The long run behaviour of solutions of Lipschitzian quantum stochastic differential equation (QSDE) with non-instantaneous impulse is studied. This is achieved by imposing some conditions on the coefficients associated with the map P. Using the fixed point approach, we show that a solution exists under the given conditions and subsequently establish Ulam's type stability. We present some examples to further justify its application

Bishop, S.A.

Ogundile, O. P.

Ogundiran, M. O.

Covenant University Repository

 http://www.iaeme.com/IJMET/index.asp 367 editor@iaeme.com International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 9, September 2018, pp. 367–374, Article ID: IJMET_09_09_040 Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=9 ISSN Print: 0976-6340 and ISSN Online: 0976-6359  © IAEME Publication Scopus Indexed  ON STABILITY OF QUANTUM STOCHASTIC DIFFERENTIAL EQUATION S. A. Bishop and O. P. Ogundile Department of Mathematics, Covenant University, Ota, Ogun State, Nigeria M. O. Ogundiran Obafemi Awolowo University, Ile-Ife,Osun State,Nigeria ABSTRACT The long run behaviour of solutions of Lipschitzian quantum stochastic differential equation (QSDE) with non-instantaneous impulse is studied. This is achieved by imposing some conditions on the coefficients associated with the map P. Using the fixed point approach, we show that a solution exists under the given conditions and subsequently establish Ulam's type stability. We present some examples to further justify its application. 2010 Mathematics Subject Classification: 81S25, 31A37. Keywords: Ulams-Hyers stability, QSDE, Non-instantaneous impulse functions, Fixed point method. Cite this Article: S. A. Bishop, M. O. Ogundiran and O. P. Ogundile, On Stability of Quantum Stochastic Differential Equation, International Journal of Mechanical Engineering and Technology, 9(9), 2018, pp. 367–374. http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=9 1. INTRODUCTION Stability of solutions of impulsive ordinary differential equations (ODEs), partial differential equations (PDEs), Functional differential equations (FDEs), etc. have been of interest to many authors [1, 4-7, 9-13]. Wang and Feckan (2013) [13], established stability results for stochastic differential equations. [4, 5, 9] established similar results when the impulse conditions are combinations of the traditional initial value problems and the short term perturbations. However, the perturbation terms in these classes of equations cannot show the dynamic change of evolution processes as it should in some applications. To address some of these limitations, Liao and Wang (2014) in [7], studied generalized Ulam-Hyers-Rassias (U-H-R) stability of solutions for a class of equations with non-instantaneous impulses and provided some examples to show their applications. Some results on existence of solution of impulsive quantum stochastic differential equations (IQSDEs) and quantum stochastic differential inclusions (QSDIs) have been established in [2, 3, 8]. So far, results on stability of these equations have not been S. A. Bishop, M. O. Ogundiran and O. P. Ogundile http://www.iaeme.com/IJMET/index.asp 368 editor@iaeme.com investigated. Considering the importance of the long run behaviour of systems in real life applications, is a motivation for this study. This paper is concerned with the study of U-H-R stability of the following QSDE (Also known as nonclassical ordinary differential equation (NODE)) with non-instantaneous impulse functions: ,0, ( ) ( , ( ))( , ), ( , ], 1,...,     , ( ) , ( ( )) , ( ]     , (0) , , : [0, ]                                               (1.1)k kk k kdt P t t t s t k mdtt q t t t st I T                     Where ( , ) ( , )( , )t P t     is well defined in [2, 3], ,  𝔻 𝔼 is arbitrary, 0 0 1 1 2 1 10 ... , :m m m mt s t s t s t s t T P I sesq             B (𝔻 𝔼) continuous, and :[ , ] .k k kq t s   Note that the sesquilinear form valued map P is assumed to be real valued since2C  , hence, the methods of [7, 12] are applicable to this setting. 