ON THE LYAPUNOV EQUATION IN BANACH SPACES AND APPLICATIONS TO CONTROL PROBLEMS

where X is an infinite-dimensional Banach space; A is a linear operator. Over the last two decades stability problem of differential equations has been extensively studied by many researchers in qualitative theory of dynamical systems, see, for example, [2, 11, 17] and the references therein. The classical Lyapunov theorem, which claims that the zero solution of linear system (1.1) is exponentially stable if and only if for every symmetric positive definite matrix P the matrix equation

1. Introduction.Consider a linear system described by differential equations of the form ẋ(t) = Ax(t), t ≥ 0, x(0) = x 0 ∈ X, ( where X is an infinite-dimensional Banach space; A is a linear operator.Over the last two decades stability problem of differential equations has been extensively studied by many researchers in qualitative theory of dynamical systems, see, for example, [2,11,17] and the references therein.The classical Lyapunov theorem, which claims that the zero solution of linear system (1.1) is exponentially stable if and only if for every symmetric positive definite matrix P the matrix equation has a symmetric positive definite matrix solution Q.This theorem plays an important role in the stability theory and there are several results and extensions of the Lyapunov theorem, which are closely related to the stability and Lyapunov equation (1.2), see, for example, [1,4,6,14].Moreover, the study of existence of solution of Lyapunov equation (1.2) allows us to obtain useful applications in obtaining stabilizability and controllability conditions for control systems.Among the well-known results related to these applications we mention the references [3,7,8,9,10,13,16].The purpose of this paper is twofold.Firstly, we establish equivalence between solvability of the Lyapunov equation and exponential stability of linear system (1.1) in a Banach space.Secondly, based on the Lyapunov theorem we establish a relationship between stabilizability and exact null-controllability of linear control systems and then give some applications to the exponential stabilizability of a class of nonlinear control systems in Banach spaces.The results of this paper can be considered as a further development of the results obtained earlier in [8,10].
The paper is organized as follows.In Section 2, we present the main notation, definitions and some auxiliary propositions needed later.Equivalence between solvability of the Lyapunov equation and exponential stability is given in Section 3. In Section 4, we give some applications to the exponential stabilizability of a class of nonlinear control systems in infinite-dimensional Banach spaces.

