# American Institute of Mathematical Sciences

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 & 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.  doi: 10.1016/S0377-0427(98)00100-9.  Google Scholar [2] C. Y. Chan, New results in quenching,, in World Congress of Nonlinear Analysts '92, (1992), 427.   Google Scholar [3] C. Y. Chan and H. G. Kaper, Quenching for semilinear singular parabolic problems,, SIAM J. Math. Anal., 20 (1989), 558.  doi: 10.1137/0520039.  Google Scholar [4] M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction diffusion system,, J. Differential Equations, 89 (1991), 176.  doi: 10.1016/0022-0396(91)90118-S.  Google Scholar [5] R. Glassey, Blow-up theorems for nonlinear wave-equations,, Math. Z., 132 (1973), 183.  doi: 10.1007/BF01213863.  Google Scholar [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.  doi: 10.1017/S0013091500023932.  Google Scholar [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.  doi: 10.2977/prims/1195191889.  Google Scholar [8] P. Lin, Quenching time optimal control for some ordinary differential equations,, preprint, ().   Google Scholar [9] P. Lin and G. Wang, Blowup time optimal control for ordinary differential equations,, SIAM J. Control Optim., 49 (2011), 73.  doi: 10.1137/090764232.  Google Scholar [10] J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).   Google Scholar [11] B. Yordanov and Q. S. Zhang, Finite-time blowup for wave equations with a potential,, SIAM J. Math. Anal., 36 (2005), 1426.  doi: 10.1137/S0036141004440198.  Google Scholar [12] Z. Zhang and B. Hu, Rate estimates of gradient blowup for a heat equation with exponential nonlinearity,, Nonlinear Anal., 72 (2010), 4594.  doi: 10.1016/j.na.2010.02.036.  Google Scholar

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##### References:
 [1] C. Bandle and H. Brunner, Blowup in diffusion equations: A survey,, J. Comput. Appl. Math., 97 (1998), 3.  doi: 10.1016/S0377-0427(98)00100-9.  Google Scholar [2] C. Y. Chan, New results in quenching,, in World Congress of Nonlinear Analysts '92, (1992), 427.   Google Scholar [3] C. Y. Chan and H. G. Kaper, Quenching for semilinear singular parabolic problems,, SIAM J. Math. Anal., 20 (1989), 558.  doi: 10.1137/0520039.  Google Scholar [4] M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction diffusion system,, J. Differential Equations, 89 (1991), 176.  doi: 10.1016/0022-0396(91)90118-S.  Google Scholar [5] R. Glassey, Blow-up theorems for nonlinear wave-equations,, Math. Z., 132 (1973), 183.  doi: 10.1007/BF01213863.  Google Scholar [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.  doi: 10.1017/S0013091500023932.  Google Scholar [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.  doi: 10.2977/prims/1195191889.  Google Scholar [8] P. Lin, Quenching time optimal control for some ordinary differential equations,, preprint, ().   Google Scholar [9] P. Lin and G. Wang, Blowup time optimal control for ordinary differential equations,, SIAM J. Control Optim., 49 (2011), 73.  doi: 10.1137/090764232.  Google Scholar [10] J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).   Google Scholar [11] B. Yordanov and Q. S. Zhang, Finite-time blowup for wave equations with a potential,, SIAM J. Math. Anal., 36 (2005), 1426.  doi: 10.1137/S0036141004440198.  Google Scholar [12] Z. Zhang and B. Hu, Rate estimates of gradient blowup for a heat equation with exponential nonlinearity,, Nonlinear Anal., 72 (2010), 4594.  doi: 10.1016/j.na.2010.02.036.  Google Scholar
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