An, P T

Chung, N Q

Le Na, P

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Available at: http://www.ictp.trieste.it/~pub−off IC/2004/24United Nations Educational Scientific and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSON PARAMETRIC DOMAIN FOR ASYMPTOTIC STABILITY WITHPROBABILITY ONE OF ZERO SOLUTION OF LINEAR ITOSTOCHASTIC DIFFERENTIAL EQUATIONSPhan Thanh An1Institute of Mathematics, Hanoi, VietnamandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,Phan Le NaVinh University, Vinh, VietnamandThe Abdus Salam International Centre for Theoretical Physics, Trieste, ItalyandNgo Quoc ChungVinh University, Vinh, Vietnam.AbstractWe describe a pratical implementation for finding parametric domain for asymptotic stabilitywith probability one of zero solution of linear Ito stochastic differential equations based onKorenevskij and Mitropolskij’s sufficient condition and our sufficient conditions. Numericalresults show that all of these sufficient conditions are crucial in the implementation.MIRAMARE – TRIESTEMay 20041thanhan@math.ac.vn1 IntroductionConsider the asymptotic stability with probability one of the zero solution of the following linearIto stochastic differential equationDdx(t) = Ax(t)dt + B()x(t)dw(t) (1)(t ≥ 0, x(0) = x0), where x(t), x0 ∈ IRn, A and D are constant n × n-matrices, D is nonsin-gular, w is a standard one dimensional Wiener process on a stochastic base (Ω, (Ft)t≥0,F , P ),perturbation matrix B() is dependent analytically on a parameter  and B(0) = 0 (see [2], [3]and [6]).Definition 1.1 (see [6]): The stability with probability one of the zero solution x(t) ≡ 0 ofequation (1) is said to be asymptotically stable in the Lyapunov sense with probability one, ifthere exists a number δ > 0 such thatP{limT→∞supt0+T≤t‖x(t)‖ = 0 | x(0) = x0, ‖x0‖ ≤ δ}= 1.Here ‖ · ‖ denotes the Euclidean norm in IRn.Assumption 1.1: The unperturbed equationDdx(t) = Ax(t)dtis asymptotically stable in the Lyapunov sense, which means that all solutions λi(i = 1, 2, . . . , n)of the equation det(A− λD) = 0 have negative real parts, i.e.,Reλi < 0, i = 1, 2, . . . , n. (2)Note that λi are eigenvalues of the matrix D−1A.We now consider equation (1) satisfying Assumption 1.1. Some conditions that are difficultto verify for asymptotical stability with probability one of zero solution of equation (1) havebeen established previously (see, for example, [6], Chapter 6). There arises two problems:Problem 1: For what easily verified algebraic conditions on the matrices A,D and B(), zerosolution of equation (1) is asymptotically stable with probability one?Problem 2: Under the algebraic conditions on the matrices A,D and B() given in Problem 1,what is the domain of ? Does it contain only one interval?Problem 1 was first solved in [2] and [3]. By using the Lyapunov function technique, acriterion for the asymptotic stability with probability one of the zero solution was given in formof a Sylvester matrix equationAT HD + DT HA + B()T HB() = −G,2where the solution H is negative definite for each symmetric positive definite matrix G. Con-sequently, a sufficient condition for the asymptotic stability with probability one of the zerosolution was given (see [2] and [3]). But, in view of computation, solving the Sylvester matrixequation is not efficient (see [4] and [10]). On the other hand, examples given in [2] and [3] showthat the domain of  contains only one interval.In this paper, Korenevskij and Mitropolskij’s criterion is revised in the form of a Lyapunovmatrix equation (Theorem 2.1), which can be solved efficiently (see [8]). Some