Preliminaries.
Let R be the set of all real numbers, X, U infinite-dimensional Banach spaces, and X * the topological dual space of X.Let y * ,x denote the value of y * ∈ X * at x ∈ X.The domain, the image, the adjoint, and the inverse operator of an operator A are denoted by Ᏸ(A), ImA, A * , and A −1 , respectively.
By L(X, U) we denote the Banach space of all linear bounded operators mapping X into U and by L 2 ([0,T ],X), and L 2 ((0,T ],U)-the Banach space of all L 2 -integrable functions on [0,T ] taking values in X, and in U, respectively.
Let Q ∈ L(X, X * ) be a duality operator.We recall that the operator Q is positive definite in X if Qx, x ≥ 0 for arbitrary x ∈ X, and Qx, x > 0 for x ≠ 0. In the case if Qx, x ≥ c x 2 for some c > 0 we say that Q is strongly positive definite.If X is a reflexive Banach space, we define the adjoint of Q as the operator Q * : X → X * .In this case, if Q = Q * we say that Q is a selfadjoint operator.Throughout, we will denote by LPD(X, X * ) and LSPD(X, X * ) the set of all linear bounded positive definite and strongly positive definite operators mapping X into X * , respectively.
Consider linear system (1.1),where A is a densely defined generator of the C 0semigroup S(t).The solution x(t, x 0 ) with the initial condition x(0) = x 0 ∈ Ᏸ(A) is given by x(t, x 0 ) = S(t)x 0 .
Associated with system (1.1) we consider the following linear control system: where A is the generator of the C 0 -semigroup S(t) on some Banach space X and B ∈ L(U, X).The class of admissible controls ᐁ for system (2.3) is defined by The classical solution x(t) satisfying initial condition x(0) = x 0 according to some admissible control u(•) ∈ ᐁ is then given by x t, u, x 0 = S(t)x 0 + t 0 S(t − s)Bu(s)ds.
(2.4) Definition 2.4.Control system (2.3) is exponentially stabilizable if there is an operator K ∈ L(X, U) such that the linear system ẋ(t) = (A + BK)x(t), t ≥ 0, is exponentially stable.Definition 2.5.Control system (2.3) is exactly null-controllable in time T > 0 if for every x 0 ∈ X there is an admissible control u(t) ∈ ᐁ such that (2.5) In other words, if we denote by Ꮿ T the set of null-controllable points in time T of system (2.3) defined by In the case A is the generator of an analytic semigroup S(t), for T > 0, we can define the operator W T ∈ L(ᐁ,X) by and we then have C T = Im W T .
Definition 2.6 (see [5]).A Banach space X * has the Radon-Nikodym property if In the sequel, we need some well-known null-controllability criteria for control system (2.3) presented in [3] for reflexive Banach spaces and then in [15] for non-reflexive Banach spaces having the Radon-Nikodym property.
Proposition 2.7 (see [3,15]).Let X, U be Banach spaces, S(t) the C 0 -semigroup of A. Assume that X * , U * have the Radon-Nykodym property.The following conditions are equivalent: (i) control system (2.3) is exactly null-controllable in time T > 0; (ii) there exists c > 0, for all x * ∈ X * : 3. The Lyapunov equation.Let X be an arbitrary Banach space, A the infinitesimal generator of the C 0 -semigroup S(t) and let P ∈ L(X, X * ).The operator Q ∈ L(X, X * ) is called a solution of the operator equation (1.2) if the following conditions hold: Note that if A is bounded, then the above equations has the standard form and it was shown in [4] that if A is exponentially stable in a Hilbert space then the Lyapunov equation has a solution.In Theorem 3.1 below we give the equivalence between the solvability of the Lyapunov equation and the exponential stability of the linear system (1.1).
Theorem 3.1.If for some P ∈ LSPD(X, X * ), Q ∈ LPD(X, X * ), the Lyapunov equation holds, then the operator A is exponentially stable.Conversely, if the generator A is exponentially stable, then for any 2) for some P ∈ LSPD(X, X * ).Let x 0 ∈ Ᏸ(A) and x(t, x 0 ) be a solution of system (1.1) with the initial condition x(0) = x 0 .For every t ≥ 0, we consider the following function: We have Since P is strongly positive definite, there exists a number c > 0 such that and hence, Integrating both sides of (3.6) over [0,t], we have and hence Letting t → ∞ we obtain +∞ 0 x s, x 0 2 ds < +∞, (3.10) which, by Proposition 2.3, implies the exponential stability of operator A.
Conversely, assume that the operator A is exponentially stable.Take any P ∈ LSPD(X, X * ).For each x 0 ∈ Ᏸ(A) and t ≥ 0 we consider the operator Q t ∈ L(X, X * ) defined by We have Note that where x(t, x 0 ) is the solution of system (1.1) with x(0) = x 0 , we then have P Ax s, x 0 ,x s, x 0 + P x s, x 0 ,Ax s, x 0 ds = t 0 d ds P x(s), x(s) = P x t, x 0 ,x t, x 0 − P x 0 ,x 0 . (3.14) Letting t → +∞ in the above relation and noting that x(t, x 0 ) → 0, we have where the operator is well defined due to the exponential stability assumption of A. Therefore, from the relation (3.15) it follows that Q satisfies the Lyapunov equation (1.2).To complete the proof, we need to show that Q is positive definite.For this, we consider (3.17) Since P ∈ LSPD(X, X * ), S(t) is nonsingular, we have Q ∈ LPD(X, X * ).The proof is complete.
Remark 3.2.Note that if X is reflexive, P is selfadjoint then Q is also selfadjoint.

