December  2016, 6(4): 653-704. doi: 10.3934/mcrf.2016019

Forward-backward evolution equations and applications

1. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States

Received  December 2015 Revised  May 2016 Published  October 2016

Well-posedness is studied for a special system of two-point boundary value problem for evolution equations which is called a forward-backward evolution equation (FBEE, for short). Two approaches are introduced: A decoupling method with some brief discussions, and a method of continuation with some substantial discussions. For the latter, we have introduced Lyapunov operators for FBEEs, whose existence leads to some uniform a priori estimates for the mild solutions of FBEEs, which will be sufficient for the well-posedness. For some special cases, Lyapunov operators are constructed. Also, from some given Lyapunov operators, the corresponding solvable FBEEs are identified.
Citation: Jiongmin Yong. Forward-backward evolution equations and applications. Mathematical Control & Related Fields, 2016, 6 (4) : 653-704. doi: 10.3934/mcrf.2016019
References:
[1]

V. A. Ambarzumyan, Diffuse reflection of light by a foggy medium, C. R. Acad. Sci. SSSR., 38 (1943), 229-232.(in Russian)  Google Scholar

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L. D. Berkovitz, Optimal Control Theory, Springer-Verlag, New York, 1974.  Google Scholar

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P. B. Bailey, L. F. Champine and P. E. Waltman, Nonlinear Two Point Boundary Value Problems, Academic Press, New York, 1968.  Google Scholar

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R. Bellman amd G. Wing, An Introduction to Invariant Imbedding, John Wiley and Sons, New York, 1975.  Google Scholar

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C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Elsevier, Amersterdam, 2006. Google Scholar

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P. M. Dower and W. M. McEneaney, Solving two-point boundary value problems for a wave equation via the principle of stationary action and optimal control, SIAM J. Control Optim., 53 (2015), 2898-2933, arXiv:1501.02006v1 [math.OC]. doi: 10.1137/130921908.  Google Scholar

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A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs, N.J., 1964.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

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Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields, 103 (1995), 273-283. doi: 10.1007/BF01204218.  Google Scholar

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D.-X. Kong and F. Wu, A new type of distributed parameter control systems: Two-point boundary value problems for infinite-dimensional dynamical systems, J. Appl. Math., 2013 (2013), Article ID 454857,3pp.  Google Scholar

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S. Lenhart and M. Liang, Bilinear optimal control for a wave equation wih viscous damping, Houston J. Math., 26 (2000), 575-595.  Google Scholar

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X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

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J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly - a four-step scheme, Probab. Theory & Related Fields, 98 (1994), 339-359. doi: 10.1007/BF01192258.  Google Scholar

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J. Ma, Z. Wu, D. Zhang and J. Zhang, On well-posedness of forward-backward SDEs - a unified approach, Ann. Appl. Probab., 25 (2015), 2168-2214. doi: 10.1214/14-AAP1046.  Google Scholar

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J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, Springer-Verlag, Berlin, 1999.  Google Scholar

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J. Muscat, Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, Springer-Verlag, 2014. doi: 10.1007/978-3-319-06728-5.  Google Scholar

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A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

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F. Riesz and B. Sz.-Nagy, Functional Analysis, Ungar, New York, 1955.  Google Scholar

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S. Stojanovic, Optimal damping control and nonlinear parabolic systems, Numer. Funct. Anal. Optim., 10 (1989), 573-591. doi: 10.1080/01630568908816319.  Google Scholar

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Y. Wang and J. Yong, A determinisitc affine quadratic optimal control problem, ESAIM COCV, 20 (2014), 633-661. doi: 10.1051/cocv/2013078.  Google Scholar

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J. Yong, Finding adapted solutions of forward-backward stochastic differential equations - method of continuation, Prob. Theory & Rel. Fields, 107 (1997), 537-572. doi: 10.1007/s004400050098.  Google Scholar

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J. Yong, Forward backward stochastic differential equations with mixed initial and terminal conditions, Trans. AMS, 362 (2010), 1047-1096. doi: 10.1090/S0002-9947-09-04896-X.  Google Scholar

[27]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[28]

Y. You, Nonquadratic optimal regulators and solution of quasi-Riccati equations, Sci. Sinica Ser. A, 30 (1987), 249-261.  Google Scholar

[29]

Y. You, A nonquadratic Bolza problem and a quasi-Riccait equation for distributed parameter systems, SIAM J. Control Optim., 25 (1987), 905-920. doi: 10.1137/0325049.  Google Scholar

[30]

Y. You, Synthesis of time-variant optimal control with nonquadratic criteria, J. Math. Anal. Appl., 209 (1997), 662-682. doi: 10.1006/jmaa.1997.5399.  Google Scholar

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Y. You, Syntheses of differential games and pseudo-Riccati equations, Abstr. Appl. Anal., 7 (2002), 61-83. doi: 10.1155/S1085337502000817.  Google Scholar