sufficient condi-tions for asymptotic stability are given in Section 2 (Corollaries 2.1, 2.3-2.5). To solve Problem2, an implementation algorithm for finding parametric intervals of  based on Corollaries 2.1-2.5is given. Finally, numerical results are described. One of them indicates that the parametricdomain contains some intervals (Example 4.1 and Table 1).2 Sufficient Conditions for Asymptotic Stability with Probabil-ity One of Zero SolutionWe recall the following known results.Proposition 2.1 (see Corollary 1, p.168 [6]): If for equation (1), there exists a twice differen-tiable positive definite function of the phase variables (a stochastic Lyapunov function) V (x)such that V (0) = 0 and the expectation E{dV/dt} of its total time derivative taken on the basisof (1) is negative, then the zero solution of (1) is asymptotically stable with probability one.Proposition 2.2 (see [1]): If Assumption 1.1 is satisfied, then for any symmetric positivedefinite n×n-matrix G there exists a unique symmetric positive definite n×n-matrix H solvingthe Lyapunov matrix equationAT HD + DT HA = −G.Korenevskij and Mitropolskij’s criterion given in [3] can be revised as follows.Theorem 2.1 ([3]): Suppose that Assumption 1.1 is satisfied and some n × n-matrix G issymmetric positive definite. Then the zero solution of equation (1) is asymptotically stable withprobability one, if and only if the matrix BT ()H0B()−G is negative definite where H0 is thesolution of the Lyapunov matrix equationAT H0D + DT H0A = −G. (3)Proof: The theorem can be seen directly from Korenevskij and Mitropolskij’s criterion [3]. Wepresent a proof (which repeats the proof in [3]) because a part of the proof is needed for the lastsection.Suppose that H0 is the solution of the Lyapunov matrix equation (3). By Proposition 2.1, H0is symmetric positive definite. Choose a suitable Lyapunov function for the perturbed equation3(1), being the quadratic form V (x) of the phase variable, as followsV (x) = (x)T DTH0Dx. (4)Hence,dV (x) = (x)T[(AT H0D + DT H0A + BT ()H0B())dt+(BT ()H0D + DT H0B())dw(t)]x.Then, the expectation E{dV/dt} of the total time derivative of (4) on the basis of (1) is equaltoE{dV (x(t))dt|x≡x}= xT (AT H0D + DTH0A + BT ()H0B())x.It follows from (3) that the matrixAT H0D + DT H0A + BT ()H0B()is negative definite. Therefore,E{dV (x(t))dt|x≡x}< 0.From Proposition 2.1, the requirement ensuring asymptotic stability is satisfied.Inversely, suppose that the zero solution of equation (1) is asymptotically stable with prob-ability one and G is a symmetric positive definite. Since Assumption 1.1 is satisfied, equation(3) has a symmetric positive definite matrix solution H0. It follows that BT ()H0B() − G isnegative definite. Because B(0) = 0, BT (0)H0B(0) −G is negative definite, too. Since B() isdependent analytically on , there exists α > 0 such that BT ()H0B()−G is negative definitefor 0 <  < α.It follows directly from Theorem 2.1 that:Corollary 2.1: Suppose that Assumption 1.1 is satisfied. Then the zero solution of equation (1)is asymptotically stable with probability one, if the matrix BT ()H1B()− I is negative definitewhere H1 is the solution of the Lyapunov matrix equationAT H1D + DTH1A = −I, (5)and I is the identity matrix.Since D is nonsingular, G: = DT D is symmetric positive definite. If the perturbation matrixB() (A, respectively) in (1) is assumed to be nonsingular then G: = B()T B() (G: = AT A,respectively) is symmetric positive definite and BT ()H0B() − G = BT ()(H0 − I)B(). Itfollows from Theorem 2.1 that:4Corollary 2.2 (see [2] and [3]): Suppose that