Controllability and stabilizability.
Consider the linear control system (2.3),where x(t) ∈ X, u(t) ∈ U. Throughout this section, we assume that X is a Banach space and X * has the Radon-Nykodym property, U is a Hilbert space and the operator A generates an analytic semigroup S(t).
A considerable development has taken place in the problem of controllability and stabilizability of linear control system (2.3), see, for example [8,10,13,16,17,15].In particular, the relationship between controllability and stabilizability was presented in [8,16] for systems in finite-dimensional spaces and it was shown that the exact nullcontrollability implies exponential stabilizability.It is obvious that all exactly nullcontrollable systems in finite-dimensional spaces are exponentially stabilizable, however the exponentially stabilizable system is, in general, not exactly null-controllable.For this, we need stronger notion of stabilizability in a sense of [14,16].Definition 4.1.Linear control system (2.3),where X, U are finite-dimensional, is completely stabilizable if for an arbitrary δ > 0 there is a matrix K such that the matrix A + BK is exponentially stable, that is, for some M > 0. It is well known that if a finite-dimensional linear control system is completely stabilizable in the above sense, then it is exactly null-controllable.The same definition is applied to infinite-dimensional control system (2.3) and a natural question is: to what extent does the complete stabilizability imply the exact nullcontrollability for infinite-dimensional control systems?In the infinite-dimensional control theory characterizations of controllability and stabilizability are complicated and therefore their relationships are much more complicated and require more sophisticated methods.
By a result of [8], if the linear control system (2.3),where X, U are Hilbert spaces, is completely stabilizable then it is exactly null-controllable in some finite time.In the spirit of [8] using the null-controllability results, Proposition 2.7, we improve the result of [8] by the following theorem.

Theorem 4.2. If linear control system (2.3) is completely stabilizable then it is exactly null-controllable in some finite time.
Proof.By [12, Proposition 8.3.1]we have for some M > 0 and α > 0. Assume that the system (2.3) is completely stabilizable, that is, for δ > α, there is a feedback control operator K : X → U such that the semigroup S K (t) generated by (A + BK), satisfies the condition for some N > 0. For every x 0 ∈ X and admissible control u(t) ∈ ᐁ, the solution x(t, x 0 ,u) of system (2.3) is given by and by the feedback control u(t) = Kx(t) this solution is also given by x t, x 0 ,u = S K (t)x 0 .(4.5) Therefore, we have Since the above relation holds for every x 0 ∈ X and using the equality (4.3), for every x * ∈ X * , the following estimate holds: and hence Setting we see that and then β(t) → (1/ √ 2δ) when t → ∞.To establish the exact null-controllability of system (2.3), we assume to the contrary that the system is not null-controllable at any time t ≥ 0. We take any ∈ (0, 1), and set (4.11) Since system (2.3) is not exactly null-controllable at any time t ≥ 0, by Proposition 2.7(iii), for that chosen number c > 0, there is x * ∈ X * such that From the above inequality, it follows that x * ≠ 0 and we can consider x * = 1.On the other hand, in view of (4.8), we have the following estimate: Letting t → +∞, and noting δ > α, the right-hand side goes to zero and we then have The last inequality contradicts the choice of numbers , c by (4.11).The system is exactly null-controllable.
In the sequel, we prove that if linear control system (2.3) is null-controllable then it is exponentially stabilizable by some linear feedback control K : X → U .Theorem 4.3.If control system (2.3) is exactly null-controllable in some finite time, then the system is exponentially stabilizable.
Proof.Assume that the system is exactly null-controllable in T > 0. The operator Q ∈ L(X * ,X) given by is, by Proposition 2.7(iv), well defined and strictly positive definite.Therefore, the inverse operator Q −1 : X → X * is also well defined.We will prove that the control system (2.3) is exponentially stabilizable by the feedback control It is enough to show that the operator Q satisfies the Lyapunov equation (1.2) in the dual space X * with Ꮽ = (A + BK) * for some P ∈ LSPD(X * ,X) and by Theorem 3.1, Ꮽ and then (Ꮽ) * = A + BK is exponentially stable.For this, we have to prove that Indeed, we consider we have Consider the relation QᏭ + Ꮽ * Q.Since Q is a selfadjoint operator, we see that From (4.24), it follows that where Therefore, for every x * ∈ X * , we have as desired.To complete the proof, we show that P is strictly positive definite.This follows from the following relations Since and using Proposition 2.7(ii), we have for some positive number c 1 > 0. The proof is complete.
Remark 4.4.It is worth noting that Theorem 4.2 was presented in [17] for the case X is a Hilbert space and the proof therein is based on the linear regulator optimization problem so that it is quite different from ours.
Example 4.5.Consider a control system of the form where l 2 is the space of all sequences β = β 1 ,β 2 ,..., with the norm and the operators A, B are given by