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A. Zettl, Sturm-Liouville Theory, AMS, Providence, RI, 2005.  Google Scholar

show all references

References:
[1]

V. A. Ambarzumyan, Diffuse reflection of light by a foggy medium, C. R. Acad. Sci. SSSR., 38 (1943), 229-232.(in Russian)  Google Scholar

[2]

L. D. Berkovitz, Optimal Control Theory, Springer-Verlag, New York, 1974.  Google Scholar

[3]

P. B. Bailey, L. F. Champine and P. E. Waltman, Nonlinear Two Point Boundary Value Problems, Academic Press, New York, 1968.  Google Scholar

[4]

R. Bellman amd G. Wing, An Introduction to Invariant Imbedding, John Wiley and Sons, New York, 1975.  Google Scholar

[5]

S. Chandrasekhar, Radiative Transfer, Dover, 1950.  Google Scholar

[6]

C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Elsevier, Amersterdam, 2006. Google Scholar

[7]

P. M. Dower and W. M. McEneaney, Solving two-point boundary value problems for a wave equation via the principle of stationary action and optimal control, SIAM J. Control Optim., 53 (2015), 2898-2933, arXiv:1501.02006v1 [math.OC]. doi: 10.1137/130921908.  Google Scholar

[8]

N. Dunford and J. T. Schwartz, Linear Operators, Part II. Spectral Theory, Selfadjoint Operators in Hilbert Spaces, John Wiley & Sons, New York, 1988.  Google Scholar

[9]

A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs, N.J., 1964.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[11]

Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields, 103 (1995), 273-283. doi: 10.1007/BF01204218.  Google Scholar

[12]

D.-X. Kong and F. Wu, A new type of distributed parameter control systems: Two-point boundary value problems for infinite-dimensional dynamical systems, J. Appl. Math., 2013 (2013), Article ID 454857,3pp.  Google Scholar

[13]

S. Lenhart and M. Liang, Bilinear optimal control for a wave equation wih viscous damping, Houston J. Math., 26 (2000), 575-595.  Google Scholar

[14]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[15]

J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly - a four-step scheme, Probab. Theory & Related Fields, 98 (1994), 339-359. doi: 10.1007/BF01192258.  Google Scholar

[16]

J. Ma, Z. Wu, D. Zhang and J. Zhang, On well-posedness of forward-backward SDEs - a unified approach, Ann. Appl. Probab., 25 (2015), 2168-2214. doi: 10.1214/14-AAP1046.  Google Scholar

[17]

J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, Springer-Verlag, Berlin, 1999.  Google Scholar

[18]

J. Muscat, Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, Springer-Verlag, 2014. doi: 10.1007/978-3-319-06728-5.  Google Scholar

[19]

S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., 37 (1999), 825-843. doi: 10.1137/S0363012996313549.  Google Scholar

[20]

Z. Páles and V. Zeidan, Generalized Jacobian for functions with infinite dimensional range and domain, Set-Valued Anal., 15 (2007), 331-375. doi: 10.1007/s11228-007-0043-y.  Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

F. Riesz and B. Sz.-Nagy, Functional Analysis, Ungar, New York, 1955.  Google Scholar

[23]

S. Stojanovic, Optimal damping control and nonlinear parabolic systems, Numer. Funct. Anal. Optim., 10 (1989), 573-591. doi: 10.1080/01630568908816319.  Google Scholar

[24]

Y. Wang and J. Yong, A determinisitc affine quadratic optimal control problem, ESAIM COCV, 20 (2014), 633-661. doi: 10.1051/cocv/2013078.  Google Scholar

[25]

J. Yong, Finding adapted solutions of forward-backward stochastic differential equations - method of continuation, Prob. Theory & Rel. Fields, 107 (1997), 537-572. doi: 10.1007/s004400050098.  Google Scholar

[26]

J. Yong, Forward backward stochastic differential equations with mixed initial and terminal conditions, Trans. AMS, 362 (2010), 1047-1096. doi: 10.1090/S0002-9947-09-04896-X.  Google Scholar

[27]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[28]

Y. You, Nonquadratic optimal regulators and solution of quasi-Riccati equations, Sci. Sinica Ser. A, 30 (1987), 249-261.  Google Scholar

[29]

Y. You, A nonquadratic Bolza problem and a quasi-Riccait equation for distributed parameter systems, SIAM J. Control Optim., 25 (1987), 905-920. doi: 10.1137/0325049.  Google Scholar

[30]

Y. You, Synthesis of time-variant optimal control with nonquadratic criteria, J. Math. Anal. Appl., 209 (1997), 662-682. doi: 10.1006/jmaa.1997.5399.  Google Scholar

[31]

Y. You, Syntheses of differential games and pseudo-Riccati equations, Abstr. Appl. Anal., 7 (2002), 61-83. doi: 10.1155/S1085337502000817.  Google Scholar

[32]

A. Zettl, Sturm-Liouville Theory, AMS, Providence, RI, 2005.  Google Scholar

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