Assumption 1.1 is satisfied and the perturbationmatrix B() is nonsingular. Then the zero solution of equation (1) is asymptotically stable withprobability one, if the matrix H2−I is negative definite where H2 is the solution of the Lyapunovmatrix equationAT H2D + DTH2A = −B()T B(). (6)Corollary 2.3: Suppose that Assumption 1.1 is satisfied and A is nonsingular. Then the zerosolution of equation (1) is asymptotically stable with probability one, if the matrix BT ()H3B()−AT A is negative definite where H3 is the solution of the Lyapunov matrix equationAT H3D + DT H3A = −AT A. (7)Corollary 2.4: Suppose that Assumption 1.1 is satisfied. Then the zero solution of equation(1) is asymptotically stable with probability one, if the matrix BT ()H4B()−DT D is negativedefinite where H4 is the solution of the Lyapunov matrix equationAT H4D + DT H4A = −DT D. (8)If A and D are assumed to be symmetric, then D−1A is symmetric, too. Hence, by (2), thematrix −D−1A is positive definite (see Theorem 7.2.1 [7]). Hence, the symmetric matrix G: =−12(D−1A + (D−1A)T ) is positive definite, too. Therefore, by Theorem 2.1, we conclude thatCorollary 2.5: Suppose that Assumption 1.1 is satisfied, A and D are symmetric. Thenthe zero solution of equation (1) is asymptotically stable with probability one, if the matrixBT ()H5B()+12(D−1A+(D−1A)T ) is negative definite where H5 is the solution of the Lyapunovmatrix equationAT H5D + DT H5A =12(D−1A + (D−1A)T ). (9)3 Parametric Domain for Asymptotic Stability with ProbabilityOne of Zero SolutionBased on Corollaries 2.1-2.5, we now present an algorithm for finding parametric domain of allpositive parameters  bounded from above by a number α > 0 such that the zero solution ofequation (1) is asymptotically stable with probability one.AlgorithmGiven: a stepsize d and α > 0, constant n× n-matrices A and nonsingular D satisfy-ing Assumption 1.1, perturbation n × n-matrix B() satisying B(0) = 0. Forsimplicity we assume that D is symmetric positive definite, n× n-matrix B()is of the formB() = (φ()bij)n×nwhere φ() is a univariate polynomial satisfying φ(0) = 0 and B = (bij)n×n.5Task: Find  ∈ [0, α] such that zero solution of equation (1) is asymptotically stablewith probability one.Step 1. Solve equations (5) and (8). If A is nonsingular (symmetric, respectively),solve equation (7) ((9), respectively). Suppose that symmetric positive definitematrix H1 (H3, H4, H5, respectively) is a solution of equation (5) ((7), (8),(9), respectively) (Proposition 2.2 ensures the existence of such H1, H3, H4 andH5). Set  = d.Step 2. If − d > α, set  = α, [0, α] is the required domain. STOP.Else, check whether BT ()H1B() − I is negative definite. If so, by Corollary2.1, the zero solution of equation (1) is asymptotically stable with probabilityone, then set  =  + d, go to Step 2.Step 3. Check whether BT ()H4B() − DT D is negative definite. If so, by Corollary2.4, the zero solution of equation (1) is asymptotically stable with probabilityone, then set  =  + d, go to Step 2.Step 4. If A is nonsingular, check whether BT ()H3B()−AT A is negative definite. Ifso, by Corollary 2.3, the zero solution of equation (1) is asymptotically stablewith probability one, then set  =  + d, go to Step 2.Step 5. If A is symmetric, then check whether BT ()H5B() +12(D−1A + (D−1A)T ) isnegative definite. If so, by Corollary 2.5, the zero solution of equation (1) isasymptotically stable with probability one, then set  =  + d, go to Step 2.Step 6. If B = (bij)n×n is nonsingular, solve equation (6) and suppose that symmetricpositive definite matrix H2 is a solution of