2. PRELIMINARIES 1.    B   is a topological vector space. 2. (𝔻 𝔼) is a complex space. 3. ( , ), ( , )C I PC IB B  are spaces of continuous and piecewise continuous functions. 4. Define 1( , ) : ( , ) : ( , )PC I PC I PC I   B B B   5. The sesquilinear equivalent forms '( , ( )) and ( ,PC I sesq D E PC I sesq (𝔻 𝔼)) Of the above spaces are defined in a similar manner with the usual supremum norm defined in [2]. Definition 2.1. A stochastic process is called a solution of Eq. (1.1) if, it satisfies the following: 00 10, (0), ,, ( ), , ( , ( )) , ( , ]                 = , ( , ( ))( , ) , [0, ];                 = , ( , ( ))) ( , ( ))( , ) , [ , ], 1,...,kk k kttk k kst q t t t t sP s s ds t tq t t P s s ds t t s k m                         Subsequently,  , , ( )) and 1, ...,t I D E k m      except otherwise stated. Next we re-frame the concept of Ulam’s type stability for the purpose of this paper. Let  ( , ) : : ( ) 0 , , 0 and ( ) ( ,PC I t t PC I sesq        B B(𝔻 𝔼)) The following inequality will be useful: On Stability of Quantum Stochastic Differential Equation http://www.iaeme.com/IJMET/index.asp 369 editor@iaeme.com 1, ( ) ( , ( )( , ) , ( , ], ( ) , ( , ( )) ( ), ( , ]                         (2.1)k kk k kdt P t t t s tdtt q t t t t t s                 Definition 2.2. Equation (1.1) is U-H-R stable with respect to , ( ))t    if we can find 0M    such that for each solution ( , ) of (2.1),PC I  B  there exists a solution ( , )PC I B  of Eq.(1.1) with ( ) ( ) ( , )),                                (2.2)y t t M t I         Eq. (1.1) has found applications in quantum stochastic control theory and quantum dynamical systems, see [2, 8].It is worth mentioning that this method will be more useful in many applications such as numerical analysis, Physics, especially when exact solutions are difficult to come  by. Definition 2.3. A stochastic process 1( , )PC I B    is a solution of (2.1) if and only if there exists a function 1( , ( ))F PC I sesq D E     and 1( ,F PC I sesq  (𝔻 𝔼)) such that i ( ) ( ), ,  and F t t t I F        ii 1, ( ) ( , ( ))( , ), ( , ]k kt P t t t s t          iii , ( ) ( , ( , ( )) ) , ( , ]k k kt q t t F t s t         Definition 2.4. Also, ( , )PC I B   if is a solution of the (2.1), then it is also a solution of the following integral   inequality: 110 010( ) ( , ( )) , ( , ]( ) (0) ( , ( )) ( ) , [0, ];( ) ( , ( )) ( , ( )) ( ) , [ , ]       (2.3)k k kt tt tk k k k kst q t t t s tt P s s ds s ds t tt q t t P s s ds s ds t s t                    We state the following established result and refer the reader to [7] and the references therein: Lemma 2.1. Let , ,v a b  be real valued piecewise continuous functions, where a   is nondecreasing. Assume the following inequality holds: 00( ) ( ) ( ) ( ) ( ), 0,iti it tv t a t b s v s ds v t t      where ( ) 0, 0, 1,..., .ib t i m    Then the following inequality also holds: 0( )1( ) ( )(1 ) , ( , ],tb s dsii iv t a t e t t t       where max , 1,..., .i i m     S. A. Bishop, M. O. Ogundiran and O. P. Ogundile http://www.iaeme.com/IJMET/index.asp 370 editor@iaeme.com 3. MAIN RESULTS We state the following useful hypotheses: 1  Let K 0 be a constant such thatpS     1 2 1 2( , ) ( , ) ,pP t P t K        For each 1 2, , .t I   B   2  For q C([t ,s ] , ), let there be constants 0 such thatk k kS L   B B   1 2 1 2( , ) ( , ) ,q t q t L           For each 1 2, , .t I   B  3 0 l 0 a constant and let ( , ) be a nondecreasing function such that                        ( ) ( ),  for each .tS Let PC Is ds l t t I    B The following result is a consequence of definition 2.1. Theorem 3.1. Let the map P   in Eq. (1.1) be continuous for each and let the hypotheses 1 3S S  hold. Then equation (1.1) has a unique solution ( , )PC I  B  provided { , 1,... } 1                                    (3.1)pkL K T k m    Proof: The proof is an adaptation of the method employed in [2].  