced to be equal to the quadrature one Id= Iq. Also working in the d-q reference frames, the works presented in [START_REF] Sturtzer | Mathematical and experimental method to obtain the inverse modeling of nonsinusoidal and saturated synchronous reluctance motors[END_REF]- [START_REF] Mademlis | Compensation of magnetic saturation in maximum torque to current vector controlled synchronous reluctance motor drives[END_REF] give the expressions of optimal currents to minimize the torque ripple. The authors of [START_REF] Sturtzer | Mathematical and experimental method to obtain the inverse modeling of nonsinusoidal and saturated synchronous reluctance motors[END_REF] and [START_REF] Ait-Gougam | Inverse modelling and pulsating torque minimization of salient pole non-sinusoidal synchronous machines[END_REF] propose an extended Park transformation to obtain optimal currents in the nonsinusoidal machine, while the authors in [START_REF] Mademlis | Compensation of magnetic saturation in maximum torque to current vector controlled synchronous reluctance motor drives[END_REF] obtain optimal currents to achieve a maximum torque-to-current condition which takes into account the effect of magnetic saturation. Based on input-output linearization, the authors in [START_REF] Hyeoun-Dong | Efficiency-optimized direct torque control of synchronous reluctance motor using feedback linearization[END_REF] and [START_REF] Zarchi | Adaptive Input-Output Feedback-Linearization-Based Torque Control of Synchronous Reluctance Motor Without Mechanical Sensor[END_REF] propose a method to obtain optimal S currents that give the constant torque and minimize the losses. Nonlinear controllers are proposed in [START_REF] Hyeoun-Dong | Efficiency-optimized direct torque control of synchronous reluctance motor using feedback linearization[END_REF] to regulate the torque by selecting the product of d-axes and q-axes torque currents as one of the output variables. The cross-coupling effects and iron losses are taken into account in [START_REF] Hyeoun-Dong | Efficiency-optimized direct torque control of synchronous reluctance motor using feedback linearization[END_REF]. Based on sliding mode control (SMC) [START_REF] Inanc | Torque ripple minimization of a switched reluctance motor by using continuous sliding mode control technique[END_REF], the value of the reference current is adjusted in order to keep the speed of the motor constant. Therefore, the torque ripple of the motor is minimized. The injection of current harmonic is proposed in [START_REF] Geun-Ho | Torque Ripple Reduction of Interior Permanent Magnet Synchronous Motor Using Harmonic Injected Current[END_REF], the disadvantage of this method is high torque ripple because the authors optimize the currents only for harmonics of ranks 5 and 7. Recently, based on direct torque control (DTC), the works in [START_REF] Zhang | An Improved Robust Field-Weakeaning Algorithm for Direct-Torque-Controlled Synchronous-Reluctance-Motor Drives[END_REF]- [START_REF] Bolognani | Online MTPA Control Strategy for DTC Synchronous-Reluctance-Motor Drives[END_REF] have proposed controlling the stator flux and generating the torque. In [START_REF] Yongchang | A Simple Method to Reduce Torque Ripple in Direct Torque-Controlled Permanent-Magnet Synchronous Motor by Using Vectors With Variable Amplitude and Angle[END_REF], the amplitude and angle of the commanding voltage vectors were derived from the errors of torque and flux. Therefore, the torque and flux-ripples are minimized. Based on torque predictive control [START_REF] Hao | Torque Ripple Reduction of the Torque Predictive Control Scheme for Permanent-Magnet Synchronous Motors[END_REF], the optimized voltage is utilized to reduce torque ripple. In that method, the voltage angle vector is determined from the output of torque and flux hysteresis controllers. Another method based on the injection of high-frequency current presented in [START_REF] Bolognani | Online MTPA Control Strategy for DTC Synchronous-Reluctance-Motor Drives[END_REF], the MTPA point can be detected because the variation in the torque based on the variation in the current angle is zero at the MTPA points. In [START_REF] Daryabeigi | Online MTPA Control Approach for Synchronous Reluctance Motor Drives Based on Emotional Controller[END_REF], the optimal currents are obtained based on emotional controller and space vector modulation (SVM) under an automatic search of the MTPA strategy. In [START_REF] Wenqing | Accurate Torque Control of Interior Permanent Magnet Synchronous Machine[END_REF], the estimated difference of d-q inductance was used to achieve MTPA control and accurate torque control. While all of these authors work only with sinusoidal machines, in this paper we work instead with non-sinusoidal SynRMs.

Unlike the approaches mentioned above, in this paper we use an adaptive technique based on Artificial Neural Networks (ANNs) [START_REF] Hoa | An investigation of Adaline for torque ripple minimization in Non-Sinusoidal Synchronous Reluctance Motors[END_REF]- [START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Non-sinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF] to obtain the optimal stator currents. These optimal currents minimize the copper losses and give exactly the electromagnetic torque desired in non-sinusoidal SynRMs. The ANNs presented in this paper is the Adaline (ADAptive LInear NEuron), which uses an online learning process based on the Widrow-Hoff algorithm. Therefore, the optimal currents are obtained online in real time. Moreover, comparing our copper losses with those from previous methods confirms the validity of the proposed method.

The remainder of this paper is organized as follows: torque calculation of the SynRM is presented in Section 2. Section 3 presents the Lagrange optimization method to obtain the optimal currents. The investigation of Adaline Neural Networks for torque and speed controllers is proposed in Section 4. Simulation and experimental results are shown in Section 5 and 6 respectively. Finally, some conclusions are given in Section 7.

Torque computation of synchronous reluctance motors

The electromagnetic torque of the SynRM is expressed as:

  ( ) 1 2 T e p T         L i i (1)
where

  T a b c i i i  i
is the stator currents vector.

p : The number of pole pairs, and : the mechanical angle.

The matrix of inductances  

( ) p L
is expressed as follows:

                  ( ) ( ) a a b a c ab b bc ac bc c L M M p M L M M M L p p p p p p p p p                      L (2)
where ( ), ( ), ( ) In the case of sinusoidal excitation, the currents vector is expressed as:  

p                                                i (3)
with  : the load angle. Therefore, in order to maximize the mean value of electromagnetic torque, the load angle is chosen as:  = 45° [START_REF] Hamiti | Modelling of a synchronous reluctance machine accounting for space harmonics in view of torque ripple minimization[END_REF].

An accurate of self and mutual inductances is necessary in the analysis of the SynRM.

Because of rotor saliency and stator windings distribution, the self and mutual inductances of a SynRM are non-sinusoidal. The electromagnetic torque produced by this machine presents the torque ripple when it is fed by sinusoidal currents [START_REF] Sturtzer | Mathematical and experimental method to obtain the inverse modeling of nonsinusoidal and saturated synchronous reluctance motors[END_REF], [START_REF] Hoa | An investigation of Adaline for torque ripple minimization in Non-Sinusoidal Synchronous Reluctance Motors[END_REF].

The measurement of the self and mutual inductances is realized on our laboratory machine test bench. The measurements are done at the stand-still [START_REF] Hamiti | Modelling of a synchronous reluctance machine accounting for space harmonics in view of torque ripple minimization[END_REF]- [START_REF] Sturtzer | Mathematical and experimental method to obtain the inverse modeling of nonsinusoidal and saturated synchronous reluctance motors[END_REF]. The results of the measurements are compared with the Finite Element Method (FEM) when using JMAG software (see Fig. 1). The comparison of the FEM and measurement results of the self and mutual inductances is shown in the Fig. 2 and Fig. 3 respectively. One can be seen that the errors between the FEM and measurement results are not significant.

The expressions of the self and mutual inductances with the significant harmonics are:

          0.
                                                                  (4)
The electromagnetic torque obtained with the sinusoidal currents for the SynRM whose parameters are presented in equation ( 4) is shown in Fig. 4. It can be noticed that the torque ripple is important (around 39% of the required value). Therefore, the optimal currents are thus required for reducing the torque ripple in this SynRM. In the next section, we will present the calculation of optimal currents to achieve MTPA strategy by means of the Lagrange optimization in order to reduce torque ripple.

Optimal currents based on Lagrange optimization

The Park's transformation is defined as:

  2 2 cos( ) cos( ) cos( ) 2 3 3 2 2 3 sin( ) sin( ) sin( ) 3 3 p p p p p p                               P (5)
Using Park's transformation and assuming the zero sequence of current to be zero lead to:

  . a d T b q c i i i i i                   i P (6)
where , d q i i are the d and q current components.

Replacing ( 6) in (1), the electromagnetic torque is rewritten as:

      1 . 2 T d d T e q q i i p T i i                          L P P (7)
Let us define the matrix [A] as follows:

        ( ) ( ) 1 ( ) ( ) 2 T p a p c p c p b p                        L A P P (8)
By replacing ( 8) in [START_REF] Bianchi | Rotor Flux-Barrier Design for Torque Ripple Reduction in Synchronous Reluctance and PM-Assisted Synchronous Reluctance Motors[END_REF], the electromagnetic torque e T becomes:

2 2 ( ) ( ) 2 ( ) e d q d q T a p i b p i c p i i            (9)
In order to obtain the desired electromagnetic torque and minimize the copper losses (MTPA strategy), we use the Lagrange optimization.

The cost function is defined as the copper losses: .( )

j s d q P R i i   (10)
where j P : the copper losses, s R : the stator resistance.

This task leads to establish a Lagrange function:

2 2 2 2 ( ) ( ( 2
)

) e q q q d d d T L i i a i b i c i i              (11)
with  : Lagrange's multiplier.

The derivations of L according to ,

d q i i respectively give: 2 ( 2 2 ) 0 q q d i b i c i            (12) 2 ( 2 2
) 0

q d d i a i c i            (13)
From [START_REF] Ait-Gougam | Inverse modelling and pulsating torque minimization of salient pole non-sinusoidal synchronous machines[END_REF], we obtain [START_REF] Hyeoun-Dong | Efficiency-optimized direct torque control of synchronous reluctance motor using feedback linearization[END_REF]:

  _ _ 1 d opt q q opt a i i i c         (14)
where _

d opt i , _
q opt i : the d and q optimal current components.

Replacing _ q opt i in (9) and choosing _

d opt i > 0: _ 2 2 2 2 2 2 ( ) ( 2
2 )

e d d opt T i i a b a c c a b b c                  (15)
From ( 12) and ( 13), we obtain  as expressed in (16):

                                 (16)
Finally, three phase optimal currents will be obtained by:

  _ _ _ _ _ . a opt d opt T b opt q opt c opt i i i i i                      P (17)
Fig. 5 shows the phase currents and d-q currents in two different strategies of machine supply: sinusoidal currents and optimal currents given by expressions ( 14)- [START_REF] Geun-Ho | Torque Ripple Reduction of Interior Permanent Magnet Synchronous Motor Using Harmonic Injected Current[END_REF]. It is obvious that in order to obtain a desired electromagnetic torque and minimize the copper losses in the non-sinusoidal SynRM, the machine stator currents are not sinusoidal to compensate the torque ripple.

Fig. 4 compares the electromagnetic torque when the machine is fed by the optimal phase currents with the one obtained by the sinusoidal currents. As can be seen, the torque ripple was totally cancelled when using the optimal phase currents. This numerical result confirms the validity of the Lagrange's method.

In next section, we will present a new method based on Adaline Neural Networks to obtain the optimal currents. Thanks to learning capacity of neural networks, the optimal currents will be obtained online and each machine's parameters estimation errors can be compensated. That is the advantages of ANNs compared with the Lagrange optimization.

Investigation of Adaline Neural Networks for torque and speed controllers

Main ideas for the torque and speed control:

From ( 14), we can rewrite _ q opt i as:

    _ _ _ , (1 ) , (
)

e e d opt q opt opt d opt a i T p i K p i T p c             (18)
where:

(1 ) ( )

opt a K p c        (19)
The principal idea for the torque and speed controllers comes from the axis q optimal currents in [START_REF] Zhang | An Improved Robust Field-Weakeaning Algorithm for Direct-Torque-Controlled Synchronous-Reluctance-Motor Drives[END_REF]. We can observe that this current results from a scalar product of two components: the first one is the optimal function ( ) opt K p , and second one is the axis d optimal current _ ( , ) d opt e i T p .

Fig. 5b) and Fig. 6 show the current _ ( , ) d opt e i T p and the optimal function is estimated with:

( ) 1 0.062sin(6 ) 0.002sin(12 ) 0.001cos( 12)

opt K p p p p         (20)
It can be noticed that ( ) opt K p contains very few harmonics and it is independent of the electromagnetic torque e T .

We can write the optimal current _ ( )

d opt i
p by the sum of the Fourier's terms:

0 1 _ ( ) ( sin( ) cos( )) ai bi N i d d d d opt i p i i ip i ip         ( 21
)
where N: the highest considered harmonic.

We determine the vector containing the optimal current _ ...

aN bN T d d d a b d d i i i i i      W (22)
The optimal current _ d opt i will be learned and synthesized by the Adaline controller. The details are presented in the Section 4.2.

Torque and speed controllers based on Adaline Neural Networks:

We can remark that the torque (or speed) errors between the desired torque ref T and the calculated torque e T in Fig. 7 is necessary for deriving the optimal current _ d opt i

. We define the input vector of Adaline as:

  0 sin( ) cos( ) ... sin( ) cos( )

T x p p N p N p      X (23)
Similarly, the weights of the Adaline are defined as:

  0 1 1 ... T a b aN bN w w w w w  W (24)
These weights are solved using an iterative linear LMS (Least Mean Square) algorithm [START_REF] Widrow | Adaptive Inverse Control[END_REF] in order to minimize the torque (or speed) errors. They are adjusted according to:

( 1)

( ) . ( ). ( )

k k k k      W W X ( 25
)
Where X is the input vector,  is a learning rate,  is the torque (or speed) errors.

The Adaline controllers have to ensure that the electromagnetic torque (or speed) of the motor converge toward the desired one. Therefore, the weights of the Adaline are also enforced to converge toward values: *  W W [START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Non-sinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF] Finally, we will obtain the optimal current _ d opt i which is given by the expression (21).

Simulation of the proposed method by Adaline Neural Networks

To confirm the validity of the proposed method presented in the previous sections, a The results of neural torque control are shown in Fig. 9 with the desired torque

2 ref T  [N.m].
The optimal currents obtained online by Adaline controllers are close to their references as shown in Fig. 9a) to reduce the torque ripple. It can be noticed that the torque ripple obtained with proposed method in Fig. 9b) is not significant (only 2% of the desired torque) and the convergence of this strategy takes approximately one electrical cycle (about 25 ms). The weights of the Adaline are also learned online and their convergence is shown in Fig. 9c).

A simulation work with a varying torque is presented in order to evaluate the dynamic response of the proposed technique. Fig. 10 shows the obtained results when we change the desired torque to 5 ref T  [N.m] at t = 0.1s. It can be seen that a good response of Adaline's method is achieved by observing the optimal currents in Fig. 10a) and the torques obtained in Fig. 10b) converging to the reference values in a short time (about 20 ms). This good result of torque control is ensured by a convergence of the Adaline network's weights as shown in Fig. 10c).

Fig. 11 shows the comparison of the copper losses between the proposed methods and previously discussed method. It can be noticed that these results are presented in [START_REF] Hoa | An investigation of Adaline for torque ripple minimization in Non-Sinusoidal Synchronous Reluctance Motors[END_REF] with the SynRM type massive rotor. It is obvious that the copper losses are not minimized when d q i i  as proposed by the authors in [START_REF] Hamiti | Modelling of a synchronous reluctance machine accounting for space harmonics in view of torque ripple minimization[END_REF] (with 5% higher than the proposed method). The simulation results demonstrate a good performance of Adaline Neural Networks for minimizing the torque ripple.

Experimental results

The experimental platform is presented in Fig. 12. A three phase SynRM is connected to a three-phase voltage source inverter. The machine parameters are given in Table I. The rotor's position and the stator currents are measured in real time by using an incremental encoder and currents sensors. The measures are sent to a dSPACE DS1104 board hosted by a personal computer. The ANN controller has been implemented using the ACE 1104 board (DSPACE)

with Matlab/Simulink.

Neural Torque Control

The neural torque control is presented in the Fig. 8 when the switch s = 1. It can be noticed that the estimated electromagnetic torque will be calculated from the measured currents and the inductances of the SynRM as shown in Fig. 2 and Fig. 3.

The experimental results in steady state with the desired torque

  0.8 N.m ref T  are
reported in Fig. 13. Fig. 13a) compares the reference currents obtained by the proposed method with an Adaline controller and the measured currents. It is shown that the resulting currents provided by the neural controller are close to their references. The non-sinusoidal currents obtained with the Adaline controller compensate the torque ripple. Fig. 13c) shows the comparison of the electromagnetic torque obtained with the optimal currents with the one derived by the sinusoidal currents. It can be noticed that the torque ripple by optimal currents is about 8% of the desired torque and 38% when the machine is fed by the sinusoidal currents.

We can remark that in this case, the currents in d-q frame from Fig. 13b) have the higher harmonics. Therefore, the Adaline controller is a good candidate to choose in order to compensate the higher harmonics to minimize the torque ripple.

In order to show the convergence of our method, we present the results with dynamic torque as shown in Fig. 14. In this case, the desired torque

  2.5 N.m e T 
(about 36% of the nominal torque is applied at time t = 0.77s. Fig. 14a) and 14b) show the electromagnetic torque and the optimal phase currents obtained with the Adaline controller respectively. It can be noticed that the torque and currents obtained with the Adaline controller converge in a short time (about 50 ms), approximately with a half electrical cycle of the machine which is acceptable in many practical applications.

The experimental performances under load reduction are also presented in Fig. 15. As can be seen, the convergence of the proposed method within a short time. . We can remark that the torque ripple by optimal currents is about 13% of the desired torque and 44% when the machine is fed by the sinusoidal currents. These experimental results show the good performance of the proposed method based on an Adaline Neural Network.