equation (6) (Proposition 2.2 ensuresthe existence of such H2). Check whether H2 − I is negative definite. If so,by Corollary 2.2, the zero solution of equation (1) is asymptotically stable withprobability one, then set  =  + d, go to Step 2.Else, [0, [ is the required inverval. STOP.Since the set of parametrics  is bounded, the algorithm stops after a finite number of steps.This algorithm is implemented by a Fortran code. It is based on the LAPACK library [5] andNAG library [9] for checking whether the matrices BT ()H1B()−I, H2−I, BT ()H3B()−AT A,BT ()H3B() − DTD or BT ()H3B() +12(D−1A + (D−1A)T ) are negative definite and theFortran-77 software package [8] for solving (5), (6), (7), (8) or (9). It can be seen that usingthe Fortran-77 software package [8] more often, the computing time increases. Hence, in thealgorithm, to check whether the points  = id with i = 1, 2, . . . belong to the required parametricdomain, Corollary 2.2 is the last corollary to be used (at step 6) after either corollaries.6Moreover, we use the Fortran code for the case n ≤ 6 since the Fortran-77 software package[8] is effective for n ≤ 6. This code is also designed for finding many intervals of the parametricdomain in [0, α].On the other hand, all of Corollaries 2.1-2.5 are crucial in the algorithm. Indeed, havingan algorithm basing only one of Corollaries 2.1-2.5, we can accordingly modify this algorithm.But no single according algorithm is best for finding the largest parametric domain. This isillustrated in Table 2.• Using the Fortran codeFor a given application, the user provides the files:- inname: gives all needed data: the stepsize d, the boundary above α of the parametricdomain, the dimension n of matrices A, D, and B as well as their values.- functeps.f: gives the univariate polynomial φ() =∑ki=1 aii in the matrix B() (ai ∈ IR).4 Examples and Numerical ResultsIn this section, our Fortran code is compiled by Intel(R) Fortran Compiler 7.0 under MandrakeLinux 9.2 and is executed on a Pentium IV processor. The examples are considered with thestepsize d = 10−5 and α = 4.Example 4.1 (given in [2]):A = −1 10 00 −1 100 0 −1D = 1 0 00 1 00 0 1B = 0 0 −10 1 01 0 0φ():= . Then, the zero solution of equation (1) is asymptotically stable with probability onewith  ∈ [0, 0.02283[ . This domain is also the largest (see [2]).If φ() =  is replaced by some φ(·), the parametric domain may contain some intervals in[0, α]. Table 1 below gives some results in cases φ():= , φ():= −2 + , φ():= −(2 − 1),andφ():= −3(− 0.3)( − 0.8)( − 1.2)( − 1.7)respectively.7Table 1: Performance Results of Example 4.1 with some φ(·)φ(·) Parametric Domain in [0, 4] [0, 0.02283[−(2 − 1) [0, 0.02284[∪ ]0.98838, 1.01123[−3(− 0.3)(− 0.8)(− 1.2)(− 1.7) [0, 1.25894[∪ ]1.69230, 1.70709[Example 4.2:A =(−1 11 −3.5)D =(59 00 5)B =(0 −41 0)φ():= . Then, the zero solution of equation (1) is asymptotically stable with probability onewith  ∈ [0, 2.20687[ .Example 4.3:A =(−1 11 −3)D =(1 00 1)B =(0 −41 0)φ():= , Then, the zero solution of equation (1) is asymptotically stable with probability onewith  ∈ [0, 0.51660[ .Example 4.4:A = −1 −1 0−1 −4 −10 −1 −3.1D = 2 0 00 1 00 0 1B = 0 −12 −16.90 1 −1−1 0 0φ():= . Then, zero of equation (1) is asymptotically stable with probability one with  ∈[0, 0.35080[ .8Example 4.5:A = −8 −1 3−1 −4 −13 −1 −3.1D = 4 0 00 1 00 0 1B = 0 −912 −0.10 1 −1−1 0 0φ():= . Then, the zero solution of equation (1) is asymptotically stable with probability onewith  ∈ [0, 0.00862[ .Table 2: Parametric Domains Corresponding to Corollaries (with φ() = )ExampleAlgorithm Parametric Domainbased only on in [0, 4]4.1Corollary 2.1 [0,0.02283[Corollary 2.2 [0,0.02283[Corollary 2.3 [0, 0.00132[Corollary 2.4 [0,0.02283[4.2Corollary 2.1 [0,2.20687[Corollary 2.2 [0, 1.46934[Corollary 2.3 [0, 1.06409[Corollary 2.4 [0, 0.19511[Corollary 2.5 [0, 0.08342[4.3Corollary 2.1 [0, 0.28767[Corollary 2.2 [0,0.51660[Corollary 2.3 [0, 0.29390[Corollary 2.4 [0, 0.28767[Corollary 2.5 [0, 0.48320[4.4Corollary 2.1 [0, 0.08118[Corollary 2.2 [0, 0.13938[Corollary 2.3 [0,0.35080[Corollary 2.4 [0, 0.04166[Corollary 2.5 [0, 0.27008[4.5Corollary 2.1 [0, 0.00484[Corollary 2.2 [0, 0.00291[Corollary 2.3 [0, 0.00282[Corollary 2.4 [0, 0.00171[Corollary 2.5 [0,0.00862[95 Concluding RemarksThe algorithm presented in Section 3 shows that computations at points  = id with i = 1, 2, . . .are completely decoupled and hence can be computated in parallel on appropriate subintervalsof [0, α]. It folllows from the proof of Theorem 2.1 that if the equation B() = 0 has positivesolutions then the parametric domain contains some open neighbourhoods of these solutions(this is illustrated in Table 1). Therefore, appropriate subintervals can be determined by thesepositive solutions.Finally, to find the largest parametric domain such that the zero solution of equation (1) isasymptotically stable with probability one is still an open problem.Acknowledgements. The authors express their deep gratitude to Professor Dr. Sc. NguyenDinh Cong for many helpful suggestions during the preparation of the paper at the AbdusSalam International Centre for Theoretical Physics (ICTP), Trieste. This work was partiallysupported by the Swedish International Development Cooperation Agency and MathematicsResearch Fellowship 2003-2004 at ICTP.References[1] Ikramov, Kh. D., Numerical Solution of Matrix Equations, Orthogonal Methods, “Nauka”,Moscow (1984) (in Russian)[2] Korenevskij, D. G., Cofficient criteria and sufficient conditions for asymptotic stability withprobability one of linear systems of Ito stochastic differential equations, Soviet Math. Dokl.,Vol. 34 (1987) No. 2, 377-380[3] Korenevskij, D. G. and Mitropolskij, Yu. A., On algebraic criteria of asymptotic stabilitywith probability one for solutions of systems of linear Ito stochastic differential and differ-ence equations not reduced to the Cauchy form; in the book “Numerical-Analytical Methodsin The Investigation of The Dynamics and Stability of Multidimensional Systems”, Math.Inst. Acad. Sci. Ukrain. SSSR, Kiev (1985) 152-169 (in Russian)[4] Lancaster, P., Explicit solutions of linear matrix equations, SIAM Rev., Vol. 12 (1970),544–566[5] Linear Algebra PACKage - http://www.netlib.org/lapack/[6] Hasminskij, R. Z., Stability of Systems of Differential Equations with Random Disturbancesof Their parametrics, Nauka, Moscow (1969). (in Russian). Engl. ed.: Stochastic Stabilityof Differential Equations, Sijthoff and Noordhoff, Alphen (1980)[7] Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge, University Press (1990)[8] Gardiner, J. D., Wette, M. R., Laub, A. J., Amato, J. J., and Moler, C. B., Algorithm 705, AFortran-77 software package for solving the Sylvester matrix equation AXBT +CXDT = E,ACM Transactions on Mathematical Software, Vol. 18 (1992) No. 2, 232-238[9] Numerical Algorithms Group - http://www.nag.co.uk/[10] Wimmer, H. and Ziebur, A. D., Solving the matrix equation Σrρ=1fρ(A)Xgp(B) = C, SIAMRev. Vol. 14 (1972), 318–323.10

On parametric domain for asymptotic stability with probability one of zero solution of linear Ito stochastic differential equations

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