We give a sketch as follows and refer the reader to the reference [2] for d e t a i l s . Transform the Eq. (1.1) to a fixed point problem by defining the map    as follows:  Let : ( ,PC I sesq(𝔻 𝔼)) ( ,PC I sesq (𝔻 𝔼)) 0( )( )( , ) , (0) ( , ( ))( , )                      +q ( , ( ))( , )                                                           (3.2)tt P s s dst t              and by the assumption 1 2( ),S S  we have 00( )( )( , ) ( )( )( , ) ( , ( ))( , ) ( , ( ))( , )                                               + ( , ( )( , ) ( , ( )( , )                                              tktpt y t P s s P s y s dsq t t q t y tK y ds L                                                                 , 1,...,                                              pyL K T k m yy        Where (0) (0).y  Showing that (3.1) is satisfied and hence,   is a con- traction operator on ( ,PC I sesq (𝔻 𝔼)) and a fixed point exists, which is a unique solution of (1.1). Next, is the main result on stability. Theorem 3.2: Let the conditions 1 2S S  and (3.1) hold. Then Eq. (1.1) is On Stability of Quantum Stochastic Differential Equation http://www.iaeme.com/IJMET/index.asp 371 editor@iaeme.com U-H-R stable. Proof: Let 0 10( , ) be a solution of Eq. (1.1). Then, ( ) , ( , ( ( )) , ( , ];                = , ( , ( ))( , ) , [0, ];                =q ( , ( )) ( ( , ( ))( , ) ( ) ,    ( ,kttk k k k ksPC It q t t t t sP s s ds t ts s P s s F s ds t s t                   B1].                 From (2.3) and S3 we get 01( ) ( , ( )) ( , ( ))                                  ( ) ,                                   +l  (t), [ , ]                   tk kstk kt q t t P s s dss dst s t             For ( , ],  we obtain                                       k kt s t  ( ) ( , ( ) ,kt q t t      And when [0, ],kt t  yields 01( ) (0) ( , ( ) ( ).Therefore, for each [ , ], we get( ) ( ) ( ) ( , ( )) ( , ( ))                       + ( , ( )) ( , ( ))                       + ( ,tk ktk kss k k kt P s s ds l tt s tt y t t q t t P s s dsq t s q s y sP s              0( )) ( , ( ))                      (1 )( ( )) ( ) ( )                       + K ( ) ( )ktskt ttss P s y s dsl t L s y ss y s ds              Applying Lemma 2.1, yields 1 1( ) ( ) (1 )( ( ))(1 ) exp( ), ( , ].                  (3.3)mt k kt y t l t L K t t s t              Moreover, for ( , ],  we obtaink kt s t   ( ) ( ) ( ) ( , ( ))                       + ( , ( )) ( , ( ))                      ( ) ( )1                      , max{ , 1,..., } 1         (3.4)1qqt y t t q t tq t t q t y tL t y tL L k mL                  S. A. Bishop, M. O. Ogundiran and O. P. Ogundile http://www.iaeme.com/IJMET/index.asp 372 editor@iaeme.com Again, for each 1[0, ]t t  , we obtain 0( ) ( ) ( ) ( ) ( )tt y t l t K s y s ds         By Gronwall’s Inequality, we get 1( ) ( ) ( )                                                    (3.5)tKt y t l t e       Hence, by putting (3.3),(3.4) and (3.5) together, we obtain 1 11( ) ( ) (1 )(1 ) ( ( ))1: ( ( ))tkK Ktmqnt y t l L e l e tLM t                      where M is as defined in Definition 2.2 This implies that equation (1.1) is generalized U-H-R stable with respect to ( , ( )).t     4. EXAMPLE Let 221[0,2], ( , ( )( , ) ( 1)( , ( ) ), , , ( , ( ))61 1 1( )exp , . Now let ( ) and ,2 6 2.2ttkI P t t e t K q t tet t L t I                         Let ,, (0) , 1.e          Considering the following problems: 22, ( ) ( , ( )( , )                    =(e -1)( , ( ) ),t (0,1], ( ) , ( , ( ))e 1                = ( ) exp , (1,2]                            (4.1)22                tktdt P t tdttt q t tt t t                  22, ( ) ( , ( )( , )                    =(e -1)( , ( ) ),t (0,1], ( ) , ( , ( ))e 1                = y( )exp , (1, 2]                            (4.2)22                tktdy t P t y