September  2014, 4(3): 289-314. doi: 10.3934/mcrf.2014.4.289

Time optimal control problems for some non-smooth systems

1. 

School of Mathematical Sciences and LMNS, Fudan University, Shanghai 200433

2. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

3. 

School of Mathematical Sciences, Fudan University, KLMNS, Shanghai, 200433

Received  September 2013 Revised  November 2013 Published  April 2014

Time optimal control problems for some non-smooth systems in general form are considered. The non-smoothness is caused by singularity. It is proved that Pontryagin's maximum principle holds for at least one optimal relaxed control. Thus, Pontryagin's maximum principle holds when the optimal classical control is a unique optimal relaxed control. By constructing an auxiliary controlled system which admits the original optimal classical control as its unique optimal relaxed control, one get a chance to get Pontryagin's maximum principle for the original optimal classical control. Existence results are also considered.
Citation: Hongwei Lou, Junjie Wen, Yashan Xu. Time optimal control problems for some non-smooth systems. Mathematical Control and Related Fields, 2014, 4 (3) : 289-314. doi: 10.3934/mcrf.2014.4.289
References:
[1]

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.

[2]

C. Y. Chan, New results in quenching, in World Congress of Nonlinear Analysts '92, Vol. I-V (Tampa, FL, 1992), de Gruyter, Berlin, 1996, 427-434.

[3]

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.

[4]

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.

[5]

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

[6]

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.

[7]

H. Kawarada, On solutions of initial-boundary problem for $u_t=u_{x x}+1/(1-u)$, Publ. Res. Inst. Math. Sci., 10 (1975), 729-736. doi: 10.2977/prims/1195191889.

[8]

P. Lin, Quenching time optimal control for some ordinary differential equations, preprint, arXiv:1209.0784v1.

[9]

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.

[10]

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

[11]

B. Yordanov and Q. S. Zhang, Finite-time blowup for wave equations with a potential, SIAM J. Math. Anal., 36 (2005), 1426-1433. doi: 10.1137/S0036141004440198.

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Z. Zhang and B. Hu, Rate estimates of gradient blowup for a heat equation with exponential nonlinearity, Nonlinear Anal., 72 (2010), 4594-4601. doi: 10.1016/j.na.2010.02.036.

show all references

References:
[1]

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.

[2]

C. Y. Chan, New results in quenching, in World Congress of Nonlinear Analysts '92, Vol. I-V (Tampa, FL, 1992), de Gruyter, Berlin, 1996, 427-434.

[3]

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.

[4]

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.

[5]

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

[6]

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.

[7]

H. Kawarada, On solutions of initial-boundary problem for $u_t=u_{x x}+1/(1-u)$, Publ. Res. Inst. Math. Sci., 10 (1975), 729-736. doi: 10.2977/prims/1195191889.

[8]

P. Lin, Quenching time optimal control for some ordinary differential equations, preprint, arXiv:1209.0784v1.

[9]

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.

[10]

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

[11]

B. Yordanov and Q. S. Zhang, Finite-time blowup for wave equations with a potential, SIAM J. Math. Anal., 36 (2005), 1426-1433. doi: 10.1137/S0036141004440198.

[12]

Z. Zhang and B. Hu, Rate estimates of gradient blowup for a heat equation with exponential nonlinearity, Nonlinear Anal., 72 (2010), 4594-4601. doi: 10.1016/j.na.2010.02.036.

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