.34)
Since A N BU = l 2 , the system is exactly null-controllable and hence the system is stabilizable.

Stabilizability of nonlinear control systems.
As an application, we study stabilizability problem of a nonlinear control system of the form ẋ = Ax + Bu + f (x,u), t ≥ 0. (5.1) In this section, we also assume that X is a Banach space and X * has the Radon-Nykodym property, U is a Hilbert space, f (x,u) : X × U → X is some given nonlinear function satisfying the following comparable condition for some a > 0, b > 0. We recall that control system (5.1) is stabilizable by a feedback control u(t) = Kx(t), K ∈ L(X, U) if the uncontrolled system ẋ(t) = (Ax + KB)x + f (x,Kx) is asymptotically stable in Lyapunov sense.
The following theorem gives a sufficient condition for stabilizability of nonlinear control system (5.1) in the case A is a stable operator.
Theorem 5.1.Assume that A is exponentially stable and the condition (5.2) is satisfied.Let P,Q ∈ LPD(X, X * ) be the operators satisfying the Lyapunov equation (1.2), where Q = Q * and Px,x ≥ α x 2 , for all x ∈ X, α > 0. The nonlinear control system (5.1) is stabilizable by the feedback control u (5.3) Proof.Let x(t, x 0 ) be any solution of system (5.1).Let Q ∈ LPD(X, X * ) be a solution of the Lyapunov equation (1.2).We consider the following function of the form V x t, x 0 = Qx t, x 0 ,x t, x 0 . (5.4) and we prove that this function is a Lyapunov function for the system (5.1).Indeed, we have (5.5) Since Q is selfadjoint, by conditions (5.2) and (5.3), we obtain the following estimate where as desired.
In Theorem 5.2, we give another sufficient condition for the stabilizability of system (5.1) in the case A is not stable, but the associated linear control system (2.3) is exactly null-controllable and the nonlinear perturbation f (•) is small enough.Theorem 5.2.Assume that the linear control system (2.3) is exactly null-controllable.The system (5.1) is stabilizable for some appropriate numbers a > 0, b > 0 satisfying the condition (5.2).
Proof.By Theorem 4.3, the linear control system (2.3) is stabilizable and then there is an operator D ∈ L(X, U) such that the operator Ꮽ = A + BD is exponentially stable.Let P,Q ∈ LPD(X, X * ) be a solution pair of the Lyapunov equation with respect to Ꮽ, where Px,x ≤ α x 2 , Q = Q * .Consider the Lyapunov function V (t,x) = Qx, x , for the nonlinear control system (5.1).By the same arguments used in the proof of Theorem 5.1, we have where (5.9) Then the nonlinear system (5.1) is stabilizable.The proof is complete.

Conclusions.
In this paper an extension of the Lyapunov equation in Banach spaces was studied.A relationship between stabilizability and exact null-controllability of linear systems in Banach spaces was established.Some applications to stabilizability problem of a class of nonlinear control systems in infinite-dimensional Banach spaces were also given.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation