A Criterion for Robust Stability with Respect to Parametric Uncertainties Modeled by Multiplicative White Noise with Unknown Intensity, with Applications to Stability of Neural Networks

. In the present paper a robust stabilization problem of continuous-time linear dynamic systems with Markov jumps and corrupted with multiplicative (state-dependent) white noise perturbations is considered. The robustness analysis is performed with respect to the intensity of the white noises. It is proved that the robustness radius depends on the solution of an algebraic system of coupled Lyapunov matrix equations.


Introduction
The stochastic systems subject both to Markovian jumps and to multiplicative white noise perturbations received a considerable attention over the last years.Relevant results include the stability of such systems, optimal control and filtering (see e.g.[4], [5], [6], [8] and their references).In the present paper a robust stabilization problem of continuous-time linear dynamic systems with Markov jumps and corrupted with multiplicative (state-dependent) white noise perturbations is considered.The robustness analysis is performed with respect to the intensity of the white noise terms.It is proved that the robustness radius depends on the solution of an algebraic system of coupled Lyapunov matrix equations.The derived results are a generalization of the ones proved in [10] for the case without Markovian jumps.The paper is organized as follows: in the next section the problem statement is presented.The third section includes some preliminary results concerning the Lyapunov operators associated to the considered class of stochastic systems.The main result is presented and proved in Section 4. In the last section the stability radius is determined for two relevant particular cases and a numerical example illustrates the theoretical developments.

The problem statement
Consider the system of stochastic linear differential equations: where {w l (t)} t≥0 , 1 ≤ l ≤ r, are one-dimensional independent standard Wiener processes defined on a given probability space (Ω, F, P); {η t } t≥0 is a homogeneous standard right continuous Markov process defined on the same probability space (Ω, F, P) and taking value in the finite set N = {1, 2, ..., N } and having the transition semigroup P (t) = e Qt , t ≥ 0, where Q ∈ R N ×N is a matrix whose elements q ij satisfy the condition For more details we refer to [1], [3], [9], [12].We also assume that {η t } t≥0 , {w l (t)} t≥0 , 1 ≤ l ≤ r, are independent stochastic processes.In (1) the matrices known, while the scalars µ l ∈ R are unknown.The system (1) can be regarded as a perturbation of the so called nominal system The perturbed system (1) emphasizes the fact that the coefficients of the nominal system are affected by parametric uncertainties modeled by state multiplicative white noise perturbations with unknown intensity µ l .Often when we refer to the perturbed system (1) we shall say that it corresponds to the vector of parameters µ = (µ 1 , µ 2 , ..., µ r ).Assuming that the nominal system (3) is exponentially stable in mean square (ESMS) we want to find necessary and sufficient conditions which will be satisfied by the parameters µ l , 1 ≤ l ≤ r such that the perturbed system (1) to be also ESMS.In the special case N = {1} (no Markov jumps) the conditions derived in this note recover those derived in [10].The concept of exponential stability in mean square of the linear stochastic systems of type ( 1) and (3) may be found in [2] and [8], respectively.

Some preliminaries
Let S N n = S n ⊗S n ⊗...⊗S n , where S n ⊂ R n×n is the subspace of symmetric matrices.Let S N n+ be the convex cone defined by Here, X(i) ≥ 0 means that X(i) is positive semidefinite, S N n+ is a closed convex cone with non empty interior.Its interior is (5) (iv) There exists X = (X(1), ..., X(N )) ∈ IntS N n+ satisfying the following system of LMIs where Then one associates the following Lyapunov operators to the nominal system (3) L : and for all X = (X(1), ..., X(N )) ∈ S N n .Is is easy to check that L is the adjoint operator of L with respect to the usual inner product on S N n : Invoking Proposition 3.20 and Theorem 3.21 from [2] (see also Theorem 3.2.2 and Theorem 3.2.4 from [8]) one deduces that the nominal system (3) is ESMS if and only if the eigenvalues of the linear operator L are located in the half plane C − .This allows us to obtain the following result.
Corollary 3.2.If the nominal system (3) is ESMS then for each H ∈ S N n the equations and have unique solutions given by n+ then the unique solutions of ( 9) and (10), respectively are in IntS N n+ i.e. and where the order relation is induced by the convex cone In this case D will be called positive matrix.Applying Theorems 2.6 and 2.7 from [7] in the special case of the ordered linear space (R d , R d + ) one obtains the following result.
Proposition 3.3.For a positive matrix D ∈ R d×d the following are equivalent: (i) ρ(D) < 1, ρ(•) being the spectral radius; (ii) There exists ψ ∈ IntR d + such that the equation

Main results
The equivalence (i) ↔ (iv) from Proposition 3.1 allows us to deduce that if the perturbed system (1) is ESMS for a value µ = (µ 1 , µ 2 , ..., µ r ) of the intensities of the white noises, then this system is ESMS for every value In this section we shall derive a set of necessary and sufficient conditions which guarantee the exponential stability in mean square of a perturbed system (1) corresponding to a set of unknown vector of intensities µ = (µ 1 , ..., µ r ).Using Proposition 3.1 one notices that the system (1) is ESMS if and only if for any H ∈ IntS N n+ the equation ( 4) has a solution X ∈ IntS N n+ .Using (6) we may rewrite (4) in the form where H = ( H(1), ..., H(N )), Further, we rewrite H in the form: where Since the nominal system (3) is necessarily ESMS if the perturbed system ( 1) is ESMS, we deduce via Corollary 3.2 and ( 16) that the solution of the equation ( 14) satisfies The i th component of this solution is Multiplying (18) on the left by b T k (i) and on the right by b k (i), one obtains and One sees that (19) is a system of rN scalars equations with rN scalar unknowns.
Based on the fact that for each integer α ∈ {1, 2, ..., rN } there exists a unique pair of natural numbers (k, i) ∈ {1, 2, ..., r} × {1, 2, ..., N } such that α = (k − 1)N + i we may write (19) as an equation of the form (13) on the space R rN .To this end we set if (k − 1)N + i = α and (l − 1)N + j = β.With these notations (19) may be written in a compact form: Since the matrix D defined by (24) depend upon the unknown parameters µ l , for each perturbed system of type (1) one may associate a matrix D = D(µ) as before.Now we are in a position to state and proof the following result.
In the space R r of the vector µ = (µ 1 , ..., µ r ) the critical vectors of noise intensities µ 0 are included in the boundary of the stability region.

Several special cases
The first special case analyzed here is r = 1 and N ≥ 2. In this case the system (1) becomes dx(t) = A(η t )x(t)dt + µb(η t )c T (η t )x(t)dw 1 (t). (34) The matrix D associated to the system (34) is D = µ 2 D 1 where for each 1 ≤ j ≤ N , (Z j (1), ..., Z j (N )) is the unique solution of the following equation on S In the special case of the system (34), the Theorem 4.1 yields to the following result.
Corollary 5.1 If b(i) = 0, ∀ 1 ≤ i ≤ N the following are equivalent: (i) The perturbed system (34) is ESMS; (ii) The nominal system (3) is ESMS and the parameter µ satisfies the condition µ 2 < 1 ρ[D1] .Remark 5.1 The previous Corollary shows that the exponential stability in mean square of the nominal system (3) is preserved for the perturbed system (34) if and only if the unknown parameter µ lies in the interval (−ρ ), which is the stability region in the case of perturbed system (34).
The second special case discussed here is r ≥ 1, N = 1.Now, the system (1) becomes The matrix D associated to the system (37) via ( 24) is for each 1 ≤ l ≤ r, Z l is the unique solution of the Lyapunov equation The result proved in Theorem 4.1 yields the next result.Remark 5.2 a) The result stated in Corollary 5.2 is just the main result derived in [10].Its discrete-time version may be found in [11].
b) Condition of the form b l = 0, 1 ≤ l ≤ r (as it is imposed in the general case in Theorem 4.1) is redundant in the case of system (37) because, if b l0 = 0 for some l 0 , it follows that the noise w l0 (t) does not affect the perturbed system.
In order to illustrate the above theoretical results one considers the dynamics of Hopfield neural network of form vi (t) = a i v i (t) + n j=1 b ij g j (v j (t)) + c i , i = 1, ..., N a i < 0 and the activation functions g i (•) are strictly increasing.Then its approximation around an equilibrium point v 0 is ẋ(t) = Ax(t) + Bf (x(t)) where x(t) = v(t) − v 0 , f (x) = g(x + v 0 ) − g(v 0 ) and where A = diag(a 1 , ..., a n ) the elements of f (•) being sector-type nonlinearities satisfying f k (x k ) (f k (x k ) − µ k x k ) ≤ 0, k = 1, ..., n.Then for the above system associate the linear approximation dx(t) = Ax(t)dt + µ