In classical torque control, except the DTC control, the references of current are obtained by an analytical formulation from the torque, for example the MTPA-based vectorial control algorithm. Thus, an overshoot of torque depends strictly to the current controller performance.

If a good design of the current control is achieved, the overshoot can be avoided. It is the case where IP (Integrate-Proportional) controllers are used and a double pole is imposed.

We can remark that the torque response of the proposed method shows an overshoot (see Figs. 9b), 10b) and 14a), both in simulation and experimental tests. Indeed, the references of current are determined through a learning process of Adaline whose the training dynamic has a significant impact on the torque response in the close loop. The parameter that modifies the dynamic of convergence is the tuning factor η. Theoretically, this parameter could vary between 0 and 1 but a study more details has been given in [START_REF] Widrow | Adaptive Inverse Control[END_REF] for choosing a good value without destabilize the system. In our work, the right values of η are given in the Fig. 17 where there are some torque responses corresponding to different values of η (with the same current controllers). We can see that the torque response based on Adaline seems like a classical PI controller where there is always a compromise between the dynamic and the overshoot for a step excitation.

Simulation and experimental results given in the paper have been obtained with η=0.002. It was a choice for a compromise between the dynamic response and the torque response time. Now, it is possible to cancel completely the overshoot by taking η smaller but the torque response time is more important as shown in the Fig. 17.

A higher value of η could leads to an instability of the drive. This is verified in Fig. 18 where we see the drive losses the control of torque with η=0.01.

Neural Speed Control

The neural speed control is shown in the Fig. 8 when the switch s = 0. A constant reference speed has been fixed at ref ω = 70 (rad/s).

Fig. 19a) shows the comparison of the speed obtained with the proposed method and the one when the machine is fed by the sinusoidal currents. We can observe that the speed pulsation of the proposed method with an Adaline is much reduced in Fig. 19b). It is obvious that there is a torque ripple when the machine is supplied by the sinusoidal currents which lead to the speed pulsation. By using the proposed method, the torque and speed ripple that is presented in the control of conventional SynRM has been solved.

Conclusions

In this paper, we have proposed a new method based on Adaline Neural Networks for minimizing the torque and speed ripple in non-sinusoidal SynRMs. Thanks to this method, we can derive the optimal stator currents that precisely give the electromagnetic torque desired while minimizing the copper losses. With their learning capacity, short convergence time and good stability, the neural networks can obtain the optimal currents online in real time. The Adaline controllers take the place of the conventional torque or speed controller to ensure that the motor's torque or speed converge toward the desired ones. Our simulation and experimental results clearly show that the torque and speed ripples have been much reduced and the convergence of the proposed method is achieved within a short time. Furthermore, comparing our copper losses with those from previous methods confirms the validity of our new approach.

  mutual inductances.

  presented non-sinusoidal SynRM. We can observe that _d opt iand opt K are the periodic functions, it is thus possible to learn and estimate them by using Adaline Neural Networks.Therefore, we propose to learn online _( , ) Adaline with an input vector composed of cos( ) kp and sin( ) kp terms with k varying from 0 to 20[START_REF] Flieller | A Self-Learning Solution for Torque Ripple Reduction for Non-sinusoidal Permanent-Magnet Motor Drives Based on Artificial Neural Networks[END_REF]. After convergence, the function ( )

Matlab/

  Simulink program is used to simulate for the non-sinusoidal SynRM and the Adaline Neural Networks. The self and mutual inductances are expressed in (4). The reference speed is fixed at 1000 [rpm] in the simulation. The value of the training parameter used for simulation is fixed to 0.002   (a justification is given in the subsection 6.1). The parameters of the machine for simulation and experimental tests are shown in Table I. The torque and speed controllers based on Adaline Neural Networks are presented in Fig. 8. It can be noticed that the current controllers shown in this figure are PI controllers.

Fig. 16

 16

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .

 1234 Fig. 1. Meshed FEM model of the studied SynRM by JMAG

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 10 .Fig. 11 .Fig. 12 .

 5678101112 Fig. 5. Comparison in two cases: with and without optimal currents (simulation results).

Fig. 13 .

 13 Fig. 13. Experimental performances with desired torque

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Experimental performances with load increasing when

Fig. 16 .Fig. 17

 1617 Fig. 16. Experimental performances under nominal load

  Fig. 17. Torque responses corresponding to different training coefficients η

Fig. 18 .

 18 Fig. 18. Torque responses corresponding to higher values of η

List of tables