September & December  2018, 8(3&4): 809-828. doi: 10.3934/mcrf.2018036

Optimal control problems for some ordinary differential equations with behavior of blowup or quenching

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

2. 

Mathematics & Science College, Shanghai Normal University, Shanghai 200234, China

* Corresponding author: Weihan Wang

Received  October 2017 Revised  January 2018 Published  September 2018

Fund Project: This work was supported in part by National Natural Science Foundation of China under grant 11701376 and 11471070, and School Foundation of Shanghai Normal University under grant SK201713.

This paper is concerned with some optimal control problems for equations with blowup or quenching property. We first study the existence and Pontryagin's maximum principle for optimal controls which have the minimal energy among all the controls whose corresponding solutions blow up at the right-hand time end-point of a given functional. Then, the same problem for quenching case is discussed. Finally, we establish Pontryagin's maximum principle for optimal controls of extended problems after quenching.

Citation: Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control and Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036
References:
[1]

H. Amann and P. Quittner, Optimal control problems with final observation governed by explosive parabolic equations, SIAM J. Control Optim., 44 (2005), 1215-1238.  doi: 10.1137/S0363012903433450.

[2]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.

[3]

C. Bandle and H. Brunner, Blowup in diffusion equations: A survey, J. Comput. Appl. Math., 97 (1998), 3-22.  doi: 10.1016/S0377-0427(98)00100-9.

[4]

E. N. Barron and W. Liu, Optimal control of the blowup time, SIAM J. Control Optim., 34 (1996), 102-123.  doi: 10.1137/S0363012993245021.

[5]

C. Y. Chan and H. G. Kaper, Quenching for semilinear singular parabolic problems, SIAM J. Math. Anal., 20 (1989), 558-566.  doi: 10.1137/0520039.

[6]

M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176-202.  doi: 10.1016/0022-0396(91)90118-S.

[7]

H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. 

[8]

R. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203.  doi: 10.1007/BF01213863.

[9]

J.-S. Guo and B. Hu, The profile near quenching time for the solution of a singular semilinear heat equation, Proc. Edinburgh Math. Soc., 40 (1997), 437-456.  doi: 10.1017/S0013091500023932.

[10]

B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4.

[11]

H. Kawarada, On solutions of initial-boundary problem for ut = uxx + 1/(1 − u), Publ. Res. Inst. Math. Sci., 10 (1974/75), 729-736.  doi: 10.2977/prims/1195191889.

[12]

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

[13]

P. Lin, Quenching time optimal control for some ordinary differential equations, J. Appl. Math., 2014 (2014), Art. ID 127809, 13 pages. doi: 10.1155/2014/127809.

[14]

P. Lin, Extendability and optimal control after quenching for some ordinary differential equations, J. Optim. Theory Appl., 168 (2016), 769-784.  doi: 10.1007/s10957-015-0858-x.

[15]

P. Lin and G. Wang, Blowup time optimal control for ordinary differential equations, SIAM J. Control Optim., 49 (2011), 73-105.  doi: 10.1137/090764232.

[16]

P. Lin and G. Wang, Some properties for blowup parabolic equations and their application, J. Math. Pures Appl., 101 (2014), 223-255.  doi: 10.1016/j.matpur.2013.06.001.

[17]

J.-L. Lions, Contrôle des Systèmes Distribués Singuliers, (French) [Control of Singular Distributed Systems], Gauthier-Villars, Montrouge, 1983.

[18]

H. Lou and W. Wang, Optimal blowup time for controlled ordinary differential equations, ESAIM Control Optim. Calc. Var., 21 (2015), 815-834.  doi: 10.1051/cocv/2014051.

[19]

H. Lou and W. Wang, Optimal blowup/quenching time for controlled autonomous ordinary differential equations, Math. Control Relat. Fields, 5 (2015), 517-527.  doi: 10.3934/mcrf.2015.5.517.

[20]

H. LouJ. Wen and Y. Xu, Time optimal control problems for some non-smooth systems, Math. Control Relat. Fields, 4 (2014), 289-314.  doi: 10.3934/mcrf.2014.4.289.

[21]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972.

show all references

References:
[1]

H. Amann and P. Quittner, Optimal control problems with final observation governed by explosive parabolic equations, SIAM J. Control Optim., 44 (2005), 1215-1238.  doi: 10.1137/S0363012903433450.

[2]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.

[3]

C. Bandle and H. Brunner, Blowup in diffusion equations: A survey, J. Comput. Appl. Math., 97 (1998), 3-22.  doi: 10.1016/S0377-0427(98)00100-9.

[4]

E. N. Barron and W. Liu, Optimal control of the blowup time, SIAM J. Control Optim., 34 (1996), 102-123.  doi: 10.1137/S0363012993245021.

[5]

C. Y. Chan and H. G. Kaper, Quenching for semilinear singular parabolic problems, SIAM J. Math. Anal., 20 (1989), 558-566.  doi: 10.1137/0520039.

[6]

M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176-202.  doi: 10.1016/0022-0396(91)90118-S.

[7]

H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. 

[8]

R. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203.  doi: 10.1007/BF01213863.

[9]

J.-S. Guo and B. Hu, The profile near quenching time for the solution of a singular semilinear heat equation, Proc. Edinburgh Math. Soc., 40 (1997), 437-456.  doi: 10.1017/S0013091500023932.

[10]

B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4.

[11]

H. Kawarada, On solutions of initial-boundary problem for ut = uxx + 1/(1 − u), Publ. Res. Inst. Math. Sci., 10 (1974/75), 729-736.  doi: 10.2977/prims/1195191889.

[12]

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

[13]

P. Lin, Quenching time optimal control for some ordinary differential equations, J. Appl. Math., 2014 (2014), Art. ID 127809, 13 pages. doi: 10.1155/2014/127809.

[14]

P. Lin, Extendability and optimal control after quenching for some ordinary differential equations, J. Optim. Theory Appl., 168 (2016), 769-784.  doi: 10.1007/s10957-015-0858-x.

[15]

P. Lin and G. Wang, Blowup time optimal control for ordinary differential equations, SIAM J. Control Optim., 49 (2011), 73-105.  doi: 10.1137/090764232.

[16]

P. Lin and G. Wang, Some properties for blowup parabolic equations and their application, J. Math. Pures Appl., 101 (2014), 223-255.  doi: 10.1016/j.matpur.2013.06.001.

[17]

J.-L. Lions, Contrôle des Systèmes Distribués Singuliers, (French) [Control of Singular Distributed Systems], Gauthier-Villars, Montrouge, 1983.

[18]

H. Lou and W. Wang, Optimal blowup time for controlled ordinary differential equations, ESAIM Control Optim. Calc. Var., 21 (2015), 815-834.  doi: 10.1051/cocv/2014051.

[19]

H. Lou and W. Wang, Optimal blowup/quenching time for controlled autonomous ordinary differential equations, Math. Control Relat. Fields, 5 (2015), 517-527.  doi: 10.3934/mcrf.2015.5.517.

[20]

H. LouJ. Wen and Y. Xu, Time optimal control problems for some non-smooth systems, Math. Control Relat. Fields, 4 (2014), 289-314.  doi: 10.3934/mcrf.2014.4.289.

[21]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972.

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