tdty ty t q t y tt t t             Where (0) (0).y   Let 1( , )y PC I B  be a solution of (4.2). Then, we find 1(.) (1, )F PC  B  and 1  such thatF B   On Stability of Quantum Stochastic Differential Equation http://www.iaeme.com/IJMET/index.asp 373 editor@iaeme.com 2 1( ) ( ), (0,1], ,1 ,2tF t e t F     22, ( ) ( , ( )( , ) ( )                    =(e -1)( , ( ) ),+ ( ), t (0,1], ( ) , ( , ( )) ( ),1e 1                = y( )exp , (0,2]                            (4.3)22     tktdy t P t y t F tdty t F ty t q t y t F tt t t                          Integrating (4.3), yields 20, ( ) , (0) (( 1)( , ( ) ) , (0,1],tsy t y e y s ds t          And 211, ( ) , ( , ( )) ( )exp ,1.22tkey t q t y t F y t t F             By Theorem 3.1, (4.1) has a unique solution which we present by 20, ( ) , (0) (( 1)( , ( ) )) , (0,1].ttt e s ds t             And 2 1, ( ) ( ) exp( ).2tt t e t      And for [0,1],t  we obtain 22 201, ( ) , ( ) .2 2tts tet y t e ds e           Again we obtain 1, ( ) , ( ) , ( ) , ( ) ,161 1                                   , ( ) , ( )6 2t y t t y t Ft y t                   And this yields 6, ( ) , ( )5t y t       Which finally result to 26 1, ( ) , ( ) , .5 2tt y t e t I             S. A. Bishop, M. O. Ogundiran and O. P. Ogundile http://www.iaeme.com/IJMET/index.asp 374 editor@iaeme.com 5. CONCLUSION This shows that the solution of Eq. (1.1) is generalized U-H-R stable with 21  and ( ) .2tt e      CONFLICT OF INTEREST The authors declare that there are no conflicts of interest. REFERENCE [1] A. Bahyrycz, J. Brzdek, E. Jablonska, R. Malejki, Ulams stability of a generalization of the Frchet functional equation, J. Math. Anal. Appl. 442 (2016) 537-553. [2] S. A. Bishop, E. O. Ayoola, J. G. Oghonyon, Existence of mild solu- tion of impulsive quantum stochastic differential equation with nonlocal conditions.  Anal.Math.Phys. 7(3)(2017) 255-265. [3] S. A. Bishop and P. E. Oguntunde, Existence of Solutions of Impulsive Quantum Stochastic Differential Inclusion, Journal of Engineering and Applied Sciences, 10(7)(2015) 181-185. [4] MF. Bota, E. Karapinar, and O. Mlesnite, Ulam-Hyers Stability Re- sults for Fixed Point Problems via Contractive Mapping in (G)-Metric Space. Abstract and Applied Analysis Volume 2013, Article ID 825-293, 6 pages. [5] N. B. Huy, T. D. Thanh, Fixed point theorems and the UlamHyers stability in non-Archimedean cone metric spaces, J. Math. Anal. Appl. 41(4) (2014) 10-20. [6] Y. H. Lee and S. M. Jung, Generalized Hyers-Ulam Stability of a Mixed Type Functional Equation. Abstract and Applied Analysis Vol- ume 2013, Article ID 472-531, 5 pages [7] Y. Liao, J. Wang, A note on stability of impulsive differential equa- tions.  Boundary Value Problems (2014), 2014:67. [8] M. O. Ogundiran, V. F. Payne, On the Existence and Uniqueness of solution of Impulsive Quantum Stochastic Differential Equation, Dif- ferential equations and control processes, N 2, (2013) 63-73. [9] Q. Wang, X.Z. Liu, Stability criteria of a class of nonlinear impulsive switching systems with time-varying delays, J. Franklin Inst. 34(9) (2012) 1030-1047. [10] J. Wang, L. Lv, Y. Zhou, New concepts and results in stability of frac- tional differential equations. Commun Nonlinear Sci Numer Simulat 17 (2012) 2530-2538. [11] J. Wang, Y. Zhou, M. Feckan, Nonlinear impulsive problems for frac- tional differential equations and Ulam stability. Computers and Math- ematics with Applications 64 (2012) 3389-3405. [12] J. Wang, M. Feckan, Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 39(5)  (2012)  258-264. [13] Y. Xu, Z. He, Stability of impulsive stochastic differential equations with Markovian switching. Appl. Mathematics Letters 35 (2014) 35-40. 

ON STABILITY OF QUANTUM STOCHASTIC DIFFERENTIAL EQUATION

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ON STABILITY OF QUANTUM STOCHASTIC DIFFERENTIAL EQUATION

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