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

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L. D. Berkovitz, Optimal Control Theory,, Springer-Verlag, (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, (1968). Google Scholar

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

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

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C. De Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions,, Elsevier, (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. doi: 10.1137/130921908. Google Scholar

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

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A. Friedman, Partial Differential Equations of Parabolic Type,, Englewood Cliffs, (1964). Google Scholar

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

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X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Birkhäuser, (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. 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. doi: 10.1214/14-AAP1046. Google Scholar

[17]

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

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J. Muscat, Functional Analysis: An Introduction to Metric Spaces,, Hilbert Spaces, (2014). doi: 10.1007/978-3-319-06728-5. Google Scholar

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

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

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

[22]

F. Riesz and B. Sz.-Nagy, Functional Analysis,, Ungar, (1955). Google Scholar

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S. Stojanovic, Optimal damping control and nonlinear parabolic systems,, Numer. Funct. Anal. Optim., 10 (1989), 573. 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. 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. 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. 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. 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. 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. 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. doi: 10.1155/S1085337502000817. Google Scholar

[32]

A. Zettl, Sturm-Liouville Theory,, AMS, (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. Google Scholar

[2]

L. D. Berkovitz, Optimal Control Theory,, Springer-Verlag, (1974). Google Scholar

[3]

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

[4]

R. Bellman amd G. Wing, An Introduction to Invariant Imbedding,, John Wiley and Sons, (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, (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. 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, (1988). Google Scholar

[9]

A. Friedman, Partial Differential Equations of Parabolic Type,, Englewood Cliffs, (1964). Google Scholar

[10]

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

[11]

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

[13]

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

[14]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems,, Birkhäuser, (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. 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. doi: 10.1214/14-AAP1046. Google Scholar

[17]

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

[18]

J. Muscat, Functional Analysis: An Introduction to Metric Spaces,, Hilbert Spaces, (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. 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. doi: 10.1007/s11228-007-0043-y. Google Scholar

[21]

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

[22]

F. Riesz and B. Sz.-Nagy, Functional Analysis,, Ungar, (1955). Google Scholar

[23]

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

[24]

Y. Wang and J. Yong, A determinisitc affine quadratic optimal control problem,, ESAIM COCV, 20 (2014), 633. 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. 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. 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. 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. 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. 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. doi: 10.1155/S1085337502000817. Google Scholar

[32]

A. Zettl, Sturm-Liouville Theory,, AMS, (2005). Google Scholar

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