National Committee of Ukraine by Theoretical and Applied Mechanics Higher School Academy of Sciences of Ukraine Taras Shevchenko National University of Kyiv Institute of Mathematics of NAS of Ukraine Institute of Mechanics of NAS of Ukraine Institute of Cybernetics of NAS of Ukraine
XVI International Conference
DYNAMICAL SYSTEM MODELLING AND STABILITY INVESTIGATION
MODELLING & STABILITY ABSTRACTS OF CONFERENCE REPORTS Kiev, Ukraine
May 29-31, 2013
ABOUT AN UNSOLVED OPTIMAL CONTROL PROBLEM FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATION
Leonid Shaikhet Donetsk State University of Management, Department of Higher Mathematics, Chelyuskintsev Street, 163a, Donetsk 83015, Ukraine e-mail:
[email protected]
Abstract: One unsolved optimal control problem for stochastic differential equation of the hyperbolic type is proposed to readers attention. This paper continues the series of the papers [1, 2] that are devoted to unsolved problems in the theory of stability and optimal control for stochastic differential and difference equations. Consider the optimal control problem for the stochastic differential equation of the hyperbolic type ∂ 2 ξ(z) ∂ 2 w(z) = a(z, ξ(z)) + B(z, ξ(z))u(z, ξ(z)) + b(z, ξ(z)) , ∂x∂y ∂x∂y ξ(x, 0) = α(x), ξ(0, y) = β(y), α(0) = β(0), with the performance criterion · ¸ Z Z 2 J(u) = E Φ(ξ(Z)) + |N (z, ξ(z))u(z, ξ(z))| dxdy .
(1)
(2)
G
Here z = (x, y) ∈ G = [0, X]×[0, Y ], Z = (X, Y ), ξ(z) ∈ Rn , a(z, q) ∈ Rn , B(z, q) ∈ {Rl → ∂2w Rn }, b(z, q) ∈ {Rm → Rn }, N (z, q) ∈ {Rl → Rd }, ∂x∂y is an n-dimensional two-parametric l n white noise, u(z, q) ∈ R is a control, q ∈ R , E is the expectation. Equations of the type of (1) have been studied, for example, in [3]. The optimal control for the problem (1), (2) has been obtained in [4] via stochastic derivatives [5, 6] and Clark’s representation [6]. In particular, from [4] it follows that for the simple scalar case of the problem (1), (2) ∂ 2 ξ(z) ∂ 2 w(z) = u(z, ξ(z)) + , ∂x∂y ∂x∂y · ¸ Z Z 2 2 J(u) = E ξ (Z) + u (z, ξ(z))dxdy ,
(3)
G
Typeset by AMS-TEX
1
2
LEONID SHAIKHET
the obtained optimal control has the form u0 (z) =
Ez D2 ζ(x, t; s, y) , Ez D1 ζ(x, y)
(4)
where Ez D1 ζ(x, y) and Ez D2 ζ(x, t; s, y) are respectively the first and the second stochastic 2 1 derivatives of the functional ζ(Z) = e 2 (1−ξ (Z)) , Ez is the conditional expectation. On the other hand the optimal control problem (1), (2) has been studied by virtue of the necessary condition of control optimality in [7], where, in particular, the optimal control of the problem (3) has been obtained in the form ξ(z) − Ez (ξ(X, y) + ξ(x, Y )) . (5) 1 + (X − x)(Y − y) Remark. For comparison note that in the case of the ordinary stochastic differential equation the optimal control of the problem " # Z T ˙ = u(t) + w(t), ξ(t) ˙ J(u) = E ξ 2 (T ) + u2 (t)dt u0 (z) =
0
has the form [8] ξ(t) . 1+T −t The problem is: to prove that the both presentations (4) and (5) of the optimal control of the problem (3) coincide and to get the optimal control in the explicit form. u0 (t) = −
References 1. Shaikhet L. About an unsolved stability problem for a stochastic difference equation with continuous time, Journal of Difference Equations and Applications, 2011, Vol.17, No.3, pp.441-444. 2. Shaikhet L. Two unsolved problems in the stability theory of stochastic differential equations with delay, Applied Mathematics Letters, 2012, Vol.25, No.3, pp.636-637. 3. Gikhman I.I. On the existence of weak solutions of a class of hyperbolic systems containing two-parameter white noise, Teoriya slutchajnikh protsessov, 1978, Vol.6, pp.3948 (in Russian). 4. Shaikhet L.E. Optimal control of a class of stochastic partial differential equations, Matematitcheskiye zametki, 1982, V.31, N.6, p.933-936, 958 (in Russian). Translated in Mathematical Notes, 1982, Vol.31, No.6, pp.471-472. 5. Skorokhod A.V. On a generalization of a stochastic integral, Teoriya veroyatnostej i ee primenenija, 1975, V.20, N.2, 223237 (in Russian). Translated in Theory of Probability and its Applications, 1975, Vol.20, No.2, pp.219-233. 6. Shevlyakov A.Yu. Clark’s formula for two-parametric Wiener process, Teoriya slutchajnikh protsessov, 1979, Vol.7, pp.114-118 (in Russian). 7. Shaikhet L. About a necessary condition of control optimality for stochastic differential equations of hyperbolic type”, Teoriya slutchajnikh protsessov, 1984, Vol.12, pp.96-101 (in Russian). 8. Kolmanovskii V.B., Shaikhet L.E. Control of Systems with Aftereffect, Translations of Mathematical Monographs, 157, American Mathematical Society, Providence, RI (1996).
Leonid Shaikhet Dr. Of the Science Donetsk State University of Management, Donetsk, Ukraine, email:
[email protected]; Shaikhet L. - About an unsolved optimal control problem for stochastic partial differential equation
ABOUT AN UNSOLVED OPTIMAL CONTROL PROBLEM FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATION Shaikhet L. Abstract: One unsolved optimal control problem for stochastic differential equation of the hyperbolic type is proposed to readers attention. This paper continues the series of the papers [1, 2] that are devoted to unsolved problems in the theory of stability and optimal control for stochastic differential and difference equations. Consider the optimal control problem for the stochastic differential equation of the hyperbolic type 2 ( z )xy = a( z, ( z )) B( z, ( z ))u( z, ( z )) b( z, ( z)) 2 w( z)xy, (1)
( x, 0) = ( x), (0, y) = ( y), (0) = (0),
with the performance criterion
J (u ) = E ( ( Z )) | N ( z, ( z ))u( z, ( z )) |2 dxdy . G
(2) Here ( z) R , z = ( x, y) G = [0, X ] [0, Y ] , Z = ( X ,Y ) , a( z , q) R n , 2 l n B( z, q) {R R } , b( z, q) {R m R n } , N ( z, q) {Rl R d } , wxy is an n-dimensional two-parametric white noise, u( z, q) Rl is a control, q R n , E is the expectation. Equations of the type of (1) have been studied, for example, in [3]. The optimal control for the problem (1), (2) has been obtained in [4] via stochastic derivatives [5, 6] and Clark's representation [6]. In particular, from [4] it follows that for the simple scalar case of the problem (1), (2) (3) 2 ( z )xy = u ( z, ( z )) 2 w( z )xy, J (u ) = E 2 ( Z ) u 2 ( z, ( z ))dxdy , G the obtained optimal control has the form (4) u0 ( z ) = Ez D2 ( x, t; s, y) Ez D1 ( x, y), 1 where Ez D ( x, y) and Ez D2 ( x, t; s, y) are respectively the first and the second stochastic derivatives of the functional (Z ) = e12(1 2 ( Z )) , Ez is the conditional expectation. On the other hand the optimal control problem (1), (2) has been studied by virtue of the necessary condition of control optimality in [7], where, in particular, the optimal control of the problem (3) has been obtained in the form u0 ( z ) = ( z ) Ez ( ( X , y) ( x, Y ))1 ( X x)(Y y). (5) Remark. For comparison note that in the case of the ordinary stochastic differential equation the optimal control of the problem n
(t ) = u (t ) w(t ),
T J (u ) = E 2 (T ) u 2 (t )dt 0
has the form [8] u0 (t ) = (t )1 T t. The problem is: to prove that the both presentations (4) and (5) of the optimal control of the problem (3) coincide and to get the optimal control in the explicit form. 1. Shaikhet L. About an unsolved stability problem for a stochastic difference equation with continuous time, Journal of Difference Equations and Applications, 2011, Vol.17, No.3, pp.441444. 2. Shaikhet L. Two unsolved problems in the stability theory of stochastic differential equations with delay, Applied Mathematics Letters, 2012, Vol.25, No.3, pp.636-637. 3. Gikhman I.I. On the existence of weak solutions of a class of hyperbolic systems containing two-parameter white noise, Teoriya slutchajnikh protsessov, 1978, Vol.6, pp.39-48 (in Russian). 4. Shaikhet L.E. Optimal control of a class of stochastic partial differential equations, Matematitcheskiye zametki, 1982, V.31, N.6, p.933-936, 958 (in Russian). Translated in Mathematical Notes, 1982, Vol.31, No.6, pp.471-472. 5. Skorokhod A.V. On a generalization of a stochastic integral, Teoriya veroyatnostej i ee primenenija, 1975, V.20, N.2, 223вЂ‖237 (in Russian). Translated in Theory of Probability and its Applications, 1975, Vol.20, No.2, pp.219-233. 6. Shevlyakov A.Yu. Clark's formula for two-parametric Wiener process, Teoriya slutchajnikh protsessov, 1979, Vol.7, pp.114-118 (in Russian). 7. Shaikhet L. About a necessary condition of control optimality for stochastic differential equations of hyperbolic type", Teoriya slutchajnikh protsessov, 1984, Vol.12, pp.96-101 (in Russian). 8. Kolmanovskii V.B., Shaikhet L.E. Control of Systems with Aftereffect, Translations of Mathematical Monographs, 157, American Mathematical Society, Providence, RI (1996). 344
